Charles Bolden stopped by JPL to highlight research being done on advanced propulsion techniques that would be used in the proposed asteroid retrieval mission.

 

At last—a textbook on category theory for scientists! And it’s free!

• David Spivak, Category Theory for Scientists.

It’s based on a course the author taught:

This course is an attempt to extol the virtues of a new branch of mathematics, called category theory, which was invented for powerful communication of ideas between different fields and subfields within mathematics. By powerful communication of ideas I actually mean something precise. Different branches of mathematics can be formalized into categories. These categories can then be connected together by functors. And the sense in which these functors provide powerful communication of ideas is that facts and theorems proven in one category can be transferred through a connecting functor to yield proofs of an analogous theorem in another category. A functor is like a conductor of mathematical truth.

I believe that the language and toolset of category theory can be useful throughout science. We build scientific understanding by developing models, and category theory is the study of basic conceptual building blocks and how they cleanly fit together to make such models. Certain structures and conceptual frameworks show up again and again in our understanding of reality. No one would dispute that vector spaces are ubiquitous. But so are hierarchies, symmetries, actions of agents on objects, data models, global behavior emerging as the aggregate of local behavior, self-similarity, and the effect of methodological context.

Some ideas are so common that our use of them goes virtually undetected, such as set-theoretic intersections. For example, when we speak of a material that is both lightweight and ductile, we are intersecting two sets. But what is the use of even mentioning this set-theoretic fact? The answer is that when we formalize our ideas, our understanding is almost always clarified. Our ability to communicate with others is enhanced, and the possibility for developing new insights expands. And if we are ever to get to the point that we can input our ideas into computers, we will need to be able to formalize these ideas first.

It is my hope that this course will offer scientists a new vocabulary in which to think and communicate, and a new pipeline to the vast array of theorems that exist and are considered immensely powerful within mathematics. These theorems have not made their way out into the world of science, but they are directly applicable there. Hierarchies are partial orders, symmetries are group elements, data models are categories, agent actions are monoid actions, local-to-global principles are sheaves, self-similarity is modeled by operads, context can be modeled by monads.

He asks readers from different subjects for help in finding new ways to apply category theory to those subjects. And that’s the right attitude to take when reading this book. I’ve found categories immensely valuable in my work. But it took effort to learn category theory and see how it can apply to different subjects. People are just starting to figure out these things, so don’t expect instant solutions to the problems in your own favorite field.

But Spivak does the best job I’ve seen so far at explaining category theory as a general-purpose tool for thinking clearly. Since I’m busy using category theory to clarify the relationships between fields like chemistry, population biology, electrical engineering and control theory, this subject is very much on my mind.

“…the story of Homo sapiens trying to stake a claim on shifting ground, flanked on both sides by beast and machine, pinned between meat and math.” (p.13)

No typo in the title, this is truly how this book by Brian Christian is called. It was kindly sent to me by my friends from BUY and I realised I could still write with my right hand when commenting on the margin. (I also found the most marvellous proof to a major theorem but the margin was just too small…)  “The most human human: What artificial intelligence teaches us about being alive” is about the Turing test, designed to test whether an unknown interlocutor is a human or a machine. And eventually doomed to fail.

“The final test, for me, was to give the most uniquely human performance I could in Brighton, to attempt a successful defense against the machines.” (p.15)

What I had not realised earlier is that there is a competition every year running this test against a few AIs and a small group of humans, the judges (blindly) giving votes for each entity and selecting as a result the most human computer. And also the most human … human! This competition is called the Loebner Prize and it was taking place in Brighton, this most English of English seaside towns, in 2008 when Brian Christian took part in it (as a human, obviously!).

“Though both [sides] have made progress, the `algorithmic’ side of the field [of computer science] has, from Turing on, completely dominated the more `statistical’ side. That is, until recently.” (p.65)

I enjoyed the book, much more for the questions it brought out than for the answers it proposed, as the latter sounded unnecessarily conflictual to me, i.e. adopting a “us vs.’em” posture and whining about humanity not fighting hard enough to keep ahead of AIs… I dislike this idea of the AIs being the ennemy and of “humanity lost” the year AIs would fool the judges. While I enjoy the sci’ fi’ literature where this antagonism is exacerbated, from Blade Runner to Hyperion, to Neuromancer, I do not extrapolate those fantasised settings to the real world. For one thing, AIs are designed by humans, so having them winning this test (or winning against chess grand-masters) is a celebration of the human spirit, not a defeat! For another thing, we are talking about a fairly limited aspect of “humanity”, namely the ability to sustain a limited discussion with a set of judges on a restricted number of topics. I would be more worried if a humanoid robot managed to fool me by chatting with me for a whole transatlantic flight. For yet another thing, I do not see how this could reflect on the human race as a whole and indicate that it is regressing in any way. At most, it shows the judges were not trying hard enough (the questions reported in The most human human were not that exciting!) and maybe the human competitors had not intended to be perceived as humans.

“Does this suggest, I wonder, that entropy may be fractal?” (p.239)

Another issue that irked me in the author’s perspective is that he trained and elaborated a complex strategy to win the prize (sorry for the mini-spoiler: in case you did  not know, Brian did finish as the most human human). I do not know if this worry to appear less human than an AI was genuine or if it provided a convenient canvas for writing the book around the philosophical question of what makes us human(s). But it mostly highlight the artificial nature of the test, namely that  one has to think in advance on the way conversations will be conducted, rather than engage into a genuine conversation with a stranger. This deserves the least human human label, in retrospect!

“So even if you’ve never heard of [Shanon entropy] beofre, something in your head intuits [it] every time you open your mouth.” (p.232)

The book spend a large amount of text/time on the victory of Deep Blue over Gary Kasparov (or, rather, on the defeat of Kasparov against Deep Blue), bemoaning the fact as the end of a golden age. I do not see the problem (and preferred the approach of Nate Silver‘s). The design of the Deep Blue software was a monument to the human mind, the victory did not diminish Kasparov who remains one of the greatest chess players ever, and I am not aware it changed chess playing (except when some players started cheating with the help of hidden computers!). The fact that players started learning more and more chess openings was a trend much before this competition. As noted in The most human human,  checkers had to change its rules once a complete analysis of the game had led to  a status-quo in the games. And this was before the computer era. In Glasgow, Scotland, in 1863. Just to draw another comparison: I like playing Sudoku and the fact that I designed a poor R code to solve Sudokus does not prevent me from playing, while my playing sometimes leads to improving the R code. The game of go could have been mentioned as well, since it proves harder to solve by AIs. But there is no reason this should not happen in a more or less near future…

“…we are ordering appetizers and saying something about Wikipedia, something about Thomas  Bayes, something about vegetarian dining…” (p.266)

While the author produces an interesting range of arguments about language, intelligence, humanity, he missed a part about the statistical modelling of languages, apart from a very brief mention of a Markov dependence. Which would have related to the AIs perspective. The overall flow is nice but somehow meandering and lacking in substance. Esp. in the last chapters. On a minor level, I also find that there are too many quotes from Hofstadter’ Gödel, Escher and Bach, as well as references to pop culture. I was surprised to find Thomas Bayes mentioned in the above quote, as it did not appear earlier, except in a back-note.

“A girl on the stairs listen to her father / Beat up her mother” C.D. Wright,  Tours

As a side note to Andrew, there was no mention made of Alan Turing’s chess rules in the book, even though both Turing and chess were central themes. I actually wondered if a Turing test could apply to AIs playing Turing’s chess: they would have to be carried by a small enough computer so that the robot could run around the house in a reasonable time. (I do not think chess-boxing should be considered in this case!)

Filed under: Books, University life Tagged: AIs, Alan Turing, bots, Brian Christian, Brighton, chatbots, entropy, Loebner Prize, poetry, The most human human, Thomas Bayes, Turing's chess, Turing's test

Note to readers: this is my best attempt to describe some issues in accelerator operations; I welcome comments from people more expert than me if you think I don’t have things quite right.

The operators of the Large Hadron Collider seek to collide as many protons as possible. The experimenters who study these collisions seek to observe as many proton collisions as possible. Everyone can agree on the goal of maximizing the number of collisions that can be used to make discoveries. But where the accelerator physicists and particle physicists might part ways over just how those collisions might best be delivered.

Let’s remember that the proton beams that circulate in the LHC are not a continuous current like you might imagine running through your electric appliances. Instead, the beam is bunched — about 10¹¹ protons are gathered in a formation that is about as long as a sewing needle, and each proton beam is made up of 1380 such bunches. As the bunches travel around the LHC ring, they are separated by 50 nanoseconds in time. This bunching is necessary for the operation of the experiments — it ensures that collisions occur only at certain spots along the ring (where the detectors are) and the experiments can know exactly when the collisions are occurring and synchronize the response of the detector to that time. Note that because there are so many protons in each beam, there can be multiple collisions each time two bunches pass by each other. At the end of the last LHC run, there were typically 30 collisions that occurred per bunch crossing.

There are several ways to maximize the number of collisions that occur. Increasing the number of protons in each bunch crossing will certainly increase the number of collisions. Or, one could imagine increasing the total number of bunches per beam, and thus the number of bunch crossings. The collision rate increases like the square of the number of particles per bunch, but only linearly with the number of bunches. On the face of it, then, it would make more sense to add more particles to each bunch rather than to increase the number of bunches if one wanted to maximize the total number of collisions.

But the issue is slightly more subtle than that. The more collisions that occur per beam crossing, the harder the collisions are to interpret. With 30 collisions happening at the same time, one must contend with hundreds, if not thousands, of charged particle tracks that cross each other and are harder to reconstruct, which means more computing time to process the event. With more stuff going on each event, the most important parts of the event are increasingly obscured by everything else that is going on, degrading the energy and momentum resolution that are needed to help identify the decay products of particles like the Higgs boson. So from the perspective of an experimenter at the LHC, one wants to maximize the number of collisions while having as few collisions per bunch crossing as possible, to keep the interpretation of each bunch crossing simple. This argument favors increasing the number of bunches, even if this might ultimately mean having fewer total collisions than could be obtained by increasing the number of protons per bunch. It’s not very useful to record collisions that you can’t interpret because the events are just too busy.

This is the dilemma that the LHC and the experiments will face as we get ready to run in 2015. In the current jargon, the question is whether to run with 50 ns between collisions, as we did in 2010-12, or 25 ns between collisions. For the reasons given above, the experiments generally prefer to run with a 25 ns spacing. At peak collision rates, the number of collisions per crossing is expected to be about 25, a number that we know we can handle on the basis of previous experience. In contrast, the LHC operators generally to prefer the 50 ns spacing, for a variety of operational reasons, including being able to focus the beams better. The total number of collisions delivered per year could be about twice as large with 50 ns spacing…but with many more collisions per bunch crossing, perhaps by a factor of three. This is possibly more than the experiments could handle, and it could well be necessary to limit the peak beam intensities, and thus the total number of collisions, to allow the experiment to operate.

So how will the LHC operate in 2015 — at 25 ns or 50 ns spacing? One factor in this is that the machine has only done test runs at 25 ns spacing, to understand what issues might be faced. The LHC operators will re-commission the machine with 50 ns spacing, with the intention of switching to 25 ns spacing later, as soon as a couple of months later if all goes well. But then imagine that 50 ns running works very well outset. Would the collision pileup issues motivate the LHC to change the bunch spacing? Or would the machine operators just like to keep going with a machine that is operating well?

In ancient history I worked on the CDF experiment at the Tevatron, which was preparing to start running again in 2001 after some major reconfigurations. It was anticipated that the Tevatron was going to start out with a 396 ns bunch spacing and then eventually switch over to 132 ns, just like we’re imagining for the LHC in 2015. We designed all of the experiment’s electronics to be able to function in either mode. But in the end, 132 ns running never happened; increases in collision rates were achieved by increasing beam currents. This was less of an issue at the Tevatron, as the overall collision rate was much smaller, but the detectors still ended up operating with numbers of collisions per bunch crossing much larger than they were designed for.

In light of that, I find myself asking — will the LHC ever operate in 25 ns mode? What do you think? If anyone would like to make an informal wager (as much as is permitted by law) on the matter, let me know. We’ll pay out at the start of the next long shutdown at the end of 2017.

The American Astronomical Society has issued a strongly worded statement against NASA's proposed elimination of its education and public outreach programs, and I agree with it.

Partnering with our friends from The Planetary Society, the Space Generation Advisory Council (SGAC), whose members hail from all over the globe, is bringing you an update on our activities and something you can join in on—at least if you are a student or young professional aged 18–35.

Posted on behalf Jane Qiu.

The Chinese Academy of Sciences (CAS) on 20 May announced the list of candidates for the biennial selection of its prestigious membership. None is attracting more intense speculation than Shi Yigoing, dean of the School of Life Sciences at Tsinghua University in Beijing, who applied for the membership unsuccessfully two years ago.

The speculation is partly fuelled by the US National Academy of Sciences’ announcement last month that it awarded membership to Shi, whose specialty is protein crystallography, for his “distinguished and continuing achievements in original research”. This has renewed a heated debate in the Chinese press and blogosphere on the criteria and selection process of Chinese academies.

In 2008, Shi gave up his professorship at Princeton University in New Jersey to take up the position as at Tshinghua. Since his return he has published a total of 12 papers in the journals Nature, Science and Cell, and more in other journals.

The main reason for the CAS 2011 decision to reject Shi’s membership application, according to Muming Poo, director of the CAS Institute of Neuroscience in Shanghai, was that candidates must make a significant contribution to China in addition to academic accomplishment. “Since Shi hadn’t been back for long, he didn’t score high in this regard,” he says.

But that might not be the whole story: Shi is also an outspoken critic of China’s science culture and institutions and a driving force of reforms, and his outspokenness has earned him enemies. In 2010, he and Rao Yi, a biologist at Peking University, wrote a fierce critique of China’s research culture and funding system in Science.

In a blog post in 2011 in response to the selection decision against Shi and other outstanding scientists, Rao wrote that the outcome is indicative of the predicament faced by many high-flying returnees from abroad. “They are perceived as threats and rejected by the scientific establishment in China,” he wrote.

The CAS congratulated Shi for his induction into its US counterpart but told Xinhua, China’s state news agency, that the two academies are independent organizations with different criteria for membership.

Critics say, however, that the incident should trigger a rethinking on whether the selection process is fair or truly rewards academic excellence. An online commentator going by the name Dmlprince wrote that it’s tragic to see scientists much less accomplished than Shi get selected as members. Another, named Qiudy, wrote that the membership system needs to be reformed for China to attract distinctive scientists from overseas.

I’ve been away from blogging for quite some time – mainly to finish a book I was working on.   The book is unrelated to particle physics, but follows a course I teach at Harvard, called Primitive Navigation.   We explore navigational techniques used by cultures like the Polynesians and Norse, in addition to looking at environmental topics like the origins of ocean currents and global weather systems.   While doing research for the book and the course, I found that humans have always been exceedingly clever in making sense of their environments and harnessing this knowledge to journey long distances.   I found that the ability of humans to develop sophisticated constructs to bring order to their environment is not limited to the lineage of Western scientific thought but is a more universal trait.

We often think of the roots of science starting with the ancient Greeks, or even further back to the Babylonians.   The canonical history is a marriage of mathematics and logic coupled with empirical observation.  The story stretches through the Arab translations of works like Euclid’s Elements during the Dark and Middle Ages, through the emergence of the scientific revolution, and culminating in the dizzying heights of modern works like quantum field theory.   This is not to say that there weren’t hiccups.   Although most scientists would dismiss astrology as quackery, astronomy and astrology were once deeply intertwined from their Western birth in Babylon through the time of Kepler.

I invite you to take a big step back and ponder the following conjecture – that Homo sapiens has always been intrinsically disposed toward scientific thinking.   This is perhaps not ‘science’ in the way we view Western science, but it still has the existence of conceptual framework on which to hang and connect observations.

In the process of doing research for the book, I interacted with a number of anthropologists who are studying the navigational schemes of Pacific Islanders.   Their work demonstrates the existence of an exceedingly sophisticated ‘toolkit’ of navigational schema that allowed them to travel huge distances across the ocean to find small target islands successfully.   Three anthropologists in particular have uncovered some amazing findings:  Cathy Pyrek, Rick Feinberg, and Joe Genz.

Most archaeological evidence points to the emergence of long-distance voyaging by a group called the Lapita people, circa 1600 BC from the Bismarck Archipelago, near New Guinea.   They built craft capable of sailing into the wind, making jumps of hundreds of miles eastward to locations like Fiji, Tonga, Tahiti and the Marquesas.   Even more astonishing was the rapid explosion of voyages of thousands of miles around AD 1000 to Hawaii and the north island of New Zealand.

In order to sail against the wind, one needs to create a sail capable of lift, like a wing and use it in combination with a hull that ‘grabs’ the water as it slices through.    The Lapita figured how to harness the complex fluid dynamics involved in lift and used it to their advantage.  In the 18^th century, Captain James Cook marveled at sophisticated design of the Polynesian voyaging canoes that allowed them to travel at speeds far in excess of Western European vessels.  It wasn’t until 1904 that physicist Ludwig Prandtl laid out the theoretical basis for lift in wings, and wasn’t until the 1970’s that this theory was applied to sails.

The clever design of voyaging canoes was only part of the innovations the Pacific Islanders.   In order to sail across vast stretches of ocean, they needed viable navigational schema.    We don’t have written records from the height of the voyaging period for Polynesians (circa AD 1000), but we do have interviews with modern day practitioners of indigenous navigational techniques that hint at the ways their ancestors crossed large stretches of ocean accurately.

Anthropologists Rick Feinberg and Cathy Pyrek from Kent State have shown how indigenous navigators in eastern Solomon Islands use a ‘navigational tool-kit’, that consists of multiple signs.   Stars that are rising or setting close to the horizon form a natural star-compass.  Their rising and setting positions allow navigators to find the ‘azimuth’ or compass heading toward a destination island.   This requires the navigator to memorize a large number of stars and become familiar with their paths across the sky at different times of the year.

While a star compass may be useful, what does a navigator do during the day or in overcast weather?   Another helpful construct is a wind-compass.  Winds blowing from different directions have different characteristics.    In the eastern Solomons, the trade winds blow from the southeast, and are marked by characteristic ‘trade wind cumulus’ clouds that only grow to heights of roughly 15,000 feet and are then truncated.   These winds mark the direction ‘tonga’, or the southeast, which corresponds to the direction of the island cluster of Tonga.   Winds from the north arrive during the winter months and are associated with variable, stormy weather.

Steady winds and storm systems can also create ocean swells that act as reliable direction indicators.  Often, multiple swells can arise – for example, the Southern Ocean produces a long swell from the south, while trade winds can create shorter wavelength swells from the east.   Even if the wind shifts, the swells retain some ‘memory’ of the winds that created them allowing the navigator to maintain a steady heading.

The above tools are useful in maintaining direction under different conditions, but there’s an inherent uncertainty in the position of a vessel, and this uncertainty grows with time.   A navigator completing a 200-mile journey may only be able to establish a position to within 20 or 30 miles.   Another trick then comes into play:  birds.   Certain birds, like pelicans and frigate birds will fly some distance out to sea to feed, and then will return to their home islands in the evening.   A sailor only has to get to within 30 miles of a target island and then observe land-based birds.   The sail is dropped and when the birds fly home in the evening, a course is set.

The navigational toolkit allows for a kind of successive approximation, where the stars, wind, and swells form a rough guide, and the presence and behavior of birds provides the final precision.

A somewhat related but unique tradition is that of wave-piloting in the Marshall Islands.  Most of us are familiar with refraction and reflection of waves, whether they’re light or sound waves.   Waves on the oceans’ surface are similar, but have some notable differences.   First waves in deep-water have a speed that is proportional to the square root of the wavelength.   Second, waves in shallow water have a speed that’s proportional to the square root of the depth.   This latter relation causes waves to refract in shallow water.   When waves get into very shallow water, they’ll often break, losing much, if not all of their energy.   On the other hand, waves impinging on a steep cliff that extends underwater will reflect with very little energy lost.   Depending on the bathymetry surrounding an island, one can get very different wave patterns produced by the interaction of an incident swell with the island.

Joe Genz from the University of Hawaii studied the tradition of Marshall Island wave piloting for his doctoral thesis.   Navigators in the Marshalls have their own language for describing characteristic wave patterns around islands. Nit in kōt is the name given to a crossing pattern of waves on the lee side of an island.   If a uniform swell impinges in the eastern shore of an island, the waves passing the north shore will be refracted inside the swell-shadow toward the south and the waves passing the south shore will be refracted into the swell-shadow toward the north.   The resulting pattern of crossing waves creates a disturbed region that’s easy to identify at distances beyond which the islands are visible.

In principle, reflected ways should also give clues to the presence of an island.  Joe made the acquaintance of one Captain Korent Joel, a native Marshall Islander who was trying to revive the tradition of wave piloting.   Joe persuaded Captain Korent to demonstrate his wave piloting technique to a group of oceanographers who deployed a set of sensitive wave buoys.  As Captain Korent left the atoll of Arno, he first pointed out the incoming swell from the east, and then the reflected swell off of Arno.

There was only one problem.   No one on the boat with Captain Korent could notice the reflections, although the dominant eastern swell was clearly visible.   Even the sensitive wave buoys couldn’t detect the presence of the reflected swell.    What was going on?    Joe wondered whether Captain Korent just thought he should be seeing a reflected swell and was making this up.

In order to put Captain Korent to a sterner test, Joe waited until he (Captain Korent) was taking a nap on in the cabin.   Joe instructed the crew to motor some 30 miles to the southwest of Arno to get to a new location.   When Captain Korent woke up, Joe told him that he had taken the boat to an undisclosed location and asked him if he could identify the direction to Arno, and the kind of wave patterns he was seeing.   Captain Korent was quite certain the Arno was to the northwest, and he was also quite correct!   So, he was reading the waves properly after all!

I met Joe in person at a conference of the Association for Social Anthropology in Oceania (ASAO) in Portland Oregon in February 2012.    Joe had some videos on his laptop of Captain Korent and shared them with me.   I downloaded them to my computer.   That evening, I watched the video where Captain Korent was pointing out the reflected swell to Joe on the boat.   This was the reflected swell that Joe couldn’t see, and the oceanographer’s buoys couldn’t detect.   Joe told me what Captain Korent was saying in Marshallese about the waves.   I do some sea kayaking, and I’m often close to the water, and am a bit of an amateur wave-watcher myself.

In my first viewing of the video, I could definitely see the incoming dominant swell from the east.   But, by the third or fourth viewing, I could see a weaker reflected swell moving at slight angle against the larger incoming swell.   When I compared my observations to what Captain Korent was saying in Marshallese, they agreed completely!  By the tenth viewing, I became 100% convinced that Captain Korent was pointing out the reflected swell correctly.

 

The next day, I called Joe over, along with Cathy Pyrek, who was also attending the ASAO conference.   I pulled up the video on my laptop and showed what I saw as the reflected swell.   Joe said, “Oh yeah, now I see it”.    I turned to Cathy and asked if she really saw it, or I was just convincing them of it, but she said,“It’s definitely there, it’s strange that everyone missed it.”

We still have much to learn about how the human mind operates, but it struck me that Captain Korent’s talents show how we’re capable of picking up very weak signals in the presence of noise.   Evidently there is more information on the surface of the ocean than the oceanographer’s buoys were capable of recording.   This is perhaps not surprising, but it’s evidence that there are different frameworks of knowledge out there that are effective and are based on empiricism.   It may not be Western, but it is a kind of science.

For further reading:

Joeseph Genz, et al., “Wave Navigation in the Marshall Islands,” Oceanography, 22, June 2009, 234-245.

Joseph Genz, “Marshallese Navigation and Voyaging: Re-learning and Reviving Indigenous Knowledge of the Ocean,” (PhD diss., University of Hawaii, 2008)

John Huth, The Lost Art of Finding Our Way, (Belknap Press, Cambridge MA, 2013).

Twitter:  @JohnHuth1

NASA has still not sent its operating plan to Congress. Rumors of the agency reprogramming away all of the additional funding to Planetary Science remains just rumors.

I stumbled across this a while ago and, with my mind emptied by a day full of meetings, I thought I’d take the opportunity to post it today along with a couple of random connections that sprang into my mind when I saw it. The process by which 32 metronomes seem to synchronize themselves in the first video might look like magic at first glance, but it’s actually based on very simple physics…

And if you want to see an explanation of how it works with rather fewer metronomes, see

which brings me onto this remarkable piece of music by György Ligeti which is called Poème Symphonique and is written for 100 metronomes placed, hopefully, on a hard surface:

All this reminds me of the legendary Geordie darts commentator Sid Waddell, who once described the ebb-and-flow of a championship darts match in the following style…

the pendulum is swinging backwards and forwards, like a metronome…

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The Planetary Society just returned from a major political advocacy trip out to D.C. what did we do and what did we achieve? What's going on with the current funding situation regarding Planetary Science and NASA at large? How does the asteroid retrieval mission help or hurt planetary exploration goals? What's the larger plan and what are the consequences if cuts continue?

Guest post by Bruce Bartlett

On Monday, David Corfield and Kobi Kremnitzer gave philosophy talks in a snazzy new building at Oxford:

• Kobi Kremnitzer, What is geometry?, 2-4pm.
• David Corfield, What might philosophy make of homotopy type theory?, 4.30-6.30pm.

The talks shared homotopy type theory as a common theme. The name “Per Martin-Löf” was mentioned a lot, which was good for me since I had always thought Martin and Löf were two separate people:

Notes are available above, but I will try to give some brief impressions.

Kobi Kremnitzer, What is geometry?

1. Introduction

He started by answering the question “Why am I giving this talk?”, and explained that he followed the pragmatic approach to philosophy of mathematics. I think then he said his approach was somehow similar to that of Wittgenstein and Carnap (but he could have been saying the exact opposite :-)), and that for him, there is no Metaphysics in the joint carving. I’m afraid this totally went over my head, but I did imagine some Oxford dons pleasantly carving a roast chicken, which started to make me hungry!

He stressed that for him mathematics is not in the business of theorem proving only, but mathematicians create new systems, new languages, and that in the correct language a problem becomes trivial… the approach of Grothendieck.

2. Categorical language Since it was a philosophy talk, he motivated categories by starting with Kripke semantics, going via posets, and then he went from posets to categories by literally “raising the bar”!

3. Crashcourse in algebraic geometry Algebraic geometry is the study of solutions to polynomial equations, like a parabola:

$$X=\{y-x^2=0\}.$$

I haven’t specified what “y” and “x” actually are, and that’s the point. We can interpret them in any ring. Hence the Grothendieck view is to think of an affine variety $X$ defined by a bunch of polynomial equations $f_1,\dots ,f_n$ as being a presheaf on the (opposite) category of rings,

$$X:\mathrm{Rings}\to \mathrm{Set},$$

defined by

$$X(R)=\{(a_1,\dots ,a_n)\in R^n:f_i(a_1,\dots ,a_n)=0\mathrm{for}\mathrm{all}i\in I\}.$$

This leads us to define the category of algebraic spaces as being nothing but the category of presheaves on ${\mathrm{Ring}}^{\mathrm{op}}$.

This category of algebraic spaces has lots of nice properties. Inside it live subcategories of objects having nice properties, such as schemes and sheaves. But Kobi stressed that it is very handy to understand them as living in this general universe of algebraic spaces.

4. What is algebra? There was a crucial idea lurking above – that a ring is a gismo $R$ which allows you to take any polynomial $f\in \mathbb{Z}[x_1,\dots ,x_n]$ and evaluate it on elements of $R$.

So – to go from algebraic to differential geometry, we could replace the concept of a “ring” with the concept of a “$C^{\mathrm{\infty }}$-ring” – this is a gismo $R$ which allows you to take any smooth function $f\in C^{\mathrm{\infty }}({ℝ}^n,ℝ)$ and evaluate it on elements of $R$!

For instance, the space of smooth functions on a manifold is a $C^{\mathrm{\infty }}$-ring… but so is $ℂ[ϵ]/{ϵ}^2$ ! So by this slight change of view, we have accomplished Leibniz’s dream – calculus and infinitesimals in the same universe.

5. Final comments He spoke a bit about: derived geometry, set-theoretic foundations, noncommutative geometry, synthetic differential geometry, elementary theory of the category of sets – have a look at the last few pages of his notes.

He stressed that ordinary set-theoretic foundations pulverizes spaces into “atomic dust” where the elements have no “cohesion” with each other… we have to put this back in by hand using topology. As a foundation, homotopy type theory will have this cohesiveness natively built in, and that is attractive to a geometer.

---

David Corfield, What might philosophy make of homotopy type theory?

David began by defining three “camps” in the philosophy of mathematics:

• Antiformalism (eg. Heidegger, Wittgenstein)
• Proformalism (eg. Russell, Carnap, Quine)
• Historical / dialectical philosophy (eg. Cassirer, Collingwood, Lautman, Polanyi, Lakatos, Shapere, MacIntyre, Friedman)

This last camp has the longest list of names, so you guessed it, David placed himself there! He quoted from Domski and Dixon, which I couldn’t understand, but I could understand this quote from his book:

Straight away, from simple inductive considerations, it should strike us as implausible that mathematicians dealing with number, function and space have produced nothing of philosophical significance in the past seventy years in view of their record over the previous three centuries. Implausible, that is, unless by some extraordinary event in the history of philosophy a way had been found to filter, so to speak, the findings of mathematicians working in core areas, so that even the transformations brought about by the development of category theory, which surfaced explicitly in 1940s algebraic topology, or the rise of non-commutative geometry over the past seventy years, are not deemed to merit philosophical attention.

He had rather sobering news for philosophers keen on getting involved with homotopy type theory:

If one is not on-board already, that’s leaving it rather late to assist with Homotopy Type Theory.

He mentioned the n-category exposition on homotopy type theory by Mike Shulman (starting here) quite a lot, and also Urs Schreiber’s explanation of how homotopy type theory natively produces the “right answer” in a differential topology problem in string theory (flux quantization condition).

He also explained how homotopy type theory gets around certain problems, such as Have you left off beating your wife? and The present king of France is bald.

He also spoke about polarity, inference, modal logic and lots of other stuff - but my understanding was severely cramped.

Finally he ended on a positive note, there is plenty of places for philosophers to get involved:

• For philosophers of the historical / dialectical persuasion, homotopy type theory is just the kind of development that needs to be written up. There is plenty to learn from the development – the role of logician-philosopher Martin-Löf, constructive type theory, category theory, homotopic mathematics, influence of physics, computer science.

• For proformalist philosophers, there are also plenty of opportunities. Look closely at intensionality, modality, polarity, the dependent type theory-language relation, and philosophy of physics.

He ended by linking to the upcoming book by Mike Shulman and co., and the nLab.

 

Filed under: Mountains, pictures Tagged: Antarctica, big wall, climbing pictures, dvd, Posing Productions, Queen Maud, Ulvetanna

Spacecraft could determine their position anywhere in the solar system to within five kilometres using signals from x-ray pulsars, say astronomers.

I’ve finished (more or less) a version of the promised article on IceCube — the giant neutrino experiment that may have made a major discovery, as announced last week, and that had an opportunity to make another a few weeks ago (though apparently nature didn’t provide).  The article is admittedly a bit rushed (darn computer trouble) and therefore a bit rough, and it also leaves out some more subtle points that may become important in the future — but I think it’s complete enough to help explain how IceCube made their most recent measurements.  As usual, please send comments and questions, and I’ll work on it further.

Here’s the link to the article.  You may also find it interesting to read more generally about how neutrinos are detected, and about the weird story of neutrino types, and how they can oscillate from one type to another as they travel.

Filed under: Astronomy, Particle Physics Tagged: astronomy, IceCube, neutrinos

Upgrades to the Facility for Advanced Accelerator Experimental Tests—including a new 10-terawatt laser—will assist in R&D for new methods of particle acceleration.

Just over a year after opening its beam to researchers from around the world, the Facility for Advanced Accelerator Experimental Tests (FACET) at SLAC is shining a little brighter. With the addition of a new 10-terawatt laser and other equipment upgrades, one of the facility’s main goals—the development of a new method of particle acceleration that boosts particles’ energy on waves of plasma—looks especially promising.

Link to e-Bulletin Issue No. 21-22/2013Link to all articles in this issue No.

AMS – First results, Dr Mercedes Paniccia, Université de Genève.   Wednesday 29 May, 11:15 a.m. Science III, Auditoire 1S081 30, quai Ernest-Ansermet, 1211 Genève 4 Abstract: The Alpha Magnetic Spectrometer is a state-of-the-art particle physics detector operating as an external module on the International Space Station. It uses the unique environment of space to study the universe and its origin by searching for antimatter, dark matter while performing precision measurements of cosmic rays composition and flux. Since its installation on May 19, 2011 it has collected over 30 billion cosmic rays of energies ranging from several hundred MeV up to few TeV. In this talk we will present the precision measurement of the positron fraction in cosmic rays in the energy range from 0.5 to 350 GeV based on 6.8 million positron and electron events collected in the initial 18 month period of operation in space. Organised by Prof. Teresa.Montaruli@unige.ch and Prof. Giuseppe.Iacobucci@unige.ch. More information here.

Title: A Posteriori Transit Probabilities
Authors: Daniel J. Stevens and B. Scott Gaudi
Institution: Department of Astronomy, The Ohio State University

Bonus Video: OSU Astronomy Coffee Brief

Transiting exoplanets are the best! Well, I think so, anyways. A transit is the dip in light we see when a planet passes between us and its host star. Transits turn exoplanets into real worlds—we can characterize their atmospheres, bulk compositions, and much else. That’s why little dips in light curves set my heart aflutter. But discovering a transiting exoplanet requires a bit of luck. After all, we can’t observe exoplanetary systems from every direction; we’re stuck on Earth or close to it, so an exoplanet’s orbit needs to be aligned just so for us to view a transit.

How do we find transiting exoplanets? There are two general methods. First, we can look at a plethora of stars at once. The Kepler mission (RIP) observed >150,000 stars continuously for four years. Because Kepler looked at so many stars, it found thousands of transiting planetary candidates. Follow-up observations are difficult, however, because the Kepler stars are relatively faint. Second, we can target planets discovered using radial velocity (RV), hoping to see a transit. These RV planets are scattered all around the sky, so we can’t observe all target stars simultaneously using a single telescope like Kepler—although a future mission (TESS) should change this! Most RV planets won’t transit, but we can extract a huge amount of information from those that do.

What’s the probability that an RV planet will transit? As they (should) teach in kindergarden, a simple estimate is that the probability of a transit is equal to the radius of the host star divided by the semi-major axis of the companion’s orbit. More compactly, P = R/a. This is because the inclination of planetary orbits should be randomly distributed from our point of view. We fit the RV data to find the semi-major axis, a. We can estimate the stellar radius, R, from the stellar spectra we used to measure the RV. Ergo, we know precisely the probability of a transit and thus how many stars we need to target to find a transiting exoplanets… right? Actually, no. As Stevens and Gaudi discuss in this paper, it’s more complicated than that.

The orientation of planetary orbits, in general, should be random. But not all orientations for planets that we detect are equally likely. RV data can’t constrain the mass of an exoplanet independently from its orbital inclination. We only measure a combination of the two, called the “minimum mass.” In reality, planetary masses aren’t distributed evenly; the abundances of super-Earths, Neptunes, and Jupiters are different. So, certain planetary masses are more likely than others and thus our RV planets are more likely to be in some orientations than others. The authors use Bayesian statistics to examine the effects of possible mass distributions on the probabilities that known RV planets will transit. They conclude that exoplanets less massive than Jupiter have up to a ~20% better chance of transiting than previously assumed. This isn’t a huge effect, but it should buoy our optimism for finding transiting exoplanets among RV targets.

The Problem

Our problem with the simple formula (P = R/a) is that it ignores an additional piece of information. Specifically, we’ve also measured the “minimum mass” of the planet with RV. This minimum mass is equal to the actual mass of the companion times the sine of its orbital inclination. Think of it this way: RV measures the back-and-forth motion of the star that we see (in the radial direction). A bigger companion will exert a bigger pull on the star, but the component of motion in the radial direction depends on the orientation of the object. For example, a companion that orbits exactly in the plane of the sky won’t cause any movement in the radial direction no matter how massive it is. At the other extreme, an orbital inclination of 90° corresponds to a system in which the orbit is precisely aligned with our line of sight—a transiting system! The mass of a transiting exoplanet is approximately equal to the measured minimum mass. For misaligned orbits, the actual mass is larger than the minimum mass.

Let’s consider a simple example. Imagine that one Earth-mass was the strict lower limit to the mass distribution; no less-massive exoplanets exist. Now, say that you discovered a RV planet with a minimum mass of 0.1 Earth-masses. You’d know the real mass must be larger than this minimum mass, so the planetary orbit must be misaligned. The probability that this exoplanet will transit is identically zero, regardless of the R and a of the system. Our a priori knowledge of the mass distribution directly affects our knowledge of the orbital alignment.

Unfortunately, we don’t know the real distribution of exoplanetary masses. In fact, that’s currently one of the major goals of exoplanet science! But that won’t stop us from speculating. Theorists run simulations of planet formation that predict different mass distributions. These distributions disagree in many respects and won’t be observationally verified for a while. However, most people believe that small planets are intrinsically more abundant than large ones, which is consistent with the Kepler results.

The Solution

At this point, you might be confused. How do we deal with all these nested probabilities? How do we really know what’s really real, in reality? Stevens and Gaudi appeal to Bayesian statistics, a theoretical framework that’s becoming increasingly popular in astrophysics. Bayes’ theorem, simply, states that the probability that a model is true given the data (the posterior) is proportional to the probability that the model would produce the data (the likelihood) times the prior probability that the model is true (the prior). If you understand the Bayesian lingo of posteriors and priors, you’ll get the gist of their paper, even if the algebra remains opaque.

Let’s translate our problem into Bayesian-speak. We care about the actual mass of the planet—that’s the posterior. We know the likelihood that a companion of a given mass will be aligned in such a way to produce the measured minimum mass (P = R/a). We need to think carefully about the prior probability that a given mass distribution exists in reality.

Figure 1 shows a sample mass distribution, assembled from simulations of the formation of planet-sized companions and from studies of the occurrence rate of stellar companions around nearby stars. Again, this is probably wrong in detail; the important point is that it’s not flat. Fig. 1 also shows the resulting distribution of minimum masses as measured with an RV survey on Earth. As you can see, the correspondence between the two distributions depends on the slope of the mass distribution. Where the abundance is relatively insensitive to mass (in the Jupiter zone), the observations basically match reality. But many minimum masses will be measured in a range where no real objects exist (brown dwarfs). Where the mass distribution is sloped, the distributions of the real masses and minimum masses are different—not dramatically so, but enough to make a difference when planning follow-up campaigns to target RV candidates.

The authors then searched the Exoplanets Orbit Database for planets discovered using RV. They computed the posterior probability that each planet would transit, given an underlying mass distribution. They compared these probabilities to those predicted from the simple R/a scaling. They tried two different distributions from the literature, both of which yielded similar results. In particular, planets are more likely to transit than the simple scaling suggests. As seen in Figure 2, a number of planets with relatively large orbital periods have probabilities of transits >10%, even though the simple scaling predicts <1% chances. In other words, a small minimum mass is more likely to be a small planet near-transit than a massive super-Jupiter on a very inclined orbit, since small planets are intrinsically more numerous than large ones. This result should make us more optimistic about conducting surveys of RV targets. Perhaps most importantly, however, this paper highlights the importance of using all your available information to plan observations!

In November, I discussed FUTs (finite unified theories) which are \(\NNN=1\) supersymmetric grand-unification-inspired versions of MSSM with the additional constraint that the divergences already cancel at the level of the effective field theory. This finiteness boils down to the vanishing of the beta-functions, some anomalous dimensions, and some relationships between the gauge and Yukawa couplings.

This condition doesn't seem to be a "must" – the divergences may very well be taken care of by the high-energy phenomena (string theory ultimately takes care of all divergences so its approximations don't have to be finite by themselves) – but it is an aesthetically intriguing condition, anyway. Now, the same authors released a new paper

Finite Theories Before and After the Discovery of a Higgs Boson at the LHC (S. Heinemeyer, M. Mondragon, G. Zoupanos)

where they calculate some new predictions and intriguing details.




They focus on the third-generation fermions and their superpartners, the Higgs sector, and the gauginos. The nicest FUTs they consider boast names such as FUTA and FUTB – the latter seem particularly attractive. They also take some LHCb results into account. In these models, \(\tan\beta\) is typically rather large, \(\mu\) is almost necessarily negative.




The spectra seem very intriguing and consistent with everything we know. Unfortunately, they're inaccessible to the LHC – or marginally accessible – and perhaps even inaccessible to ILC/CLIC. I like the representative table of a FUTB model here:\[

\begin{array}{|l|l||l|l|}
\hline
m_b(M_Z) & 2.74 &&
m_t & 174.1 \\ \hline
m_h & 125.0 &&
m_A & 1517 \\ \hline
m_H & 1515&&
m_{H^\pm} & 1518 \\ \hline
m_{\tilde t_1} & 2483 &&
m_{\tilde t_2} & 2808 \\ \hline
m_{\tilde b_1} & 2403 &&
m_{\tilde b_2} & 2786 \\ \hline
m_{\tilde \tau_1} & 892 &&
m_{\tilde \tau_2} & 1089 \\ \hline
m_{\tilde\chi_1^\pm} & 1453 &&
m_{\tilde\chi_2^\pm} & 2127 \\ \hline
m_{\tilde\chi_1^0} & 790 &&
m_{\tilde\chi_2^0} & 1453 \\ \hline
m_{\tilde\chi_3^0} & 2123 &&
m_{\tilde\chi_4^0} & 2127 \\ \hline
m_{\tilde g} & 3632 && {\rm masses}& {\rm in}\,\GeV
\\ \hline
\end{array}

\] You see that the LSP is the lightest neutralino below \(800\GeV\). Staus are just somewhat heavier, \(900\GeV\) and \(1100\GeV\). Both sbottoms and stops fit the pattern that the lightest and heaviest one is at \(2500\GeV\) and \(2800\GeV\), respectively. The second lightest neutralino and the lightest chargino sit at \(1450\GeV\), the remaining four faces of the God particle find themselves above \(1500\GeV\) while the heavier chargino and the heaviest two neutralinos are above \(2100\GeV\). Finally, the gluino is above \(3600\GeV\).

Particularly the last figure is rather high (in a broader ensemble of models they analyze, the masses may go up to \(10\TeV\) or so). We would have trouble to see such a gluino for years. But this model or at least similar models may be right. From a theoretical viewpoint, I see absolutely no preference when I compare models with gluinos at \(1200\GeV\) and \(3600\GeV\). Some people become very emotional and start to say that one of them has to be right or wrong or its rightness or wrongness means something a priori. Well, it just doesn't. Nature doesn't give a damn whether it's easy or hard for us to observe the superpartners. Once we observe them, many new things start to be clear. If we don't observe them, we are still extremely far from ruling out supersymmetry – and nice special supersymmetric models such as FUTB in this paper.

Its not my – or other humans' – job to rate the beauty of the values of particle physics parameters that emerge from Nature's decisions. It's Her job. Nevertheless, I must say that I would find a spectrum like the table above – or many other tables – elegant. It would probably mean that all these obnoxious idiots who like to say bad things about SUSY could remain loud for many more years. That's an annoying vision from a personal viewpoint but it can't change anything about the reality and it is less important than the actual beauty and physical near-inevitability that is carried by supersymmetry at some scale. If the known – mostly theoretical – evidence makes two models equally plausible and elegant, then one is obliged to love both of them equally, regardless of the fact that one of them may be much more accessible to the experiments. I view this commandment as a part of the scientific integrity.

Posted on behalf of Brian Owens.

After a two-year, US$41-million upgrade, the venerable Alvin submersible is about to return to sea.

On 25 May, the research ship Atlantis will leave the Woods Hole Oceanographic Institution (WHOI) in Massachusetts with Alvin on board, bound for Astoria, Oregon. After a series of US Navy certification cruises in September and a scientific-verification cruise in November, Alvin will return to full service in December studying the deep ocean off the US Pacific Northwest.

The main improvement in this first phase of the Alvin upgrade is the new titanium sphere where the sub’s three-person crew sits (see Nature’s feature story ‘Deep-sea research: Dive master‘). It is 18% bigger than the previous sphere and has two extra windows and high-definition cameras, giving the scientists a better view of the deep ocean. It also has more comfortable seats. In addition, the manipulator arms have longer reach, and the sample-collection basket can carry twice as much weight — up to 181 kilograms.

Even though the new sphere was designed to travel to depths of 6,500 metres, Alvin will still be limited to its old depth of 4,500 metres after the first phase of the upgrade. Holding it back from greater depths are battery limitations, says Susan Humphries, who is in charge of the upgrade programme at WHOI. Alvin uses lead–acid batteries, which do not provide enough power for longer, deeper dives. Lithium-ion batteries would be better, but are considered to have too great a risk of fire for now. “In a few years, once the battery technology has matured, we’ll complete phase two,” says Humphries. She hopes that within five years, when Alvin is scheduled for regular maintenance, the problem will be solved.

In the meantime, ocean scientists are eager to get back below the waves. Over its five-decade career, Alvin has been responsible for revealing some of the deep ocean’s biggest surprises, including the famous ecosystems powered by hydrothermal vents rather than sunlight. Julie Huber, a microbiologist at the Marine Biological Laboratory, also in Woods Hole, has been on three Alvin dives in the past. She is looking forward to the new exploration opportunities, but sounds a note of caution: “I want to wait for them to have 50 safe dives under their belt before I go back.”

Disclosure: Brian Owens is in Woods Hole as part of the Logan Science Journalism Fellowship at the Marine Biological Laboratory.

On Saturday I decided to have a bit of simple relaxation at home, and sit on the patio with my notepad and some pencils and draw a likeness. I'd not done any practice from images for a while, and frankly my pencil work was very rusty and needed a workout. So I dug out this month's issue of a sewing magazine that I subscribe to (what?! well, it's a long story... let's move on) that happens to sometimes have interestingly lit and well reproduced photos of faces and sketched for a while. It was fun (even with the slightly flawed outcome). (Click for a larger view.) -cvj

In the plane to Birmingham, I was reading this recent arXived paper by Minh-Ngoc Tran, Michael K. Pitt, and Robert Kohn. The adaptive structure of their ACMH algorithm is based upon two parallel Markov chains, the former (called the trial chain) feeding the proposal densities of the later (called the main chain), bypassing the more traditional diminishing adaptation conditions. (Even though convergence actually follows from a minorisation condition.) These proposals are mixtures of t distributions fitted by variational Bayes approximations. Furthermore, the proposals are (a) reversible and (b) mixing local [dependent] and global [independent] components. One nice aspect of the reversibility is that the proposals do not have to be evaluated at each step.

The convergence results in the paper indeed assume a uniform minorisation condition on all proposal densities: although this sounded restrictive at first (but allows for straightforward proofs), I realised this could be implemented by adding a specific component to the mixture as in Corollary 3. (I checked the proof to realise that the minorisation on the proposal extends to the minorisation on the Metropolis-Hastings transition kernel.) A reversible kernel is defined as satisfying the detailed balance condition, which means that a single Gibbs step is reversible even though the Gibbs sampler as a whole is not. If a reversible Markov kernel with stationary distribution ζ is used, the acceptance probability in the Metropolis-Hastings transition is

α(x,z) = min{1,π(z)ζ(x)/π(x)ζ(z)}

(a result I thought was already known). The sweet deal is that the transition kernel involves Dirac masses, but the acceptance probability bypasses the difficulty. The way mixtures of t distributions can be reversible follows from Pitt & Walker (2006) construction, with  ζ  a specific mixture of t distributions. This target is estimated by variational Bayes. The paper further bypasses my classical objection to the use of normal, t or mixtures thereof, distributions:  this modelling assumes a sort of common Euclidean space for all components, which is (a) highly restrictive and (b) very inefficient in terms of acceptance rate. Instead, Tran & al. resort to Metropolis-within-Gibbs by constructing a partition of the components into subgroups.

Filed under: Statistics Tagged: ACMH, adaptive MCMC methods, arXiv, minorisation, t distribution, variational Bayes methods

Yeah. Scary, right? I woke up one morning to this result (see earlier posts here, here, and here) from a night of an intensive computer run. It was not meant to be a straight line, but pretty close to it, so I knew that something was wrong with my code. Took me a good long while to trace the problem, but I did in the end. My signal was being swamped by both [...]

I've been doing my part in trying to convey the fact that the overwhelming majority of the applications for accelerators have nothing to do with particle physics. Here is another example, the use of accelerators to generate x-rays for baggage scanners.

Two SLAC physicists with decades of particle accelerator experience helped a Silicon Valley company design and build X-ray devices that scan cargo containers for nuclear materials and other hazards. A version of this screening system is now in commercial use, and on May 16, the company received national recognition for its successful development from the federal Small Business Administration. 

There are a lot more examples of this that many of you come in contact with almost everyday. You use or come in contact with particle accelerators more often than you realize.

Also note that this is another example where research that was meant for the study of fundamental physics, found an application elsewhere as an off-shoot.

Zz.

I'm trying very hard not to roll my eyes (it's getting difficult to pop them back into the sockets), but hey, if it is done tongue-in-cheek like the first movie, it might just be entertaining enough.

It appears that the new Ghostbusters movie will tackle spacetime, or at least, something that threatened our spacetime.

Dan Akroyd told the ever-famous radio personality about the story of Ghostbusters 3:

“It’s based on new research that’s being done in particle physics by the young men and women at Columbia University. Basically, there’s research being done that I can say that the world or the dimension that we live in, our four planes of existence, length, height, width and time, become threatened by some of the research that’s being done. Ghostbusters – new Ghostbusters – have to come and solve the problem.”

Oooookay! Another creation of black holes in a particle collider that could destroy our universe, perhaps?

It sounds like it could be a storyline for a Marvel movie. We'll just have to see how they play this out.

Zz.

A bid by two Swiss billionaires to turn a mothballed pharma campus into the hub for a huge new biotechnology initiative has taken a major step forward today.

The Campus Biotech project announced that it had bought the 45,000-square-metre Sécheron site in Geneva, formerly occupied by Merck Serono, for an undisclosed amount.

The École Polytechnique Fédérale de Lausanne and the University of Geneva will create a new institute for bio- and neuroengineering on the site, backed with a donation of 100 million Swiss francs (US$103 million) from the Wyss Foundation. Other research groups from the two universities are also expected to join the initiative.

Plans for the site were announced last year, but at that point the purchase of the real estate had not been secured (see ‘Cash injection set to revive Swiss drug site’).

Campus Biotech has been bankrolled by Hansjörg Wyss, the philanthropist who established the Wyss Foundation, and the Bertarelli family, headed by philanthropist Ernesto Bertarelli, whose grandfather founded Serono, a pharmaceutical company sold to Merck KGaA in 2007.

In statement, Ernesto Bertarelli said: “We are absolutely delighted to be moving forward with Campus Biotech. A central element of our plan is the creation of a Wyss Institute focusing on bio- and neuro-engineering. We have been much encouraged by the wide support for our project which we believe will bring immense value to the Geneva Lake region and Switzerland as a whole.”

Leroy Hood, head of a prominent research institute in Seattle, Washington, violated conflict-of-interest rules when he reviewed a friend’s grant, California’s stem-cell agency disclosed in a letter to the state legislature.

The 2 April letter was first reported by the California Stem Cell Report, an independent blog that covers the California Institute for Regenerative Medicine (CIRM), a US$3-billion agency established in 2004.

Hood, a DNA-sequencing pioneer who is president of the Institute for Systems Biology, reviewed a $24-million application to CIRM that included Irving Weissman, a stem-cell scientist at Stanford University in California. Hood and Weissman are good friends and own a ranch together in Montana — a fact that Hood did not disclose when he completed a conflict of interest disclosure.

It was Hood’s first stint reviewing a CIRM grant, and he was not aware of the agency’s conflict-of-interest rules, the letter said. The application (for a genomics data centre and three research projects) was unsuccessful, and Hood’s conflict-of-interest violation did not break any laws. Hood did not review the part of the application that directly involved his pal, and the National Medal of Science winner was not involved in the decision to reject the application. Another grant reviewer raised the potential conflict.

CIRM’s president, Alan Trounson, has been a guest at the ranch, and the stem-cell scientist was the one who recruited Hood to become a reviewer for the agency, CIRM told California Stem Cell Report.

A December 2012 US Institute of Medicine review of CIRM called for changes at the agency to address conflicts of interest on the agency’s board, among other issues (see ‘Scientific panel recommends changes to California’s stem cell institute’.

I came across this last night and thought I would share it with you. It’s the preamble to Edgar Allan Poe‘s famous short story The Murders in the Rue Morgue, which is arguably the first-ever work in the genre of detective fiction. The piece is a bit dated (especially by the reference to the (now) discredited pseudoscience of phrenology, but Poe nevertheless says some very interesting things about a topic that I have returned to a number of times on this blog: the interplay between analysis and synthesis (and between deductive and inductive reasoning) involved not only in detective stories but also in card games and – I would contend – in the scientific method generally. I  agree with Poe when he says that the most fascinating part of such endeavours is the poorly understood yet vital element of intuition, that creative spark of ingenuity that sets apart a true genius, but am not sure about his contention that it is closely related to the analytic aspect. Anyway, see what you think…

–o–

IT is not improbable that a few farther steps in phrenological science will lead to a belief in the existence, if not to the actual discovery and location, of an organ of analysis. If this power (which may be described, although not defined, as the capacity for resolving thought into its elements) is not, in fact, an essential portion of what late philosophers term ideality, then there are, indeed, many good reasons for supposing it a primitive faculty. That it may be a constituent of ideality is here suggested in opposition to the vulgar dictum (founded, however, upon the assumptions of grave authority) that the calculating and discriminating powers (causality and comparison) are at variance with the imaginative — that the three, in short, can hardly co-exist. But, although thus opposed to received opinion, the idea will not appear ill-founded when we observe that the processes of invention or creation are strictly akin with the processes of resolution — the former being nearly, if not absolutely, the latter conversed.

It cannot be doubted that the mental features discoursed of as the analytical, are, in themselves, but little susceptible of analysis. We appreciate them only in their effects. We know of them, among other things, that they are always to their possessor, when inordinately possessed, a source of the liveliest enjoyment. As the strong man exults in his physical ability, delighting in such exercises as call his muscles into action, so glories the analyst in that moral activity which disentangles.  He derives pleasure from even the most trivial occupations bringing his talent into play. He is fond of enigmas, of conundrums, of hieroglyphics; exhibiting in his solutions of each a degree of acumen which appears to the ordinary apprehension præternatural. His results, brought about by the very soul ­and essence of method, have, in truth, the whole air of intuition.

The faculty in question is possibly much invigorated by mathematical study, and especially by that highest branch of it which, unjustly, and merely on account of its retrograde operations, has been called, as if par excellence, analysis.  Yet to calculate is not in itself to analyse. A chess-player, for example, does the one without effort at the other.  It follows that the game of chess, in its effects upon mental character, is greatly misunderstood. I am not now writing a treatise, but simply prefacing a somewhat peculiar narrative by observations very much at random; I will, therefore, take occasion to assert that the higher powers of the reflective intellect are more decidedly and more usefully tasked by the unostentatious game of draughts than by all the elaborate frivolity of chess. In this latter, where the pieces have different and bizarre motions, with various and variable values, that which is only complex is mistaken (a not unusual error) for that which is profound. The attention is here called powerfully into play. If it flag for an instant, an oversight is committed, resulting in injury or defeat. The possible moves being not only manifold but involute, the chances of such oversights are multiplied; and in nine cases out of ten it is the more concentrative rather than the more acute player who conquers. In draughts, on the contrary, where the moves are unique and have but little variation, the probabilities of inadvertence are diminished, and the mere attention being left comparatively unemployed, what advantages are obtained by either party are obtained by superior acumen. To be less abstract — Let us suppose a game of draughts, where the pieces are reduced to four kings, and where, of course, no oversight is to be expected. It is obvious that here the victory can be decided (the players being at all equal) only by some recherché movement, the result of some strong exertion of the intellect. Deprived of ordinary resources, the analyst throws himself into the spirit of his opponent, identifies himself therewith, and not unfrequently sees thus, at a glance, the sole methods (sometimes indeed absurdly simple ones) by which he may seduce into miscalculation or hurry into error.

Whist has long been noted for its influence upon what is termed the calculating power; and men of the highest order of intellect have been known to take an apparently unaccountable delight in it, while eschewing chess as frivolous. Beyond doubt there is nothing of a similar nature so greatly tasking the faculty of analysis. The best chess-player in Christendom may be little more than the best player of chess; but proficiency ­ in whist implies capacity for success in all those more important undertakings where mind struggles with mind. When I say proficiency, I mean that perfection in the game which includes a comprehension of all the sources (whatever be their character) whence legitimate advantage may be derived. These are not only manifold but multiform, and lie frequently among recesses of thought altogether inaccessible to the ordinary understanding. To observe attentively is to remember distinctly; and, so far, the concentrative chess-player will do very well at whist; while the rules of Hoyle (themselves based upon the mere mechanism of the game) are sufficiently and generally comprehensible. Thus to have a retentive memory, and to proceed by “the book,” are points commonly regarded as the sum total of good playing. But it is in matters beyond the limits of mere rule that the skill of the analyst is evinced. He makes, in silence, a host of observations and inferences. So, perhaps, do his companions; and the difference in the extent of the information obtained, lies not so much in the falsity of the inference as in the quality of the observation. The necessary knowledge is that of what to observe. Our player confines himself not at all; nor, because the game is the object, does he reject deductions from things external to the game. He examines the countenance of his partner, comparing it carefully with that of each of his opponents. He considers the mode of assorting the cards in each hand; often counting trump by trump, and honor by honor, through the glances bestowed by their holders upon each. He notes every variation of face as the play progresses, gathering a fund of thought from the differences in the expression of certainty, of surprise, of triumph or of chagrin. From the manner of gathering up a trick he judges whether the person taking it can make another in the suit. He recognises what is played through feint, by the air with which it is thrown upon the table. A casual or inadvertent word; the accidental dropping or turning of a card, with the accompanying anxiety or carelessness in regard to its concealment; the counting of the tricks, with the order of their arrangement; embarrassment, hesitation, eagerness or trepidation — all afford, to his apparently intuitive perception, indications of the true state of affairs. The first two or three rounds having been played, he is in full possession of the contents of each hand, and thenceforward puts down his cards with as absolute a precision of purpose as if the rest of the party had turned outward the faces of their own.

The analytical power should not be confounded with simple ingenuity; for while the analyst is necessarily ingenious, the  ingenious man is often remarkably incapable of analysis. I have spoken of this latter faculty as that of resolving thought into its elements, and it is only necessary to glance upon this idea to perceive the necessity of the distinction just mentioned. The constructive or combining power, by which ingenuity is usually manifested, and to which the phrenologists (I believe erroneously) have assigned a separate organ, supposing it a primitive faculty, has been so frequently seen in those whose intellect bordered otherwise upon idiocy, as to have attracted general observation among writers on morals. Between ingenuity and the analytic ability there exists a difference far greater indeed than that between the fancy and the imagination, but of a character very strictly analogous. It will be found, in fact, that the ingenious are always fanciful, and the truly imaginative never otherwise than profoundly analytic.

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An Australian team unveils the fundamental building block of a scalable quantum computer that could be embedded in today’s silicon chips.

Back in the late 90s, a physicist in Australia put forward a design for a quantum computer. Bruce Kane suggested that phosphorus atoms embedded in silicon would be the ideal way to store and manipulate quantum information.

Applications of category theory are described by Julie Rehmeyer in ScienceNews under the banner

One of the most abstract fields in math finds application in the ‘real’ world.

Now, how about applications in the real world?

Škoda is not just a carmaker; it is producing happy drivers. And you may see that even the engines in the factory are having a great time.

In the same way, Anthony Zee – as Zvi Bern noticed – decided to make many readers fall in love with the physics of general relativity by having written this wonderful tome, Einstein Gravity in a Nutshell. Bern said that the goal wasn't to create new experts but Zee corrected him that he wanted to make the readers fall in love so deeply that they may dream about becoming experts, too. And the clearly enthusiastic Anthony had to enjoy the writing of the book, too.

I received this large, almost 900-page scripture on Einstein's theory yesterday. Obviously, I haven't read the whole book yet but I may have spent more time with it than most readers (more than zero) so that I can tell you why you should buy it and what philosophy, style, and content you may expect.




It's a book addressed to a wide variety of readers, including very young ones (perhaps college freshmen and bright high school students) and amateur physicists. Experienced physicists and professionals may find some gems or at least entertainment in the book, too. Because of this goal, the book starts with elementary things such as the units including \(G,c,\hbar\) and Planck units, relativity even in classical physics, as well as basics of curved spaces, differential geometry, and so on.




The style is witty and somewhat dominated by words – and amusing titles. You may find lots of philosophical and historical remarks and stories from Anthony's professional life but the physics is always primary. And I mean physics, not rigorous mathematics. Tony is focusing on objects, phenomena, and their measurable and calculable quantities and the purpose of physics is to understand them and calculate them. So he spends almost no time with various picky issues – whether a function has to be smooth; whether one should use one fancy word from abstract mathematics or another. In fact, he considers the suppressed role of rigorous maths to be a part of the "shut up and calculate" paradigm that he subscribes to.

In some sense, you could say that the approach resembles the Feynman Lectures on Physics. It is very playful and the author is always careful to tell you think that are still fun and stop elaborating on details when he could start to bore you. So the book (probably) keeps its fun status at every place (it's true for the portions I have read). But Anthony Zee manages to penetrate much more deeply into general relativity with this strategy.

Once he goes through all the basics – which allow a beginner to start with the subject almost from scratch but which seem very entertaining for a reader who doesn't really need such introductions anymore – and he answers all the FAQs on tensors and lots of other things, he offers some of the simplest derivations of Einstein's equations and is ready to apply them.

It's useful to know what concepts are considered primary starting points by the author. I would say that Zee is elevating the concept of symmetries and the action – the latter allows us to formulate most dynamical laws in classical and quantum physics really concisely (although we know perfectly consistent quantum systems that don't seem to have any nice action; and the action always assumes that we prefer a particular classical limit of a quantum theory – and the classical limit isn't necessarily unique).

Concerning the applications, some of the historically important applications that were designed to verify the theory are suppressed. But you get very close to the cutting edge, including the general-relativistic aspects of topics that are hot in the contemporary high-energy theoretical physics and the cosmological/particle-physics interface. So you may actually learn advanced topics about black holes including some Hawking radiation (including the numerical prefactors of the temperature; but the author doesn't go extremely far here; note that amusingly enough, the Hawking radiation is even discussed in an introductory chapter); large and warped extra dimensions; de Sitter and anti de Sitter space including a discussion of conformal transformations (although it doesn't seem like a full-fledged textbook on AdS/CFT); topological field theories; Kaluza-Klein theory (with extra spatial dimensions) and braneworlds; Yang-Mills theory (there's lots of electromagnetism in the earlier chapters); even twistor theory; discussions on the cosmic inflation and the cosmological constant problem; and heuristic thoughts on quantum gravity (some of them are more heuristic than the state-of-the-art allows us; but Zee's philosophy is that textbook shouldn't be composed exclusively of the totally established stuff ready to be carved in stone).

Using lots of witticisms and clever analogies, Zee also proves some things you wouldn't expect – e.g. that Hades isn't inside the Earth. The equivalence principle is compared to the decision of all airlines, regardless of the size (and the size of their aircraft), to fly between two distant cities along the same path on the map. Witty and apt.

Anthony is convinced that most authors are explaining things in unnecessarily complicated ways – in some cases, perhaps, they want to look smart by looking incomprehensible. That's not Zee's cup of tea. He enjoys to simplify things as much as possible (but not more than that). And he loves to formulate things so that the reader is led to the conclusion that things are simple and make sense, after all. For example, there is a fun introduction to the least action principle (light isn't stupid enough not to know the best path) and we learn that "after Lagrange invented the Lagrangian, Hamilton invented the Hamiltonian". It makes sense, doesn't it?

There's a lot to find in the book. Some readers say that the book is less elementary than Hartle's book but more elementary than Carroll's. Maybe. Anthony is more playful and less formal but there are aspects in which he gets further than any other introductory textbook of GR.

The book is full of notes, a long index, and simply clever exercises. The illustrations are pretty and professional. If you are buying books to see photographs of attractive blonde women with toys, you won't be disappointed, either.

Because the book is really extensive and even the impressions it has made on your humble correspondent in the single day are numerous, I have to resist the temptation to offer you examples, excerpts etc. because that could make this blog entry really long by itself. Instead, I recommend you once again to try the book.

Here's a video "tutorial" of what Supersymmetry is.



Zz.

As I was checking the recent stat postings on arXiv, I noticed the paper by Chen and Xie entitled inference in Kingman’s coalescent with pMCMC.  (And surprisingly deposited in the machine learning subdomain.) The authors compare a pMCMC implementation for Kingman’s coalescent with importance sampling (à la Stephens & Donnelly), regular MCMC and SMC.  The specifics of their pMCMC algorithm is that they simulate the coalescent times conditional on the tree structure and the tree structure conditional on the coalescent times (via SMC). The results reported in the paper consider up to five loci and agree with earlier experiments showing poor performances of MCMC algorithms (based on the LAMARC software and apparently using independent proposals).  They show similar performances between importance sampling and pMCMC. While I find this application of pMCMC interesting, I wonder at the generality of the approach: when I was introduced to ABC techniques, the motivation was that importance sampling was deteriorating very quickly with the number of parameters. Here it seems the authors only considered one parameter θ. I wonder what happens when the number of parameters increases. And how pMCMC would then compare with ABC.

Filed under: Books, Statistics, University life Tagged: ABC, Gibbs sampling, importance sampling, Kingman's coalescent, pMCMC, population genetics, simulation, SMC

What is dark matter made of?

We almost know that its mass should be dominated by a new light particle species that is heavy enough so that it moves rather slowly relatively to the speed of light ("cold" dark matter). Because dark matter isn't gone yet, such a particle must be stable or almost exactly stable – lifetime in billions of years, to say the least.



The lightest particle carrying a "new type of charge" is the best explanation why it's stable. By far the most popular clarification what this new charge is is the R-parity, a new "sign" introduced by SUSY. All the known particles in the Standard Model have the R-parity equal to \(+1\) which is why the Standard Model interactions never produce individual superpartners whose R-parity is \(-1\). For the Standard Model particles, the R-parity may be written as \((-1)^{3B-3L+2J}\) where the odd coefficients may be replaced by any odd integers (although some values may be more correct for the exotic particles).




The stable particle is then the lightest particle with the R-parity equal to \(-1\), the lightest superpartner, the so-called LSP: it's almost all the supersymmetric models' candidate for the WIMP, the new dark matter particle. If you look just superficially, it may be the superpartner of any elementary particle in the Standard Model. The R-parity has just two possible values (signs) so it forms a group isomorphic to \(\ZZ_2\). The number of R-parity-odd particles in a physical object is therefore conserved modulo two. In particular, they may be pair-created at the colliders and they're expected to pair-annihilate in the outer space.

However, if the LSP were a superpartner of a charged particle, it would be charged, too. And charged particles interact with the electromagnetic field. They may be seen. They're not dark. I am sort of convinced that one may construct viable theories of the observed data with non-dark matter as well – with "superheavy hydrogen atoms" that have a different charged particle as the nucleus, for example, pretend to be hydrogen, but store much more mass – but let's assume that the dark matter is dark, indeed.




In the literature, most models assume or conclude that the LSP is either the gravitino – the spin-3/2 superpartner of the graviton (in that case, the R-parity may be broken because the gravitino is still nearly stable due to the gravitational i.e. very weak character of its interactions) – or, much more frequently, the lightest neutralino. The neutralino is the spin-1/2 superpartner of an electrically neutral Standard Model elementary particles. Because there are several such particles (components of the W-boson, photon, and the Higgs field), there are several neutralinos. The mass of such neutralinos is given by a matrix and the "sharply defined mass" eigenstates are general superpositions of the photino, zino, and higgsinos – four neutralinos.

Physical technicalities usually force us to adopt the mass of hundreds of gigaelectronvolts for such a neutralino if it is the LSP. Such a "relatively heavy" neutralino couldn't explain the positive hints in the dark-matter direct search experiments that indicate the existence of a sub-\(10\GeV\) WIMP. The SUSY phenomenological literature is really obsessed by the idea that the LSP has to be a neutralino. It almost looks like a Christian dogma and different cliques seem to fight whether the neutralino LSP is mostly a wino (Catholic) or a bino (Protestant) or a higgsino (Orthodox). I am afraid that the certainty that the LSP has to be a neutralino is one of the not quite justified assumptions that are adopted because of group think, not because of solid evidence.

Now, building on various previous papers such as this 2006 paper and the authors' 2012 paper (PRD), Ki-Young Choi (Korea) and Osamu Seto (Japan) propose an attractive label for this new hypothetical \(8.6\GeV\) or so dark matter particle: a light Dirac right-handed sneutrino.

Light Dirac right-handed sneutrino dark matter

What does it mean? The adjective "light" means that its mass is supposed to be much smaller than hundreds of gigaelectronvolts. The word "sneutrino" says that it's the superpartner of a neutrino; the prefix "s-" may be interpreted not just as "the superpartner of" but also as "scalar" because these superpartners of spin-1/2 fermions are scalar fields and particles, i.e. spin-0 particles. The adjective "right-handed" means that it's the superpartner of a right-handed neutrino – all the neutrinos we directly observe today are left-handed (while the antineutrinos are right-handed but we are not talking about any antineutrinos with \(L=-1\) here at all). Finally, the adjective "Dirac" means that the neutrino is assumed to have mostly Dirac (and not Majorana) masses, coming from the bilinear product of the left-handed and right-handed neutrino components. Note that the adjective "Dirac" says nothing about the sneutrino per se; we first say that the neutrino is Dirac and then add the "s-" prefix. How the neutrinos get their mass is important for the sneutrinos, too.

The Asian authors claim that they can write down a model of this sort that is pretty much consistent with all the data – positive and perhaps even negative (XENON100) direct search experiments; experiments measuring the invisible width of the Z-boson and the Higgs boson (including the "effective number of neutrino species").

It would be sort of fair if the superpartner of the most invisible Standard Model particle – a neutrino – became the first visible superpartner, albeit still a "dark one". ;-)



Don Lincoln of the Fermilab attempted to explain supersymmetry in 6 minutes today. See the video above. By the way, 10,000 papers on SUSY is probably a huge underestimate. It depends what you count but over 100,000 papers may be mentioning supersymmetry.

Researchers have come a step closer to understanding how and why the earthquake and tsunami that devastated Japan in 2011 were so surprisingly big. Temperature sensors installed in the fault last year now show that friction between the rocks during the quake was an order of magnitude smaller than previously assumed.

The magnitude-9 Tohoku earthquake shocked the research community by setting a record for the greatest amount of slip ever seen in a fault: some 40–80 metres. No one could explain how or why this happened. In late 2011, a group of researchers mounted a ‘rapid response’ effort to investigate (see ‘Drilling ship to probe Japanese quake zone’).

In the spring of 2012, they managed to install a suite of 55 temperature sensors more than 850 metres into the fault, which itself lies under 6,900 metres of water. Creating an observatory at those depths was in itself a record-breaking achievement. The project faced many challenges: bad weather delayed the installation, shifts in the fault could have crushed the instruments and an earthquake in December could have buried the observatory with landslides. But the team managed to retrieve their sensors on 26 April.

“Amazingly, it seems like the experiment might have actually worked,” says team member Emily Brodsky of the University of California, Santa Cruz. She and a colleague presented their preliminary results at the Japan Geoscience Union Meeting on 19 May.

The temperature measures show how heat dissipated from the fault over time, enabling the researchers to extrapolate back to the moment of the earthquake and to see how much frictional heat was generated during the shift. From this they calculated the coefficient of friction for the fault, and found it to be an order of magnitude lower than the conventional value that has been used since the 1970s. That lower number means less friction.

The result supports the theory that the friction during an earthquake can be dramatically different from the friction during quiet times, perhaps because water in clays is heated by a quake’s shaking, then expands and jacks open the fault. Brodsky says that there are hints that this finding could be generalized to other faults.

The result is consistent with experiments being conducted by Brodsky’s collaborator Kohtaro Ujiie of the University of Tsukuba, who has been trying to recreate the pressure and temperature conditions of this fault in the lab. Both groups hope to publish their results soon.

You have probably heard that the Kepler Space Telescope, which has discovered various surprising planetary systems as well as small planets in the habitable zones of their stars, suffered a major technical failure last week.  According to the Kepler Mission Manager Update, engineers determined that reaction wheel #4 failed to restart after the spacecraft entered a self-preserving “safe mode.”  In this post, I will discuss what happened to the spacecraft and how this will affect our ability to find Earth-sized planets.  The short story is that the space telescope probably won’t detect any new Earth-sized planets, but we can continue discovering such planets in the wealth of existing data.

What are the reaction wheels anyway, and why does Kepler need them?

The reaction wheels, capable of spinning at 4,000 rpm, generate the angular momentum necessary to control and stabilize the telescope pointing.  They are a mechanical type of gyroscope manufactured by Goodrich Corporation and contracted for the Kepler Space Telescope by Ball Aerospace.

The detection of Earth-sized planets requires extremely precise pointing of the telescope.  This is because the transit of an Earth-sized planet across a sun-sized star results in a change in the star’s brightness of 1/10,000 (or 100 parts per million).  Kepler was designed to detect changes in brightness of 20 parts per million, which meant it was capable of finding Earth-sized planets and smaller.  To save on bandwidth, Kepler only downlinks data from the pixels associated with 156,000 target stars out of the millions of stars in the Kepler field.  Data from an “aperture” of pixels around each target star are downlinked to Earth, and computer programs on Earth measure the brightness of the star based on the light that hit the pixels in the aperture.  If the telescope pointing is not good enough to keep the target stars in their respective apertures on the pixels, it is impossible to measure the brightness of those stars with a precision of 20 parts per million.

Can the pointing of the spacecraft be restored?

Not to the precision required to detect changes in brightness of 20 parts per million.  The Mission Manager Update offers the possibility of using the thrusters to help the telescope maintain its pointing, but it seems unrealistic to expect a crude thruster to achieve the precision of a gyroscope.  Although it might be possible to point Kepler well enough to do some new science, it seems unlikely that Kepler will be able to continue discovering Earth-sized planets.

Can we still find Earth-sized planets (in the habitable zone and elsewhere)?

Yes!  Kepler has already given us a plethora of data that will feed hungry scientists for years to come.  Buried within the 156,000 light curves of stars spanning four years are many of the tell-tale “dips” indicating the transits of planets–including small ones.  The current challenge is to process the data in a manner that teases these tiny signals out of the noise and unwanted signals of stellar variability and spacecraft systematics, much like finding a needle in a haystack.  The game of detecting Earth-sized planets has moved from the telescope to the computer, and I invite all of you enthusiastic data processors to take up the challenge!  You can find and download the light curves of the Kepler stars from the NexSci Exoplanet Archive.

Other types of observational follow-up present major opportunities for confirming and characterizing Earth-sized planets given the existing Kepler data.  Radial velocity measurements of host stars, stellar spectra, adaptive optics imaging, and speckle interferometry help rule out false positive scenarios and better characterize the planets and their stars.  Additionally, transit timing variations in the Kepler data can help characterize the architectures multi-planet systems.  Describing each of these follow-up methods is outside the scope of this article, but I encourage you to leave a comment if you are curious about them.

What are the major repercussions of having only four years of data instead of eight?

The “extended” Kepler Mission was supposed to run for eight years.  (The original mission design was for 3.5 years, and Kepler exceeded this!)  Eight years would have yielded twice as much data on Earth-sized planets, resulting in a square-root-of-two improvement in our ability to extract their transit signals from the noise.

The length of the mission corresponds to how “far-out” we can probe stellar systems.  For instance, since Earth has a one-year orbital period, we would expect to see three transits of an Earth-analog (enough to identify it as a planet candidate) in three years. However, a transiting Mars-analog (with an orbital period of 1.88 years) would transit two or three times in four years, but would certainly transit three times in eight years.  So the four-year Kepler Mission can find and characterize the occurrence of exoplanets out to the orbital period of Earth, but an eight-year mission could have probed as far out as the orbital period of Mars.

 

Lee Smolin has a new book out, Time Reborn: From the Crisis in Physics to the Future of the Universe. His previous subtitle lamented “the fall of a science,” while this one warns of a crisis in physics, so you know things must be pretty dire out there.

I’m not going to do a full-fledged review of the book, which gives Lee’s argument for why “time” needs to be something more than just a label on spacetime or a parameter in an evolution equation, but a distinct fundamental piece of reality with respect to which the laws of physics and space of states can change. (Sabine Hossenfelder does offer a review.) There are also suggestions as to how this paradigm-changing viewpoint gives us new ways to talk about economics and social problems.

Over at Edge, John Brockman has posted an interview with Lee, and is accumulating responses from various interested parties. I did contribute a few words to that, which I’m reproducing here.

---

Time and the Universe

Cosmology and fundamental physics find themselves in an unusual position. There are, as in any area of science, some looming issues of unquestioned importance: how to reconcile quantum mechanics and gravity, and the nature of dark matter and dark energy, to name two obvious ones. But the reality is that particle physicists, gravitational physicists, and cosmologists all have basic theories that work extraordinarily well in the regimes to which we have direct access. As a result, it is very hard to make progress; we know our theories are not absolutely final, but without direct experimental contradictions to them it’s hard to know how to do better.

What we have, instead, are problems of naturalness and fine-tuning. Dark energy is no mystery at all, if we are simply willing to accept a cosmological constant that is 120 orders of magnitude smaller than its natural value. We take fine-tunings to be clues that something deeper is going on, and try to make progress on that basis. Sadly, these are subtle clues indeed.

“Time” is something that physicists understand quite well. Quantum gravity remains mysterious, of course, so it’s possible that the true status of time in the fundamental ontology of the world is something that remains to be discovered. But as far as how time works at the level of observable reality, we’re in good shape. Relativity has taught us how to deal with time that is non-universal, and it turns out that’s not such a big deal. The arrow of time—the manifold differences between the past and future – is also well-understood, as long as one swallows one giant fine-tuning: the extreme low entropy of the early universe. Given that posit, we know of nothing in physics or cosmology or biology or psychology that doesn’t fit into our basic understanding of time.

But the early universe is a real puzzle. Is it puzzling enough, as Smolin suggests, to demand a radical re-thinking of how we conceive of time? He summarizes his view by saying “time is real,” but by “time” he really means “the arrow of time” or “an intrinsic directedness of physical evolution,” and by “real” he really means “fundamental rather than emergent.” (Opposing “real” to “emergent” is an extremely unfortunate vocabulary choice, but so be it.)

This is contrary to everything we think we understand about physics, everything we think we have learned about the operation of the universe, and every experiment and observation we have ever performed. But it could be true! It’s always a good idea to push against the boundaries, try something different, and see what happens.

I have two worries. One is that Smolin seems to be pushing hard against a door that is standing wide open. With the (undeniably important) exceptions of the initial-conditions problem and quantum gravity, our understanding of time is quite good. But he doesn’t cast his work as an attempt to (merely) understand the early universe, but as a dramatic response to a crisis in physics. It comes across as a bit of overkill.

The other worry is the frequent appearance of statements like “it seems to me a necessary hypothesis.” Smolin seems quite content to draw sweeping conclusions from essentially philosophical arguments, which is not how science traditionally works. There are no necessary hypotheses; there are only those that work, and those that fail. Maybe laws change with time, maybe they don’t. Maybe time is fundamental, maybe it’s emergent. Maybe the universe is eternal, maybe it had a beginning. We’ll make progress by considering all the hypotheses, and working hard to bring them into confrontation with the data. Use philosophical considerations all you want to inspire you to come up with new and better ideas; but it’s reality that ultimately judges them.

 

Filed under: Mountains, pictures Tagged: Antarctica, climbing pictures, Posing Productions, Queen Maud, Ulvetanna

Former biology teacher Felicia Svoboda shows Fermilab visitors the ins and outs of doing science.

“Everyone hold up your thumb,” says Felicia Svoboda to a group of high school students, lifting her hand and watching them follow suit. “Ten trillion neutrinos just passed through your thumb. Did you feel it?” Some of the students shake their heads or raise their eyebrows curiously.

Svoboda’s job as a Fermilab docent is to get tour groups curious about particle physics. On this day, she leads a group of students from Illinois’ Rochelle Township High School around displays on the 15th floor of Fermilab’s iconic main building, Wilson Hall.

Guest post by Emily Riehl

Whether we grow up to become category theorists or applied mathematicians, one thing that I suspect unites us all is that we were once enchanted by prime numbers. It comes as no surprise then that a seminar given yesterday afternoon at Harvard by Yitang Zhang of the University of New Hampshire reporting on his new paper “Bounded gaps between primes” attracted a diverse audience. I don’t believe the paper is publicly available yet, but word on the street is that the referees at the Annals say it all checks out.

What follows is a summary of his presentation. Any errors should be ascribed to the ignorance of the transcriber (a category theorist, not an analytic number theorist) rather than to the author or his talk, which was lovely.

Prime gaps

Let us write $p_1,p_2,\dots$ for the primes in increasing cardinal order. We know of course that this list is countably infinite. A prime gap is an integer $p_{n+1}-p_n$. The Prime Number Theorem tells us that $p_{n+1}-p_n$ is approximately $\log (p_n)$ as $n$ approaches infinity.

The twin primes conjecture, on the other hand asserts that

$$\underset{n\to \mathrm{\infty }}{\mathrm{liminf}}(p_{n+1}-p_n)=2$$

i.e., that there are infinitely many pairs of twin primes for which the prime gap is just two. A generalization, attributed to Alphonse de Polignac, states that for any positive even integer, there are infinitely many prime gaps of that size. This conjecture has been neither proven nor disproven in any case. These conjectures are related to the Hardy-Littlewood conjecture about the distribution of prime constellations.

The strategy

The basic question is whether there exists some constant $C$ so that $p_{n+1}-p_n<C$ infinitely often. Now, for the first time, we know that the answer is yes…when $C=7\times {10}^7$.

Here is the basic proof strategy, supposedly familiar in analytic number theory. A subset $H=\{h_1,\dots ,h_k\}$ of distinct natural numbers is admissible if for all primes $p$ the number of distinct residue classes modulo $p$ occupied by these numbers is less than $p$. (For instance, taking $p=2$, we see that the gaps between the $h_j$ must all be even.) If this condition were not satisfied, then it would not be possible for each element in a collection $\{n+h_1,\dots ,n+h_k\}$ to be prime. Conversely, the Hardy-Littlewood conjecture contains the statement that for every admissible $H$, there are infinitely many $n$ so that every element of the set $\{n+h_1,\dots ,n+h_k\}$ is prime.

Let $\theta (n)$ denote the function that is $\log (n)$ when $n$ is prime and 0 otherwise. Fixing a large integer $x$, let us write $n\sim x$ to mean $x$ ≤ $n<2x$. Suppose we have a positive real valued function $f$—to be specified later—and consider two sums:

$$S_1=\sum _{n\sim x}f(n)$$ $$S_2=\sum _{n\sim x}(\sum _{j=1}^k\theta (n+h_j))f(n)$$

Then if $S_2>(\log 3x)S_1$ for some function $f$ it follows that ${\sum }_{j=1}^k\theta (n+h_j)>\log 3x$ for some $n\sim x$ (for any $x$ sufficiently large) which means that at least two terms in this sum are non-zero, i.e., that there are two indices $i$ and $j$ so that $n+h_i$ and $n+h_j$ are both prime. In this way we can identify bounded prime gaps.

Some details

The trick is to find an appropriate function $f$. Previous work of Daniel Goldston, János Pintz, and Cem Yildirim suggests define $f(n)=\lambda (n{)}^2$ where

$$\lambda (n)=\sum _{d\mid P(n),d<D}\mu (d)\left(\log \right(\frac{D}{d}){)}^{k+\ell }P(n)=\prod _{j=1}^k(n+h_j)$$

where $\ell >0$ and $D$ is a power of $x$.

Now think of the sum $S_2-(\log 3x)S_1$ as a main term plus an error term. Taking $D=x^{\vartheta }$ with $\vartheta <\frac{1}{4}$, the main term is negative, which won’t do. When $\vartheta =\frac{1}{4}+\omega$ the main term is okay but the question remains how to bound the error term.

Zhang’s work

Zhang’s idea is related to work of Enrico Bombieri, John Friedlander, and Henryk Iwaniec. Let $\vartheta =\frac{1}{4}+\omega$ where $\omega =\frac{1}{1168}$ (which is “small but bigger than $ϵ$”). Then define $\lambda (n)$ using the same formula as before but with an additional condition on the index $d$, namely that $d$ divides the product of the primes less that $x^{\omega }$. In other words, we only sum over square-free $d$ with small prime factors.

The point is that when $d$ is not too small (say $d>x^{1/3}$) then $d$ has lots of factors. If $d=p_1\cdots p_b$ and $R<d$ there is some $a$ so that $r=p_1\cdots p_a<R$ and $p_1\cdots p_{a+1}>R$. This gives a factorization $d=rq$ with $R/x^{\omega }<r<R$ which we can use to break the sum over $d$ into two sums (over $r$ and over $q$) which are then handled using techniques whose names I didn’t recognize.

On the size of the bound

You might be wondering where the number 70 million comes from. This is related to the $k$ in the admissible set. (My notes say $k=3.5\times {10}^6$ but maybe it should be $k=3.5\times {10}^7$.) The point is that $k$ needs to be large enough so that the change brought about by the extra condition that $d$ is square free with small prime factors is negligible. But Zhang believes that his techniques have not yet been optimized and that smaller bounds will soon be possible.

Earlier this year, in common with other media, this blog pledged its support to a campaign to save the Royal Institution from financial oblivion. In doing so I may have given the impression that the Royal Institution is a venerable and  highly esteemed organization dedicated to the task of bring science closer to the public and inspiring future generations with its exciting range of outreach activities, including its famous public lectures.

However, in the light of the Royal Institution’s recent decision to trademark the phrase “Christmas Lectures” , I now realize that this was misleading and in fact the Royal Institution is just another rapaciously self-serving organization, run by small-minded buffoons, which is dedicated to nothing but its own self-aggrandizement. It has further become clear that the RI will do anything it can to maintain its cushy existence in a  fancy property in Mayfair to the detriment of all  outreach activities elsewhere, and  should therefore be shut down immediately as a threat to the future health of UK science.

Moreover, as a protest, this blog calls upon all University science departments in the United Kingdom to organize their own series of  Christmas Lectures Yuletide Discourses  under the title Not the Royal Institution Christmas Lectures, beginning each presentation with a lengthy preamble describing the unpleasant and idiotic actions of the Royal Institution and explaining why its Christmas Lectures® should be boycotted.

I hope this clarifies the situation.

P.S. For more blog outrage, see here, here , here…. (cont., p94).

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Can cultural factors be more important than censorship in shaping Chinese surfing habits? Two researchers argue that a new study of the way global websites cluster together supports this idea.

• Title: Herschel Observations of Gas and Dust in the Unusual 49 Ceti Debris Disk
• Authors: A. Roberge, I. Kamp, B. Montesinos, W. R. F. Dent, G. Meeus, J. K. Donaldson, J. Olofsson, A. Moor, J.-C. Augereau, C. Howard, C. Eiroa, W.-F. Thi, D. R. Ardila, G. Sandell, P. Woitke
• First Author’s Institution: NASA Goddard Space Flight Center

Background

Young stars are surrounded by a protoplanetary disk from which planets form. Over time, the gas in the disk dissipates, leaving behind the solid material (planets, dust, and everything in between) in a debris disk. A star is typically 10 million years (Myr) old when the transition from the protoplanetary disk phase to the debris disk phase takes place — although this age varies from star to star.

The star called 49 Ceti does not fit nicely into this picture. Previous observations have found two dust belts around 49 Ceti, an inner warm component and an outer cold component — a pattern common in the debris disk phase. However, carbon monoxide (CO) gas has also been detected in this disk, a feature usually only found in protoplanetary disks. Estimates of 49 Ceti’s age range from 9 to 61 Myr, with the most recent measurement at 40 Myr (in general, determining the age of a star can be difficult business). One explanation for these discrepancies is that 49 Ceti has an unusually long-lived protoplanetary disk. Another theory is that the CO gas is not primordial, but is being continuously generated from evaporating comets or colliding planetary material.

To get a handle on 49 Ceti’s true nature, the authors of this paper (including fellow Astrobites writer Jessica Donaldson) studied it in the far-IR and sub-mm (wavelengths where debris disk dust emission is typically strongest) with the Herschel Space Observatory. They obtained photometry to model the spectral energy distribution (SED) of the disk, one resolved image to constrain the size of the disk, and spectra to search for emission lines from gas.

Photometry and SED

The authors image 49 Ceti at 70 and 160 microns with Herschel’s Photodetecting Array Camera and Spectrometer (PACS) instrument and at 250, 350, and 500 microns with the Spectral and Photometric Imaging Receiver (SPIRE) instrument.  All but one of the images were unresolved spatially, meaning the shape and size of the image are not determined by the true shape and size of the disk, but by the diffraction properties of the telescope. So although the shape and size of the disk itself could not be determined from these data, the total amount of light at each of these wavelengths (the photometry) could be measured.

The authors then create an SED of 49 Ceti using this new photometry plus data from the literature at wavelengths both longer and shorter than their Herschel measurements. The SED is shown in Figure 1. They fit the SED with a model consisting of the star plus one or two blackbody emission profiles representing the dust belts in thermal equilibrium, with temperatures set by their distance from the star. The blackbody curves are modified to suppress emission at very long wavelengths because grains emit radiation inefficiently at wavelengths much larger than their own size. The authors find that one blackbody cannot fit data at all wavelengths consistently, but two blackbodies (at temperaures of 175 and 62 K) fit the data nicely.

Based on these temperatures, the authors estimate that the rings are located at 11 and 84 AU from the star, but these distances are only lower estimates because realistic dust grains can be hotter than their equilibrium temperature, and thus seem closer to the star than they actually are. Nevertheless, this confirms that the dust around 49 Ceti is confined to two discrete rings, a feature of debris — not protoplanetary — disks.

Resolved 70 Micron Image

The image at 70 microns with PACS was spatially resolved. Resolved images are very useful because they allow a disk’s true size to be determined; an SED can only estimate the minimum disk size (as discussed in the previous section).

To verify that their image is truly resolved, the authors also observe the star Alpha Boo, which does not have a disk, at 70 microns to see how the telescope diffracts light from an unresolved source (this is known as the telescope’s point spread function, or PSF). The image of 49 Ceti is shown in the left panel of Figure 2, and it is clearly more extended than the image of Alpha Boo, shown in the insert. To undo the effect of the telescope’s diffraction, the authors deconvolve their image of 49 Ceti using the PSF measured from Alpha Boo, obtaining a truer image of 49 Ceti’s disk, shown in the right panel of Figure 2.

This image traces the cold outer dust component of 49 Ceti’s disk. They find that the disk extends to 200 AU, larger than the estimate from the SED. Previous studies found that the CO gas in this disk was also coming from 200 AU, suggesting that the gas may be arising from the same location as the solid material.

Gas Emission Features 

The authors search for specific gases in 49 Ceti’s disk by taking spectra with PACS at wavelengths where these gases emit. Specifically, the authors search for CO, H2O, DCO+, O I, and C II. (The roman numerals refer to the ionization state of the atomic gas, I means the atom is neutral, II means the atom has lost one electron).

The only gas they detect is C II, emitting at 158 microns. Protoplanetary disks commonly show emission from O I at 63 microns, so the fact that this gas was not detected in 49 Ceti suggests that the gas in this disk is not simply leftover from the protoplanetary disk phase. The detection of the ionized carbon and the non-detection of the oxygen are shown in Figure 3. 49 Ceti is one of only two known disks in which C II is detected while O I is not (the other is around the star HD 32297).

Modelling the Disk

The authors create physical models of the disk in an attempt to explain their observations, and they start with a model that assumes 49 Ceti has a long-lived protoplanetary disk. They use a computer code that includes various heating and cooling mechanisms for dust and gas in the disk, as well as chemical and physical reactions involving dust and numerous gas species. Despite tweaking the model’s many parameters, the authors could not match their data, most importantly the presence of of C II and CO (known from the literature) and the absence of O I.

The authors are not able to create a model of this same complexity to test the second scenario — that the gas is being generated from comets — because not enough is know about all the sources of potential carbon and oxygen gas. However, by estimating the rate at which CO is destroyed in the disk (based on the expected flux of molecule-breaking ultraviolet radiation), they use the known amount of CO to determine how fast new CO is being produced. At this rate, a large comet would be depleted in 0.4 to 32 Myr. So this explanation is plausible, given 49 Ceti’s range of possible ages.

Conclusions

So what is 49 Ceti? Two theories existed originally: it is either a long-lived protoplanetary disk or a gas-generating debris disk. These new Herschel observations point strongly towards the second interpretation. If gas truly comes from evaporating comets or colliding planetary material, these observations could be used to learn more about the composition of pre-planet material and the nature of the planet-formation process.

The Herschel mission has recently come to an end (the all-important coolant is depleted), but the work of analyzing and interpreting the data will continue, so expect new results from Herschel to keep hitting astro-ph. And while Herschel has passed, astronomers are looking to ALMA and JWST (in the sub-mm and infrared, respectively) to further our understanding of circumstellar disks and the formation of planets.

Completing an action-packed trilogy that began with quantum mechanics and picked up speed with the Higgs boson, here I am talking with Brady Haran of Sixty Symbols about the arrow of time. If you’d like something more in-depth, I can recommend a good book.

Will there be more? You never know! The Hitchhiker’s Guide to the Galaxy started out as a trilogy, and look what happened to that. (But I promise no prequels.)

Okay, so there hasn’t been an official IceCube press release on this, not until the paper finishes collaboration review and is posted on the Arxiv, but there have been some talks showing neutrino events observed by IceCube which are almost certainly astrophysical in origin. Short version, neutrino astronomy is now a real thing. We are observing the universe in photons (ever since we looked up at the night sky, and starting with Galileo with increasingly sophisticated instruments) and also in neutrinos (which travel undisturbed from deep within the astrophysical objects, reflecting the nuclear interactions deep within).

There’s a nice Gizmodo article with interesting comments.

University of Wisconsin news item.

Phys.org coverage of the news item.

The BBC news article.

Nature blog entry.

New Scientist entry written by our friend Anil who got to visit IceCube during construction.

Since the middle of last week, the news are spread around and there are Russian, Spanish, and French language versions (at a minimum!) of the news. Previously, only the neutrinos from Supernova 1987A had been seen from beyond the sun and the Earth’s atmosphere. Analysis is still ongoing, so this isn’t a final result by any means, but it is a proof-of-functionality of the IceCube detector and of neutrino astronomy.

Addition:

Scientific American’s article includes good quotes from the three Wisconsin-Madison postdocs who led the analysis, Nathan, Claudio, and Naoko.

 

Not much time to post these days, what with one thing and another, but music is always a good standby. In fact I’ve had this at the back of my mind for some time; hearing it on the radio last week gave me the nudge I needed to post it. I always feel a but uncomfortable about posting just a movement from a classical piece, but I think it is justifiable in this case. This is the 3rd Movement of String Quartet No. 15 (in A minor) by Ludwig van Beethoven (Opus 132).

The third movement is headed with the words

Heiliger Dankgesang eines Genesenen an die Gottheit, in der lydischen Tonart

I take the liberty of translating the first two words, using my schoolboy German, as “A Holy Song of Thanksgiving”; Beethoven wrote the piece after recovering from a very serious illness which he had feared might prove fatal. The movement begins in a mood of quiet humility but slowly develops into a sense of hope and deeply felt joy. The most remarkable  thing about this movement to me, though,  is that the music possesses the same restorative powers that it was written to celebrate. This music has a therapeutic value all of its own.

I don’t know if William Wordsworth (of whose poetry I am also extremely fond) ever had the chance to hear Beethoven’s Quartet No. 15 , and in Tintern Abbey he was writing about the therapeutic power of nature rather than music, but surely the  ”tranquil restoration” described in that poem is exactly the feeling  Beethoven achieves in his music:

These beauteous forms,
Through a long absence, have not been to me
As is a landscape to a blind man’s eye:
But oft, in lonely rooms, and ‘mid the din
Of towns and cities, I have owed to them
In hours of weariness, sensations sweet,
Felt in the blood, and felt along the heart;
And passing even into my purer mind,
With tranquil restoration: — feelings too

Of unremembered pleasure: such, perhaps,
As have no slight or trivial influence
On that best portion of a good man’s life,
His little, nameless, unremembered, acts
Of kindness and of love. Nor less, I trust,
To them I may have owed another gift,
Of aspect more sublime; that blessed mood,
In which the burthen of the mystery,
In which the heavy and the weary weight
Of all this unintelligible world,

Is lightened: — that serene and blessed mood,
In which the affections gently lead us on, —
Until, the breath of this corporeal frame
And even the motion of our human blood
Almost suspended, we are laid asleep
In body, and become a living soul:
While with an eye made quiet by the power
Of harmony, and the deep power of joy,
We see into the life of things.

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The results of a third-party investigation of Rossi's E-CAT reactor have appeared on the Cornell arxiv, and the conclusions of the tests are at the very least startling:

read more

The hydrogen wavefuction is one of the few systems that we can solve analytically. That is why we teach them in undergraduate QM classes. Yet, the ability to actually view such wavefunction isn't trivial and is part of the fundamental aspect of QM.

This latest work looks at the nodal structure of a hydrogen atomic orbitals using photoionization. In the process, the authors have provided a significant step in developing a "quantum microscope".

Writing in Physical Review Letters, Aneta Stodolna, of the FOM Institute for Atomic and Molecular Physics (AMOLF) in the Netherlands, and her colleagues demonstrate how photoionization microscopy directly maps out the nodal structure of an electronic orbital of a hydrogen atom placed in a dc electric field. This experiment—initially proposed more than 30 years ago—provides a unique look at one of the few atomic systems that has an analytical solution to the Schrödinger equation. To visualize the orbital structure directly, the researchers utilized an electrostatic lens that magnifies the outgoing electron wave without disrupting its quantum coherence. The authors show that the measured interference pattern matches the nodal features of the hydrogen wave function, which can be calculated analytically. The demonstration establishes the microscopy technique as a quantum probe and provides a benchmark for more complex systems.

The link above provides free access to the paper.

Zz.

You'd think that we physicists are immune to having a yard sale. That's the look that I got over the weekend when some people found out that I was a physicist.

Over this past weekend, I decided that I've had enough crap.... er ... stuff in the house that needs to go. So I decided to do a garage/yard sale, since that was the weekend that my city designated as our zipcode-wide garage sale days. I set up stuff very early on Friday, and it went through till mid afternoon yesterday.

So it is your typical yard sale in many ways - old stuff, used stuff, but also a few new stuff that never been opened. But I guess what made it rather unusual are all the science/math books that I had for sale. Some people who walked by the stacks of books did a double take when they see books on math, physics, spectroscopy, etc. A person even picked up the infamous Abromowitz and Stegun's "Handbook of Mathematical Functions", looked at me, and said "Really?!"

And I replied "Really!".

A few people inquired what I do for a living, and of course, many of them have never met a physicist (at least, not that they know of). A person even asked if I work at CERN! :)

Anyway, the 3-day yard sale weekend was tiring, but it was a lot of fun. Made more than $300 selling stuff that I no longer want. Whatever's left will be donated.

And no, the Abromowitz/Stegun book didn't get sold, as with the rest of the math/science books, even though I was selling them for 50 cents a piece! If you are in the Chicago area, look out for those on sale at your nearest charity resell stores in the near future!

Zz.

The ability to copy electronic code makes one-time pads vulnerable to hackers. Now engineers have found a way round this to create a system of cryptography that is invulnerable to electronic attack.

One-time pads are the holy grail of cryptography—they are impossible to crack, even in principle.

Best intentions (put nose to the grindstone, get these papers finished) notwithstanding, I do have a few more public lectures and whatnot coming up over the next few weeks. Would love to see you there! And if not, I recently did an episode of the Rationally Speaking podcast with Massimo Pigliucci and Julia Galef, where we talked about naturalism, science, philosophy, and other things I’m marginally qualified to speak on.

Wednesday May 22: I’m giving a public talk on the arrow of time at UC Davis. This is in the midst of a conference on the early universe, which should also be fun.

Wednesday May 29: I’ll be talking with Jim Holt, author of Why Does the World Exist?, at the LA Public Library. It’s possible this is will be sold out, but I think they’re going to tape it.

Sunday June 2: I’m the keynote speaker at the American Humanist Society annual conference in San Diego. 10:30 a.m. on a Sunday, so this one might be easier to get into! In fact you can get in for free even if you didn’t register for the conference, by following these simple steps:

1. Go to the website here: http://ahacon13.eventbrite.com/#
2. Click the orange “Enter Promotional Code” link.
3. Enter FREECON in the field that appears and click Apply.
4. The list of items should then include the free “Free to the Public: Matt Harding & Sean Carroll” option.
5. Choose that one (and any others) and then complete the registration.

Thursday June 6: Opening night at the Seattle Science Festival features Brian Greene, Adam Frank, and me, under the stern but fair moderation of Jennifer Ouellette. Adam and I will give short talks, and Brian will show us the West Coast premiere of the multimedia performance Icarus at the Edge of Time.

Wednesday June 12: I’m giving a public lecture at Fermilab on particles, fields, and the future of physics. It’s part of the Fermilab Users’s Meeting, as well as a workshop on the International Linear Collider. Not sure if I’ve ever given a public talk that will have so many people ready to correct my mistakes.

After a couple more trips in July, my calendar actually does clear up, and I can look forward to uninterrupted vistas of productivity. Watch out!

..is the relative position of the two teams in 16th and 17th place in the final Premiership table!

Of course, it would have been more satisfying if Sunderland had finished one place lower but then you can’t have everything!

Anyway, that’s the Premiership over for another season. Time to concentrate on the cricket. If the Ashes Tests producing anything like today’s play against New Zealand then it should be an exciting summer!

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There’s been some new progress on graph Laplacians!

As a mathematical physicist, I’ve always been in love with the Laplacian:

   

It shows up in many of the most fundamental equations of physics: the wave equation, the heat equation, Schrödinger’s equation… and Poisson’s equation:

which says how the density of matter, affects the gravitational potential

As I’ve grown interested in network theory, I’ve gotten more and more interested in ‘graph Laplacians’. These are discretized versions of the Laplacian, where we replace 3-dimensional space by a ‘graph’, meaning something like this:

You can get a lot of interesting information about a graph from its Laplacian. You can also set up discretized versions of all the famous equations I mentioned.

The new progress is a simple algorithm for very efficiently solving Poisson’s equation for graph Laplacians:

• Jonathan A. Kelner, Lorenzo Orecchia, Aaron Sidford, Zeyuan Allen Zhu, A simple, combinatorial algorithm for solving SDD systems in nearly-linear time.

Here’s a very clear explanation of the general idea, conveying some sense of why it’s so important, without any nasty equations:

• Larry Hardesty, Short algorithm, long-range consequences, MIT News, 1 March 2013.

It begins:

In the last decade, theoretical computer science has seen remarkable progress on the problem of solving graph Laplacians — the esoteric name for a calculation with hordes of familiar applications in scheduling, image processing, online product recommendation, network analysis, and scientific computing, to name just a few. Only in 2004 did researchers first propose an algorithm that solved graph Laplacians in “nearly linear time,” meaning that the algorithm’s running time didn’t increase exponentially with the size of the problem.

At this year’s ACM Symposium on the Theory of Computing, MIT researchers will present a new algorithm for solving graph Laplacians that is not only faster than its predecessors, but also drastically simpler.

This animation shows two different “spanning trees” for a simple graph, a grid like those used in much scientific computing. The speedups promised by a new MIT algorithm require “low-stretch” spanning trees (green), in which the paths between neighboring nodes don’t become excessively long (red).

I can’t beat this article at its own game… except to clarify that ‘solving graph Laplacians’ means solving Poisson’s equation with a graph Laplacian replacing the usual Laplacian.

So, let me just supplement this article with the nasty equations saying what a graph Laplacian actually is. Start with a graph. More precisely, start with a simple graph. Such a graph has a set of vertices and a set of edges such that

which says the edges are undirected, and

which says there are no loops.

The graph Laplacian is an operator that takes a function on the vertices of our graph,

and gives a new such function as follows:

The version of Poisson’s equation for this graph Laplacian is thus

But I should warn you: this operator has eigenvalues that are less than equal to zero, like the usual Laplacian People often insert a minus sign to make the eigenvalues ≥ 0.

There is a huge amount to say about graph Laplacians! If you want, you can learn more here:

• Michael William Newman, The Laplacian Spectrum of Graphs, Masters Thesis, Department of Mathematics, University of Manitoba, 2000.

I’ve been learning about some of their applications here:

• Ernesto Estrada, The Structure of Complex Networks: Theory and Applications, Oxford University Press, Oxford, 2011.

I hope sometime to summarize a bit of this material and push the math forward a bit. So, it was nice to see new progress on efficient algorithms for doing computations with graph Laplacians!

...more precisely screwing string theory...

The 5,250+ TRF blog entries discuss various topics, mostly scientific ones, including minor advances. However, there isn't any text on this website that would talk about matrix string theory (inpendently found 2 months later by a herald who inaugurated the new Dutch king and an ex-co-author of mine along with two twins).



If you search for the closest topic, you will find one article about Matrix theory published a year ago and a supplement about membranes in Matrix theory that was added a week later.




But now we want to talk about matrix string theory. It's a version of Matrix theory. Much like Matrix theory – or M(atrix) Theory – describes M-theory in 11 dimensions (which has no strings), matrix string theory describes type IIA or heterotic \(E_8\times E_8\) string theory in \(d=10\). So it's a stringy version of Matrix theory; or string theory formulated in a matrix form.

The discovery of matrix string theory was important for several reasons. First, it was an important confirmation of the ability of the Matrix theory concept to define the dynamics of string/M-theory in many situations; and it was the first time when we had a complete, non-perturbative definition of a string theory.




What do I mean by this comment? Before Matrix theory, all calculations in string theory would be organized as Taylor expansions in \(g_s\), the string coupling. All amplitudes would be written as \(A_0 + A_1 g_s + A_2 g_s^2\dots\), and so on. However, not every function may be expanded in this way and the general amplitudes in quantum field theory or string theory can't. For example, \(\exp(-C/g_s^2)\) has a Taylor expansion whose terms vanish (because all higher-order derivatives of this function at \(g_s=0\) vanish) even though the function was non-vanishing.

In this sense, a complete definition was absent. One could have even believed that the existence or consistency of string theory was just a perturbative illusion. Matrix string theory was the first "constructive proof" that string theory is well-defined even non-perturbatively. In the type IIA case, one had a definition for any \(g_s\). In the \(g_s\to\infty\) limit, one could easily show that the theory reduces to Matrix theory, the matrix model for M-theory; in the \(g_s\to 0\) limit, one could prove – and this is the main achievement of the matrix string theory founding papers – that the dynamics reproduces the states and interactions of type IIA string theory as we had known them from the perturbative approaches.

Formal and informal derivations of the matrix string Lagrangian

Matrix theory is formulated in terms of the following Hamiltonian\[

H = P^- = \frac{N}{2} {\rm Tr}\zav{ \Pi_i^2 - [X_i,X_j]^2 +{\rm fermionic} }

\] which is interpreted as a light-cone component \(P^- = (P^0-P^{10})/\sqrt{2}\) of the spacetime energy-momentum vector. Well, the original Matrix theory paper by BFSS (Banks, Fischler, Shenker, Susskind) talked about the "infinite momentum frame" and various "highly boosted limits". But one could easily go to the limit and rewrite the quantities in the light-cone gauge. I was always baffled how a paper by Lenny could have become well-known just because it made this self-evident point. My papers (written before Susskind) always took the light-cone gauge as an obvious fact, for granted, and I am confident that everyone who followed the Green-Schwarz machinery from the early 1980s (these physicists preferred to calculate things in the light-cone gauge at that time) had to immediately see that the more natural and more right way to interpret the BFSS model was the light-cone gauge and not just some half-baked "infinite momentum frame".

But let me avoid these discussions. I will assume that the reader has no problem with null combinations of spacelike and timelike components of the energy-momentum vector and realizes that they are often natural combinations to consider.

The Hamiltonian above also contains fermionic, Yukawa-like terms of the form \({\rm Tr}(\theta\gamma_i [X_i,\theta])\) needed for supersymmetry (and various related crucial cancellations) and all the fields are \(N\times N\) matrices chosen for the matrix model to respect the \(U(N)\) gauge symmetry; yes, all physically allowed states must be invariant under the whole \(U(N)\) group.

In the previous articles, I tried to explain why this quantum mechanical model whose fields are "large matrices", generalizations of the usual non-relativistic operators \(X_i,P_i\), contains multi-graviton states, their superpartners, and large membranes: it has all the objects it needs to agree with the physical spectrum of M-theory in 11 dimensions.

Now, we want to compactify M-theory on a circle. M-theory on \(S^1\times \RR^{10}\) has been known to be equivalent to type IIA string theory in 10 dimensions (from the very first paper by Witten that introduced M-theory: the equivalence of the low-energy limits had been known for 10 years before that Witten's paper). What do we have to do with the matrix model to see all the physics of type IIA string theory?

There was some confusion about this question in the original BFSS paper on Matrix theory. The authors tended to believe that their exact Hamiltonian contains "the whole Hilbert space" of string/M-theory in all of its backgrounds. However, it wasn't the case. The moduli are modes with \(P^-=0\) and they correspond to excitations of the \(U(0)\) matrix model. The BFSS matrix model has no degrees of freedom for \(N=0\) so there are no ways to change the moduli. Consequently, the model may only describe one particular superselection sectors – the states of string/M-theory that respect the asymptotic form of the spacetime that looks like one in 11-dimensional M-theory (with one light-like direction compactified on a "long" circle).

To see type IIA string theory, i.e. the states in a different superselection sector of string/M-theory, we need to construct a different matrix model. What is it?

At the end of 1997, Ashoke Sen and especially Nathan Seiberg proposed a straightforward way to derive the BFSS matrix model and its compactifications from a limiting procedure combined with some widely believed dualities in string/M-theory. It's a clever (and superior) derivation that allows us to derive matrix models that are gauge theories; as well as matrix models that aren't just "ordinary" gauge theories but their novel UV completions such as the \((2,0)\) theory in \(d=6\) and little string theory.

However, if we want to find a matrix model for a compactification of M-theory on \(T^k\) and the dimension \(k\) of the torus isn't greater than three, it's enough to use the formal "gauge theory assuming" derivation I used at the beginning of 1997. How does it work?

One develops (your humble correspondent developed) a more general procedure to "orbifold a matrix model". The compactification on a circle is an orbifold by the group isomorphic to \(\ZZ\) composed of translations by \(2\pi R n\) in the direction of the circular dimension. To find the matrix description of the orbifold, we need to enhance \(N\) sufficiently and constrain the matrices of this "enhanced BFSS model" in a way that says that "the matrices transformed by elements of the orbifold group are gauge conjugations of the original ones".

This may sound complicated but the example of the compactification, an important one, makes it rather clear what I mean. The BFSS model has matrices with elements such as \(X^i_{mn}\) where \(m,n=1,2,\dots N\) are the gauge indices. We need the set of values of these indices to be infinitely greater. So we replace these matrix degrees of freedom by \(X^i_{mn}(\sigma,\sigma')\) where \(\sigma\in(0,2\pi)\) with periodic boundary conditions (a circular set of possible values of this "index") is a continuous counterpart of the index \(m\) and similarly for \(\sigma'\) and \(n\).

Now the group \(\ZZ\) of the translations in the direction \(X^9\) has a generator, a translation by \(2\pi R_{9}\), and we identify it with the conjugation by \(\exp(i\sigma)\), a gauge transformation matrix that only acts on the continuous \(\sigma\) indices. Because the translation doesn't physically act on the bosons \(X^1\dots X^8\) and their momenta \(\Pi^i\), the condition "physical transformation equals gauge transformation" says that these matrices are simply functions of one \(\sigma\) because they impose \(\sigma=\sigma'\), or demand \(\delta(\sigma-\sigma')\) in the kernel, along the way. Similarly, \(X^9\) has an extra \(\delta'(\sigma-\sigma')\) term on the right hand side so this matrix gets promoted to the covariant derivative \(D_\sigma\). Again, what used to be the degrees of freedom in \(X^9(\sigma)\) get reinterpreted as the component \(A_\sigma\) of a gauge field.

It may sound incomprehensible or difficult or abstract but I don't find it constructive to spend too much time with that. When you do these operations properly, you will find out that the matrix model for type IIA string theory is a 1+1-dimensional gauge theory with the same group \(U(N)\) as the BFSS model compactified on \(S^1\times\RR\) where the \(S^1\) part of the infinite cylinder arises from the \(\sigma\) "continuous index" we had to add. This 1+1-dimensional gauge theory has a dimensionful parameter \(g_{YM}^2\). The formal procedure "physical transformation defining the orbifold equals gauge transformation of the matrices" even tells us how the coupling \(g_{YM}^2\) depends on the length of the circle \(2\pi R_9\) in the compactification of M-theory. Together with some analyses of the interactions in the resulting matrix model, we may derive that \(R_9/l_{Pl,11}\sim g_s^{3/2}\).

But let's not be too acausal. So far, we have derived the matrix model for type IIA string theory. It looks like the integral of the BFSS Hamiltonian over the circle \(\sigma\) except that the component \(X^9\) of the bosonic fields is replaced by the covariant derivative \(D_9\) involving the 1+1-dimensional gauge field. The original BFSS matrix model may be viewed as the compactification of the 10-dimensional (non-renormalizable) supersymmetric gauge theory to 0+1 dimensions. When we're compactifying the dimensions of the M-theory we want to describe by a matrix model, we must decompactify the spatial dimensions that were dimensionally reduced in the BFSS matrix model to start with. For type IIA string theory in ten dimensions, we must decompactify one (add the single "continuous index" \(\sigma\)). This operation is the opposite of dimensional reduction and because in chemistry, the opposite of reduction is oxidation, this procedure to construct higher-dimensional versions of the BFSS model to describe lower-dimensional vacua of M-theory is sometimes jokingly called the dimensional oxidation. ;-)

Minimizing the energy

Just to be sure: we have "derived" that type IIA string theory in ten dimensions at any coupling is completely equivalent to the maximally supersymmetric \(U(N)\) gauge theory in 1+1 dimensions whose "world volume" has one infinite timelike dimension and one circular, compact spacelike dimension. To get rid of the effects of the compactification of the light-like dimension, we need to take the large \(N\) limit.

In some sense, this is a very modest generalization or variation of the original BFSS claim. I became totally certain that this matrix model is the right one. This certainty is probably necessary for one to be sufficiently motivated to study its physics a bit more closely. So I started with that.

If the 1+1-dimensional gauge theory is the full type IIA string theory, including its D-branes, type IIA supergravity at low energies, black holes, and many other things, it should contain what type IIA string theory is known to contain. For example, it must contain the strings. They must also be able to split and join.

Diagonal in a basis that may change

A general Hamiltonian defines the energy in a quantum mechanical model. All states may be written as superpositions of energy eigenstates. However, some states are more interesting than others: the low-energy eigenstates of the Hamiltonian. Because energy tends to dissipates, physical systems generally like to "drop" to their low-lying states. That's why the low-lying states, starting from the ground state (lowest-eigenvalue eigenstate of the Hamiltonian), are the most important ones.

In other words, the first step in trying to understand the physics of a Hamiltonian in a quantum mechanical theory is to try to help Nature to minimize the energy. How do we do it with the matrix model for matrix string theory?

Let's consider the bosons only; the fermions add additional degrees of freedom, terms in the zero-point energy (that mostly cancel some bosonic terms that would destroy a consistent spacetime interpretation of the physics if they remained uncancelled), and other details. If you assume that fermions play this peaceful, calming, generalizing role, you may say that the important physics is already contained in the bosons.

How do we minimize the energy carried by the bosonic parts of the Hamiltonian? The matrix string Hamiltonian contains \(\int \dd \sigma\,{\rm Tr}(\Pi_i^2)\) times a coefficient. Clearly, this is minimized if the momenta \(\Pi_i(\sigma)\) are zero. More realistically, these matrices may be approximately diagonal and the diagonal entries \(\Pi^i_{nn}(\sigma)\) will behave as the degrees of freedom \(\pi_i(\sigma)\) defined on a Green-Schwarz string. Soon we will see what happens with the extra \(n\) etc.

The off-diagonal entries of \(\Pi^i\) as well as the same entries of \(X^i\) behave like W-bosons of a sort, massive degrees of freedom, and at low energies, the wave function is almost required to be proportional to the ground states wave function as a function of these off-diagonal entries.

More interestingly, we want to minimize the term \({\rm Tr}\zav{-[X_i,X_j]^2}\) in the energy, too. The minus sign has to be there because for each \(i,j\), the commutator is anti-Hermitian so its square is negatively definite, not positively definite. How do we minimize it? Clearly, it will be smaller if the eight matrices \(X^i\) commute with each other. (Quantum mechanically, the wave function will be concentrated near the points on the configuration space where they commute with each other.)

If they commute with each other, it means that we can simultaneously diagonalize them. In other words, we can write\[

X^i(\sigma) = U(\sigma) X^i_{\rm diag}(\sigma) U^{-1}(\sigma).

\] The matrix \(U\) may be assumed to be unitary because Hermitian matrices are diagonalized in an orthonormal basis. The matrix with the "diag" subscript on the right hand side is diagonal. But an important detail is that \(U(\sigma)\) must be allowed to be arbitrary because the energy minimization tells us nothing about the basis in which all the \(X^i\) matrices are diagonal.

And that makes a difference because \(U(\sigma)\) doesn't have to be periodic with the period of \(2\pi\). Only the total field \(X^i(\sigma)\) of the gauge theory has to be periodic. However, the transformation \(U(\sigma)\) to the basis in which \(X^i(\sigma)\) is diagonal may undergo a nontrivial monodromy if we change \(\sigma\) by \(2\pi\). The matrix \(X^i_{\rm diag}(0)\), for example, was constrained by our rules to be diagonal but the matrix \(U(0)\) that (via conjugation) brings a given \(X^i(\sigma)\) to the diagonal form is "almost unique" but not quite. First, one may add some \(N\) phases on the diagonal of \(U\).

Second, and this is more important here, the matrix \(U\) may be multiplied by a permutation matrix! If a matrix is diagonal in a certain basis, it is diagonal in a permutation of this basis, too! So we must consider more general matrices \(U(\sigma)\) that are continuous functions of \(\sigma\) but that obey\[

U(\sigma+2\pi) = U(\sigma) P

\] where \(P\) is a permutation matrix. In combination with some continuous but also aperiodic diagonal matrices \(X^i_{\rm diag}\), such a unitary matrix may still produce an energy-minimizing, periodic field \(X^{i}(\sigma)\). This is the key subtlety not to be overlooked if you want to understand physics of matrix string theory.

What is this fact good for?

It's easy to see how the \(U(N)\) matrix model, the two-dimensional gauge theory, contains \(N\) "short strings". The degrees of freedom of each such short string is carried by the diagonal entries of \(X^i(\sigma)\). There are \(N\) such entries along the diagonal. However, we also need "long strings"; the length of the \(\sigma\) coordinate space has been known to be proportional to the light-cone momentum \(P^+\) to everyone who was familiar with the light-cone gauge string theory.

This \(P^+\) is quantized, equal to \(N/R\), because the null coordinate \(X^-\) is compactified on a circle of radius \(R\) (we want to send \(R\to\infty\) to get rid of this semi-unphysical compactification which also forces us to send \(N\to\infty\) to keep \(P^+\) fixed). And we know how to find strings with \(P^+=1/R\) i.e. with the \(N=1\) unit of the light-like longitudinal momentum.

However, the permutation business tells us how to find the "long strings" with \(P^+=N/R\) for any positive integer \(N\). You pick an eigenvalue of \(X^i\) along the diagonal; trace it as you continuously change \(\sigma\) from \(0\) to \(2\pi\); and when you reach \(\sigma=2\pi\), this eigenvalue doesn't connect to the original one at \(\sigma=0\). Instead, it will connect to a different one and only if you increase \(\sigma\) by \(2\pi N\), you may return to the original function because \(N\) basis vectors participate in a cycle of the permutation (used in the boundary conditions for \(U(\sigma)\).

(The "long strings" were also called "screwing strings" by your humble correspondent because the monodromy bringing the eigenvalue to a new level every time you get around the circle looks like a screw. I didn't know what the verb "screw" had meant informally. But this informal meaning of "screwing" is one of the reasons why the incorrect name "matrix string theory" became more frequently used than the technically correct name "screwing string theory". Incidentally, note that "matrices" and "nuts [waiting for screws]" are translated by the same Czech word, "matice".)

Because every permutation may be decomposed into a product of circular cycles, we see that every low-energy state in matrix string theory is composed of several strings with arbitrary values of \(P^+=N/R\). The permutation defines a "sector" of matrix string theory. The decomposition into the sector is just an artifact of the low-energy approximation; there is no sharp "barrier" between the sectors as they're continuously connected on the configuration space of the 1+1-dimensional gauge theory.

One may also derive the origin of some other subtle conditions. For example, the bosonic/fermionic states of the long strings obey the right statistics because the permutations that interchange the whole long strings are elements of the \(U(N)\) gauge group that must keep all physical states invariant. However, one may also derive the \(L_0=\tilde L_0\) condition for each separate string as the gauge invariance under the generator of the \(ZZ_k\) cyclic group that defines the cyclical permutations associated with a given string. Well, this is really equivalent to \(L_0-\tilde L_0 \in k\ZZ\) but for large values \(k\), all values except for \(L_0-\tilde L_0=0\) will correspond to string states of a high energy and will not belong to the low-energy spectrum.

Merging and splitting strings: jumping in between the permutation sectors

I have already said that in the low-energy limit, it looks like the Hilbert space is composed of sectors labeled by permutations in \(S_N\subset U(N)\). Each cycle that such a permutation is composed of corresponds to one "long string" – an ordinary type IIA string – present in the configuration.

At the same time, matrix string theory allows you to continuously switch between different "sectors". This corresponds to changing the permutation or, equivalently, the decomposition of the total longitudinal momentum \(P^+\) to the individual strings.

The most elementary operation changing a permutation is the composition of this permutation with an extra transposition (of two pieces of the string; or two eigenvalues). The low-energy approximation of the gauge theory's (matrix model's) Hamiltonian will involve the list of the allowed sectors and the free Hamiltonian for the individual strings that match the free type IIA string theory. However, the gauge theory isn't quite free so there will also be corrections and those may change the sector (the permutation). Those that only add one transposition will be the leading ones and they will correspond to nothing else than the usual splitting or merging of strings, a three-closed-string vertex.

We know that the gauge theory is supersymmetric so the interactions will have to preserve the same supersymmetry. DVV showed that the form of the splitting/merging leading interaction is essentially unique. But even without knowing its form, I could have derived – using a trick using the assumption that the large \(N\) limit is universal and independent of \(R\), the light-like radius – how the coefficient of the three-string vertex depends on the radius \(R_9\) of the coordinate we compactified to get the matrix model of type IIA string theory out of the BFSS model for M-theory. (There are two radii compactified here which are often labeled as \(R_9\) and \(R_{11}\). People who don't understand the logic of matrix string theory may confuse them. The exchange of these two radii that is effectively used in the construction was also called the 9/11 flip and be sure that it was before my PhD defense on 9/11/2001.)

The DVV description of the permutations

In March 1997, DVV who were much more familiar with the standard machinery of two-dimensional conformal field theories described the free-string limit of the gauge theory by a concise term: the symmetric orbifold CFT. It means a CFT – a linear (not non-linear, in this case) sigma model on \(\RR^{8N}/S_N\) where \(S_N\) is the permutation group exchanging the \(N\) copies of the 8-dimensional transverse space.

They also wrote down the explicit form of the three-string interaction vertex (leading interaction) emerging in this limit in terms of spin fields and twist fields, fixed a mistake in my not quite correct derivation of the level-matching \(L_0=\tilde L_0\) condition, and added some comments about the appearance of the D0-branes (short strings with the electric field etc.).

Higher-order terms in the Hamiltonian

The transposition of two eigenvalues is just the simplest among the extra permutations that may change the sector. In reality, the matrix model for string theory predicts all the complicated permutations (cycles with 3 elements or any number of elements), too. One may guess a natural Ansatz how these terms look like at any order in \(g_s\). We wrote these formulae with Dijkgraaf – a paper showing that the matrix string Hamiltonian is corrected at every order and how (these extra high-order terms produce contact terms interactions that are needed for the consistency of the light-cone gauge string theory but they may be largely circumvented in the usual covariant calculations based on moduli spaces of Riemann surfaces). This particular paper remained almost unknown, one of the numerous testimonies of the fact that in the 21st century, the interest in technical things such as "filling the gaps in the only non-perturbative definition of type IIA string theory we have" was dropping to zero. In 2003, people were already much more excited with philosophical gibberish such as the anthropic lack of principle and fabricated "technical evidence" that it applies in string theory.

I won't proof-read this text because I am afraid that its technical character will shrink its readership close to an infinitesimal number that can't justify the extra work needed for proofreading.

Hi all, and welcome to the return of the undergrad research posts! For those who don’t remember this series: this is where we feature the research that you’re doing. If you’ve missed the previous installments, you can find them under the “Undergraduate Research” category here.

What does this series mean for you? We want to hear from you! Whether you’ve done an REU project, you’re working on your senior thesis, or you’ve recently started a research project in between homework sets — if you’re an undergrad doing research, we’d love to hear about it.

You can share what you’re doing by clicking on the “Your Research” tab above (or by clicking here) and using the form provided to submit a brief (fewer than 200 words) write-up of your work. The target audience is one familiar with astrophysics but not necessarily your specific subfield, so write clearly and try to avoid jargon. Feel free to also include either a visual regarding your research or else a photo of yourself!

We look forward to hearing from you!

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Halston Lim and Jason Liang
Halston and Jason did this work jointly at the North Carolina School of Science and Mathematics.

Neutrinos, fundamental particles of the Standard Model of particle physics, can provide unique information about the internal processes of opaque high-energy astrophysical events. The ability of neutrinos to travel vast distances through matter is a crucial advantage of neutrino astronomy over optical astronomy. We demonstrated how neutrinos can be used to study the properties of failed supernovae (fSN) and black hole-neutron star mergers (BHNSM), thus providing a valuable contribution to neutrino astronomy. fSN neutrino detection would result in the first observation of black hole formation, while neutrinos from BHNSM could be used to determine if BHNSM are progenitors of short-period gamma-ray bursts, some of the most energetic events in the universe. By calculating the observed neutrino signal in various current and proposed detectors, we determined the detectability of fSN and BHNSM and demonstrated how the observed neutrino signal can provide information about the temperature and average energy of the neutrinos at the source. We also showed how these emission characteristics can then provide further information about the production of heavy metals in fSN and BHNSM. Our results confirm that neutrino observations of galactic fSN and BHNSM are feasible and provide fundamental groundwork for future research on fSN and BHNSM.

 

Andrew Emerick
Andrew is a graduating senior at the University of Minnesota. He worked on this project for his Honors thesis under Dr. Tom Jones and Dr. David Porter, using the resources of the Minnesota Supercomputing Institute. Andrew will be entering graduate school this fall at Columbia University, pursuing a doctorate in Astronomy with an intended focus in computational astrophysics.

Galaxy clusters are the largest gravitationally bound objects in the Universe, containing hundreds to thousands of individual galaxies. A majority of the baryonic matter in a cluster is contained within the intracluster medium (ICM): a hot, diffuse plasma that is interspersed throughout the galaxy cluster. The ICM is host to many phenomena, some of which can be used as key diagnostics, such as its often strong X-ray and radio emission. By studying the radio emission, we know that the ICM contains weak, cluster wide magnetic fields, but we do not understand well where they came from, or how they grew to the strength that is observed. One means to study the problem is to simulate the detailed microphysics of the interactions between the magnetic field and the “weather” of the ICM. We study the evolution of a weak, non-uniform magnetic field in a turbulent plasma, focusing on the details as to how turbulence amplifies a magnetic field. We focus primarily on the early evolution, and concern ourselves with how various magnetic field conditions can affect how the magnetic field grows over time, while fixing the nature of the turbulence. This study provides insight which can improve the accuracy of cosmological scale models of galaxy clusters. In addition, we know at some point information of the magnetic field conditions will be erased in the course of the ICM’s evolution. This study serves to help pinpoint exactly when that occurs, and thus if it could be possible to extract that information from potential observations.

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Many thanks to Halston, Jason and Andrew, as well as everyone else who has recently submitted contributions! Look for more undergraduate research posts in the future — this series will continue once a month.

 

The Pan-Andromeda Survey (PAndAS) continues to be a gold-mine for science. We're squeezing it hard to get out key results, but next year, the data will become public and everyone can have a looksie and write their own paper.

Here we have another paper by ANU astronomer, Dougal Mackey. Dougal's expertise is understanding the globular clusters orbiting the Andromeda galaxy, especially the distant clusters. He published a really nice piece of work recently which showed that these distant globulars are not just scattered randomly about Andromeda, but are more likely to be sitting on the stellar substructure we see. This substructure is the tidal debris from smaller galaxies that have fallen in and been shredded, meaning that the globulars are immigrants, having been born outside Andromeda, but joining the halo when their parent galaxy is destroyed; this is galactic cannibalism in action.

This new paper is about a particular cluster of stars orbiting Andromeda, named PAndAS-48 (who says astronomers aren't imaginative when it comes to naming things!). While this cluster was initially observed with the Canada-France-Hawaii Telescope (CFHT) as part of PAndAS, this paper presents new observations with the Hubble Space Telescope.

While the CFHT, at 3.6m, is larger than Hubble (2.5m), the lack of an atmosphere means we get much sharper images, and hence can see a lot fainter. Here's images from CFHT (left) compared to Hubble (right).
Nice! We actually observed the cluster in a couple of photometric bands with Hubble, which allowed us to make a colour-magnitude diagram; as you know, stars are not randomly scattered in such a picture, but sit on sequences that are driven by stellar evolution. What do we see?
For those in the know, yes, the faintest stars in there are around 28^th magnitude!

In there, we can see the Red Giant Branch and Horizontal Branch, and that allows us to understand lots of things about the globular, such as how far away it is and what stage it is at in terms of its evolution.

We can also measure the distribution of stars, and measure the shape of the clusters.
So, what is this cluster of stars? Is it a dwarf galaxy, dominated by dark matter? or a globular cluster, which are thought not to contain dark matter? It's actually very hard to tell. This piccy illustrates the issue.
The picture is pretty self-explanatory; size is along the bottom in parsecs, and brightness is up the side. The dots are colour-coded in terms of how elliptical they are.  The squares on the right are dwarf galaxies; they tend to be big and elliptical. The dots on the left are globular clusters, which tend to be small and circular (but notice that they can be of the same brightness as the dwarfs).

Where's PAndAS-48? It's the point with a circle around it, stubbornly right between the two populations! In fact, the ultimate conclusion is that we don't know what it is. If it is one or the other, then there are problems. But that's cool too!

 It is worth noting that PAndAS-48 appears to sit on the vast thin plane of satellites orbiting Andromeda, which makes it even more intriguing, but we haven't got it's velocity so can't confirm if it is orbiting in the same sense. But if it is, it will be extra cool.

As ever, the more we learn, the more questions we have. Yay!!

Well done Dougal!

It has been a while since I shared a snippet of the book project with you, so here's an update: Yesterday I completed a short burst of activity in which I re-did two pages in a story that were just horrible to behold. This is a panel form one of the pages. I'm pleased [...]

The best of the rest from the Physics arXiv preprint server

Performance of a Remotely Located Muon Radiography System to Identify the Inner Structure of a Nuclear Plant

 

Lisa and I had dinner with Gregory Benford and his wife when I visited U.C. Irvine a couple of weekends ago, and he raised an interesting point. So far, radio searches for extraterrestrial life have only seen puzzling brief signals – not long transmissions. But what if this is precisely what we should expect?

A provocative example is Sullivan, et al. (1997). This survey lasted about 2.5 hours, with 190 1.2 minute integrations. With many repeat observations, they saw nothing that did not seem manmade. However, they “recorded intriguing, non-repeatable, narrowband signals, apparently not of manmade origin and with some degree of concentration toward the galactic plane…” Similar searches also saw one-time signals, not repeated (Shostak & Tarter, 1985; Gray & Marvel, 2001 Gray, 2001). These searches had slow times to revisit or reconfirm, often days (Tarter, 2001). Overall, few searches lasted more than hour, with lagging confirmation checks (Horowitz & Sagan, 1993). Another striking example is the “WOW” signal seen at the Ohio SETI site…

That’s a quote from a paper Benford wrote with his brother and nephew:

• Gregory Benford, James Benford, and Dominic Benford, Searching for cost optimized interstellar beacons.

They claim the cheapest way a civilization could communicate to lots of planets is a pulsed, broadband, narrowly focused microwave beam that scans the sky. So, for anyone receiving this signal, there would be a lot of time between pulses. That might explain some of the above mysteries, or this one:

As an example of using cost optimized beacon analysis for SETI purposes, consider in detail the puzzling transient bursting radio source, GCRT J17445-3009, which has extremely unusual properties. It was discovered in 2002 in the direction of the Galactic Center (1.25° south of GC) at 330 MHz in a VLA observation and subsequently re-observed in 2003 and 2004 in GMRT observations (Hyman, 2005, 2006, 2007). It is a pulsed coherent source, with the ‘burst’ lasting as much as 10 minutes, with 77-minute period. Averaged over all observations, Hyman et al. give a duty cycle of 7% (1/14), although since some observations may have missed part of bursts, the duty cycle might be as high as 13%.

Even if these are red herrings, it seems very smart to figure out the cheapest ways to transmit signals and use that to guess what signals we should look for. We can easily make the mistake of assuming all extraterrestrial civilizations who bother to send signals through space will be willing to beam signals of enormous power toward us all the time. That could be true of some, but not necessarily all.

The cost analysis is here:

• James Benford, Gregory Benford, Dominic Benford, Messaging with cost optimized interstellar beacons.

and you can see a summary in this talk by Gregory’s brother James, who works on high-power microwave technologies:

I've lived in Australia for thirteen years, but in the way that Sting was an English Man in New York, I have never quite felt "Australian", rather, I am a Welsh Man in Sydney. Anyway, I still feel very British, and am a fan of British TV (apart from a few highlights, Australian TV is generally bilge).

Anyway, I've always loved a good murder mystery, and I like Midsomer Murders, even though they have changed the lead character (and the new chief inspector was actually a criminal in a previous episode). The premise of Midsomer's is simple; a cop in the quite fictional county of Midsomer solves murders. However, the show has been running for 15 years, and there seems to have been an awful lot of murders (although the murder rate is considerably lower than Honduras!). To keep the stories going, murders are set in, quite often, bizzarre circumstances.

A recent episode, Written in the Stars, focused on the intrigue and mystery at a research observatory at Midsomer University (up until this point, I don't think there had been mention of a university in the county). With usual stereotypical fashion, we have a mean professor, who is ready to steam-roller anybody to build his reputation, and a young genius who is writing her thesis (on the Heisenberg uncertainty principle) and threatens to dethrone the evil professor.

As part of her research, she needs to look at an eclipse (go figure) and the murder mayhem ensues. That's not the bad physics (but doesn't help).

Here's the young genius at work, presenting her work in the dome of a telescope (not sure why she is not in an office or lecture room).
Someone has gone to great effort to fill the board with lots of scientific squiggles. It's not, however, gibberish. I'm not sure if they used a text book, or wikipedia, but there are some correct things there.

However,  something annoyed me. Zooming in on the board, what do we see?
Plank's constant! Argh!! You'd think that our young genius who has written a thesis on quantum mechanics and is presenting her research to evil and nasty professor could spell Planck's name correctly. But there is more! Whoever wrote the squiggles got the symbol, h, correct, and even the value, 1.054 x 10^-27, correct, but they completely screwed up the units (that's too painful to go into) and what this number actually is is ħ ,which is Planck's constant divided by 2π.

Why would they bother going to the effort of writing something semi-correct, but pay so little attention that they make a mess of it? Why not just do it right? Don't they realise that professors of astrophysics might be watching?

One other thing that annoyed me is that they did the "astronomers only do their work inside telescope domes" thing
We don't. We have offices like everyone else. And even when we are at the telescope, we are in the control room, not freezing our bottoms off in the dome.

Before finishing, I think it's worth noting that the observatory actually used in the show is actually a university observatory. It is the University of London Observatory at Mill Hill
Even though I was a student at the University of London, I never used this observatory, although I did visit there when I was looking for a PhD position. However, the observatory is not in the picturesque county of Midsomer, but is next to the A1 in North West London.
Like a lot of observatories around the world, it was build outside of a city, but the cities have grown around them.

Anyway, the murderer was not the evil astrophysicist..... It was actually the friendly professor of Quantum Physics! I'm sure his knowledge of the uncertainty principle will help him in prison.

This essay makes a point that is only implicit in most of my other essays–namely that scientists are arro—oops that is for another post. The point here is that science is defined not by how it goes about acquiring knowledge but rather by how it defines knowledge. The underlying claim is that the definitions of knowledge as used, for example, in philosophy are not useful and that science has the one definition that has so far proven fruitful. No, not arrogant at all.

The classical concept of knowledge was described by Plato (428/427 BCE – 348/347 BCE) as having to meet three criteria: it must be justified, true, and believed. That description does seem reasonable. After all, can something be considered knowledge if it is false? Similarly, would we consider a correct guess knowledge? Guess right three times in a row and you are considered an expert –but do you have knowledge? Believed, I have more trouble with that: believed by whom? Certainly, something that no one believes is not knowledge even if true and justified.

The above criteria for knowledge seem like common sense and the ancient Greek philosophers had a real knack for encapsulating the common sense view of the world in their philosophy. But common sense is frequently wrong, so let us look at those criteria with a more jaundiced eye. Let us start with the first criteria: it must be justified. How do we justify a belief? From the sophists of ancient Greece, to the post-modernists and the-anything-goes hippies of the 1960s, and all their ilk in between it has been demonstrated that what can be known for certain is vanishingly small.

Renee Descartes (1596 – 1960) argues in the beginning of his Discourse on the Method that all knowledge is subject to doubt: a process called methodological skepticism. To a large extend, he is correct. Then to get to something that is certain he came up with his famous statement: I think, therefore I am.  For a long time this seemed to me like a sure argument. Hence, “I exist” seemed an incontrovertible fact. I then made the mistake of reading Nietzsche[1] (1844—1900). He criticizes the argument as presupposing the existence of “I” and “thinking” among other things. It has also been criticized by a number of other philosophers including Bertrand Russell (1872 – 1970). To quote the latter: Some care is needed in using Descartes’ argument. “I think, therefore I am” says rather more than is strictly certain. It might seem as though we are quite sure of being the same person to-day as we were yesterday, and this is no doubt true in some sense. But the real Self is as hard to arrive at as the real table, and does not seem to have that absolute, convincing certainty that belongs to particular experiences. Oh, well back to the drawing board.  

The criteria for knowledge, as postulated by Plato, lead to knowledge either not existing or being of the most trivial kind. No belief can be absolutely justified and there is no way to tell for certain if any proposed truth is an incontrovertible fact.  So where are we? If there are no incontrovertible facts we must deal with uncertainty. In science we make a virtue of this necessity. We start with observations, but unlike the logical positivists we do not assume they are reality or correspond to any ultimate reality. Thus following Immanuel Kant (1724 – 1804) we distinguish the thing-in-itself from its appearances. All we have access to are the appearances. The thing-in-itself is forever hidden.

But all is not lost. We make models to describe past observations. This is relatively easy to do. We then test our models by making testable predictions for future observations. Models are judged by their track record in making correct predictions–the more striking the prediction the better. The standard model of particle physics prediction of the Higgs[2] boson is a prime example of science at its best. The standard model did not become a fact when the Higgs was discovered, rather its standing as a useful model was enhanced.  It is the reliance on the track record of successful predictions that is the demarcation criteria for science and I would suggest the hallmark for defining knowledge. The scientific models and the observations they are based on are our only true knowledge. However, to mistake them for descriptions of the ultimate reality or the thing-in-itself would be folly, not knowledge.

 

There’s going to be a “thematic trimester” in Paris starting next spring:

• Semantics of proofs and certified mathematics, Institut Henri Poincaré, April 22nd - July 11th, 2014, organized by Pierre-Louis Curien, Hugo Herbelin, Paul-André Melliès.

If you like applications of category theory to logic and computer science, there should be a lot for you here!

The basic layout is this:

• Week 1 — Kick-off: Formalisation in mathematics and in computer science
• Week 3 — Workshop 1: Formalization of mathematics in proof assistants, organized by Georges Gonthier and Vladimir Voevodsky.
• Week 6 — Workshop 2: Constructive mathematics and models of type theory, organized by Thierry Coquand and Thomas Streicher.
• Week 8 — Workshop 3: Semantics of proofs and programs, organized by Thomas Ehrhard and Alex Simpson.
• Week 10 — Workshop 4: Abstraction and verification in semantics, organized by Paul-André Melliès and Luke Ong.
• Week 12 — Workshop 5, Certification of high-level and low-level programs organized by Christine Paulin and Zhong Shao.

A lot of people I know will attend parts of this, such as Jean Benabou, Marcelo Fiore, Dan Ghica, André Joyal, Samuel Mimram, and Bas Spitters. And that makes me happy, because Paul-André Melliès has invited me to spend up to a month attending this series of workshops, perhaps in two 2-week stretches. With a little luck I’ll be able to actually do this.

(My wife Lisa Raphals has gotten invited to Erlangen for the spring of 2014, meaning roughly April 1 - June 1. If she and I succeed in getting leaves of absence, I’ll go with her, and then take some trips to nearby places. Since I split my time between the Wild West and the Far East, Paris seems nearby to Erlangen to me. I also have vague invitations to IHES, Prague and Berlin which I might try to take advantage of. And if you have a luxurious villa in northern Italy or the French Riviera, let me know.)

Title: Galaxy pairs in the Sloan Digital Sky Survey – VI. The orbital extent of enhanced star formation in interacting galaxies

Authors: David R. Patton, Paul Torrey, Sara L. Ellison, J. Trevor Mendel, and Jillian M. Scudder

First author’s institution: Department of Physics and Astronomy, Trent University, Canada

Ok, this is no big surprise: enviroment affects star formation in galaxies. Observations have long shown that the star formation rate (SFR) is strongly enhanced when two galaxies merge or simply interact, with strongest enhancements found in the closest galaxy pairs, such as coalescing galaxies, or systems observed near to the first pericentre passage. Enhancements in star formation result in bluer colours and lower metallicities, i.e. characteristic features of young stellar populations, and spectacular objects such as luminous infrared galaxies.

However, a question is still open, as you can guess from the title of today’s astrobite: what is the orbital extent of enhanced star formation in interacting galaxies? At which projected separation of the two galaxies does it disappear? This Letter aims at investigating the enhancement of star formation as a function of the separation in galaxy pairs. The issue is addressed in two complementary ways: from an observational perspective, analyzing galaxy pairs from the Sloan Digital Sky Survey (SDSS), and from a theoretical perspective, studying the outputs of numerical simulations of galaxy mergers.

First, a large sample of ~600,000 galaxies from the SDSS is considered, which have secure spectroscopic redshift between 0.02 and 0.2, and total stellar mass estimated from photometry. For each galaxy, the closest neighbour is singled out, by requiring that it has 1) the smallest projected separation from the galaxy, 2) a rest-frame relative velocity lower than 1000 km/s, and 3) a stellar mass which is not excessively different (a factor of 10) from that of the galaxy.

Then, based on previous measurements of the SFR (see the catalogue in Brinchmann et al 2004), only star-forming galaxies are selected from the sample, without any special requirement on the SFR of their neighbours. In this way, also “mixed” galaxy pairs are included in the resulting sample, which contains ~211,000 star forming galaxies. For each of these galaxies, the authors determine a statistical “control sample” which matches each galaxy in both physical properties (stellar mass, redshift) and environment (local density, isolation), but does not necessarily contain star forming galaxies. The details of the procedure adopted to identify such control samples are deferred to a subsequent paper.

The bottom panel of Figure 1 shows, as a function of projected distance, the mean SFR of all the paired galaxies (blue) and of their statistical control samples (red). The ratio of these two quantities, which is defined as the “enhancement in star formation” is plotted in the top panel, where the inset plot shows its behaviour at even larger values of the projected separation. This figure nicely shows that star formation is enhanced in interacting galaxies, that such enhancement is stronger at the smallest separations, especially less than 20 kpc, and finally that the enhancement in SFR extends to larger separations than what was previously thought, being visible out to projected separations of ~ 150 kpc. In particular, it is found that the 66% of the enhanced star formation in galaxy pairs occurs at separations greater than 30 kpc.

Takeaway message: an enhancement in star formation is not only limited to strongly interacting galaxies with a very close companion, but also to wide galaxy pairs.

Now, are these findings consistent with the predictions from numerical simulations of interacting galaxies? In order to answer to this question, the authors investigate a suite of ad-hoc simulations of galaxy mergers run with the N-body/SPH code GADGET.

The simulated galaxy pairs are simple binary systems, where the stellar masses of the two initial galaxies is set to match the median stellar mass and mass-ratio of the observed SDSS sample. The simulated mergers span a significant set of five values of orbital eccentricities, five values of impact parameters, and three values of merger disc orientation, not limiting the galaxy orbits to low values of eccentricities and to small values of impact parameters. In total, 75 (5 x 5 x 3) orbital configurations for galaxy mergers are explored, and each one can be observed from a random set of viewing angles and at random times during the orbital evolution.

The authors compute the mean SFR over the 75 orbital configurations, observing each orbit from random orientations and at random moments during the merging history. Of course, these random times imply many different values of projected separations. This measurement of SFR is then translated into a measurement of SFR enhancement by normalizing by the SFR of the same galaxy evolved in isolation.

Figure 2 shows the mean enhancement in star formation rate computed from galaxy merger simulations (black), and the extremely small error bars are due to the average over many orbit orientations. The curve showing the same data derived from galaxy pairs in SDSS is overlaid in blue. Remarkably, the two curves, hence the two different approaches, yield a similar result: an enhancement in SFR is observed out to large projected distances ~150 kpc, though stronger in the SDSS data. In the simulations, the enhancement is a result of starburst activity triggered at the first pericentre passage, which persists as the galaxies move to wider separations.

Hence, the authors can safely conclude that interaction-induced star formation is not only limited to those galaxies which have a close companion, but rather it affects a larger variety of galaxies.

 

An artist honors the people and science of the CMS collaboration.

There’s a new splash of color at Point Five, the home of CMS detector on the Large Hadron Collider. Five vivid banners drape the gray walls of the complex, lending the warehouse a cathedral-like atmosphere. Arranged in a line, they pull the viewer’s gaze from panel to panel to land on a true-to-scale photo of the detector itself, magnificently displayed on the back wall. 

I was sent or came across a few interesting links that relate to things covered on this blog and/or of general scientific interest.

It was announced yesterday that the European Physical Society 2013 High Energy Physics Prize was awarded to the collaboration of experimental physicists that operate the ATLAS and CMS experiments that discovered a type of Higgs particle, with special mention to Michel Della Negra, Peter Jenni, and Tejinder Virdee, for their pioneering role in the development of ATLAS and CMS.  Jenni and Virdee are both at the LHCP conference in Barcelona, which I’m also attending, and it has been a great pleasure for all of us here to be able to congratulate them in person .

One thing that came up a couple of times regarding weather forecasting (for instance, in forecasting the path of Hurricane Sandy) is that the European weather forecasters are doing a much better job of predicting storms a week in advance than U.S. forecasters are.  And I was surprised to learn that one of the the main reasons is simple: U.S. forecasters have less computing power than their European counterparts, which sounds (and is) ridiculous.  The new director of the U.S. National Weather Service, Louis Uccellini, has been successful in his goal of improving this situation, as reported here.  [Thanks to two readers for pointing me to this article.]

One of the possible interpretations of the new class of high-energy neutrinos reported by IceCube (see yesterday’s post) is that they come from the slow decay of a small fraction of the universe’s dark matter particles, assuming those particles have a mass of a couple of million GeV/c². [That's much heavier than the types of dark matter particles that most people are currently looking for, in searches that I discussed in a recent article.]  I didn’t immediately mention this possibility (which is rather obvious to an expert) because I wanted a couple of days to think about it before generating a stampede or press articles.  But, not surprisingly, people who were paying more attention to what IceCube has been up to had recently written a paper on this subject.  [Here's an older, related paper, but at much lower energy; maybe there are other similar papers that I don't know about?]  At the time these authors wrote this paper, only the two highest energy neutrinos — which have energies that, within the uncertainties of the measurements, might be equal (see Figure 2 of yesterday’s post) — were publicly known.  In their paper, they predicted that (just as any expert would guess) in addition to a spike of neutrinos, all at about 1.1 million GeV, one would also find a population of lower-energy neutrinos, similar to those new neutrinos that IceCube has just announced. So yes, among many possibilities, it appears that it is possible that the new neutrinos are from decaying dark matter.  If more data reveals that there really is a spike of neutrinos with energy around 1.1 million GeV, and the currently-observed gap between the million-GeV neutrinos and the lower-energy ones barely fills in at all, then this will be extremely strong evidence in favor of this idea… though it will be another few years before the evidence could become convincing.  Conversely, if IceCube observes any neutrinos near but significantly above 1.1 million GeV, that would show there isn’t really a spike, disfavoring this particular version of the idea.

Regarding yesterday’s post, it was pointed out to me that when I wrote “The only previous example of neutrinos being used in astrophysics occurred with the discovery of neutrinos from the relatively nearby supernova, visible with the naked eye, that occurred in 1987,” I should also have noted that neutrinos were and are used to understand the interior of the sun (and vice versa).  And you could even perhaps say that atmospheric neutrinos have been used to understand cosmic rays (and vice versa.)

In sad news, in the “all-good-things-must-come-to-an-end” category, the Kepler spacecraft, which has brought us an unprecedented slew of discoveries of planets orbiting other stars, may have reached the end of the line (see for example here), at least as far as its main goals.  It’s been known for some time that its ability to orient itself precisely was in increasing peril, and it appears that it has now been lost.  Though this has occurred earlier than hoped, Kepler survived longer than its core mission was scheduled to do, and its pioneering achievements, in convincing scientists that small rocky planets not unlike our own are very common, will remain in the history books forever.  Simultaneous congratulations and condolences to the Kepler team, and good luck in getting as much as possible out of a more limited Kepler.

Filed under: Astronomy, LHC News, Particle Physics, Science and Modern Society Tagged: astronomy, cms, DarkMatter, Higgs, LHC, neutrinos, weather

One of the topics I am working on is how the standard model of particle physics can be extended. The reason is that it is, intrinsically, but not practically, flawed. Therefore, we know that there must be more. However, right now we have only very vague hints from experiments and astronomical observations how we have to improve our theories. Therefore, many possibilities are right now explored. The one I am working on is called technicolor.

A few weeks ago, my master student and I have published a preprint. By the way, a preprint is a paper which is in the process of being reviewed by the scientific community, whether it is sound. They play an important role in science, as they contain the most recent results. Anyway, in this preprint, we have worked on technicolor. I will not rehearse too much about technicolor here, this can be found in an earlier blog entry. The only important ingredient is that in a technicolor scenario one assumes that the Higgs particle is not an elementary particle. Instead, just like an atom, it is made from other particles. In analogy to quarks, which build up the protons and other hadrons, these parts of the Higgs are called techniquarks. Of course, something has to hold them together. This must be a new, unknown force, called techniforce. It is imagined to be again similar, in a very rough way, to the strong force. Consequently, the carrier of this fore are called technigluons, in analogy to the gluons of the strong force.

In our research we wanted to understand the properties of these techniquarks. Since we do not yet know if there is really technicolor, we can also not be sure of how it would eventually look like. In fact, there are many possibilities how technicolor could look like. So many that it is not even simple to enumerate them all, much less to calculate for all of them simultaneously. But since we are anyhow not sure, which is the right one, we are not yet in a position where it makes sense to be overly precise. In fact, what we wanted to understand is how techniquarks work in principle. Therefore, we just selected out of the many possibilities just one.

Now, as I said, techniquarks are imagined to be similar to quarks. But they cannot be the same, because we know that the Higgs behaves very different from, say, a proton or a pion. It is not possible to get this effect without making the techniquarks profoundly different from the quarks. One of the possibilities to do so is by making them a thing in between a gluon and a quark, which is called an adjoint quark. The term 'adjoint' is referring to some mathematical property, but these are not so important details. So that is what we did: We assumed our techniquarks should be adjoint quarks.

The major difference is now what happens if we make these techniquarks light and lighter. For the strong force, we know what happens: We cannot make them arbitrarily light, because they gain mass from the strong force. This appears to be different for the theory we studied. There you can make them arbitrarily light. This has been suspected since a long time from indirect observations. What we did was, for the first time, to directly investigate the techniquarks. What we saw was that when they are rather heavy, we have a similar effect like for the strong force: The techniquarks gain mass from the force. But once they got light enough, this effect ceases. Thus, it should be possible to make them massless. This possibility is necessary to make a Higgs out of them.

Unfortunately, because we used computer simulations, we could not really go to massless techniquarks. This is far too expensive in terms of the time needed to do computer simulations (and actually, already part of the simulations were provided by other people, for which we are very grateful). Thus, we could not make sure that it is the case. But our results point strongly in this direction.

So is this a viable new theory? Well, we have shown that a necessary condition is fulfilled. But there is a strong difference between necessary and sufficient. For a technicolor theory to be useful it should not only have a Higgs made from techniquarks, and no mass generation from the techniforce. It must also have more properties, to be ok with what we know from experiment. The major requirement is how strong the techniforce is over how long distances. There existed some indirect earlier evidence that for this theory the techniforce is not quite strong enough for sufficiently far distances to be good enough. Our calculations have again a more direct way of determining this strength. And unfortunately, it appears that we have to agree with this earlier calculations.

Is this the end of technicolor? Certainly not. As I said above, technicolor is foremost an idea. There are many possibilities how to implement this idea, and we have just checked one. Is it then the end of this version? We have to agree with the earlier investigations that it appears so in this pure form. But, in fact, in this purest form we have neglected a lot, like the rest of the standard model. There is still a significant chance that a more complete version could work. After all, the qualitative features are there, it is just that the numbers are not perfectly right. Or perhaps just a minor alteration may already do the job. And this is something where people are continuing working on.

"New Physics can appear at any moment but it is now conceivable that no new physics will show up at the LHC"

Guido Altarelli, LHC Nobel Symposium, May 15th 2013

It is funny reading the above quote if you are one who "conceived" that the LHC could find no new physics 7 years ago, as demonstrated by where I put my money...

In these days is ongoing LHCP 2013 (First Large Hadron Collider Physics Conference) and CMS data seem to point significantly toward new physics. Their measurements on the production modes for WW and ZZ are agreeing with my recent computations (see here) and overall are deviating slightly from Standard Model expectations giving

Note that Standard Model is alive and kicking yet but looking at the production mode of WW you will read

in close agreement with results given in my paper and improved respect to Moriond that was . The reason could be that: Higgs model is a conformal one. Data from ZZ yield

that is consistent with the result for WW mode, though. I give here the full table from the talk

For the sake of completeness I give here also the same results from ATLAS at the same conference that, instead, seems to go the other way round obtaining overall and this is already an interesting matter.

At CMS, new physics beyond the Standard Model is peeping out and, more inteestingly, the Higgs model tends to be a conformal one. If this is true, supersymmetry is an inescapable consequence (see here). I would like to conclude citing the papers of other people working on this model and that will be largely cited in the foreseeable future (see here and here).

Marco Frasca (2013). Revisiting the Higgs sector of the Standard Model arXiv arXiv: 1303.3158v1

Marco Frasca (2010). Mass generation and supersymmetry arXiv arXiv: 1007.5275v2

T. G. Steele, & Zhi-Wei Wang (2013). Is Radiative Electroweak Symmetry Breaking Consistent with a 125 GeV
Higgs Mass? Physical Review Letters 110, 151601 arXiv: 1209.5416v3

Krzysztof A. Meissner, & Hermann Nicolai (2006). Conformal Symmetry and the Standard Model Phys.Lett.B648:312-317,2007 arXiv: hep-th/0612165v4

Filed under: Particle Physics, Physics Tagged: ATLAS, CERN, CMS, Conformal Standard Model, Higgs particle, High-energy physics conferences

Suppose $X$ is a topological space and $U\subseteq X$ is an open subset, with closed complement $K=X\setminus U$. Then $U$ and $K$ are, of course, topological spaces in their own right, and we have $X=U\bigsqcup K$ as a set. What additional information beyond the topologies of $U$ and $K$ is necessary to enable us to recover the topology of $X$ on their disjoint union?

Recall that the subspace topologies of $U$ and $K$ say that for each open $V\subseteq X$, the intersections $V\cap U$ and $V\cap K$ are open in $U$ and $K$, respectively. Thus, if a subset of $X$ is to be open, it must yield open subsets of $U$ and $K$ when intersected with them. However, this condition is not in general sufficient for a subset of $X$ to be open — it does define a topology on $X$, but it’s the coproduct topology, which may not be the original one.

One way we could start is by asking what sort of structure relating $U$ and $K$ we can deduce from the fact that both are embedded in $X$. For instance, suppose $A\subseteq U$ is open. Then there is some open $V\subseteq X$ such that $V\cap U=A$. But we could also consider $V\cap K$, and ask whether this defines something interesting as a function of $A$.

Of course, it’s not clear that $V\cap K$ is a function of $A$ at all, since it depends on our choice of $V$ such that $V\cap U=A$. Is there a canonical choice of such $V$? Well, yes, there’s one obvious canonical choice: since $U$ is open in $X$, $A$ is also open as a subset of $X$, and we have $A\cap U=A$. However, $A\cap K=\varnothing$, so choosing $V=A$ wouldn’t be very interesting.

The choice $V=A$ is the smallest possible $V$ such that $V\cap U=A$. But there’s also a largest such $V$, namely the union of all such $V$. This set is open in $X$, of course, since open sets are closed under arbitrary unions, and since intersections distribute over arbitrary unions, its intersection with $U$ is still $A$.

Let’s call this set $i_{*}(A)$. In fact, it’s part of a triple of adjoint functors $i_{!}⊣i^{*}⊣i_{*}$ between the posets $O(U)$ and $O(X)$ of open sets in $U$ and $X$, where $i^{*}:O(X)\to O(U)$ is defined by $i^{*}(V)=V\cap U$, and $i_{!}:O(U)\to O(X)$ is defined by $i_{!}(A)=A$. Here $i$ denotes the continuous inclusion $U\hookrightarrow X$.

Now we can consider the intersection $i_{*}(A)\cap K$, which I’ll also denote $j^{*}{i}_{*}(A)$, where $j:K\hookrightarrow X$ is the inclusion. It turns out that this is interesting! Consider the following example, which is easy to visualize:

• $X={ℝ}^2$.
• $U=\{(x,y)\mid x<0\}$, the open left half-plane.
• $K=\{(x,y)\mid x\ge 0\}$, the closed right half-plane.

If an open subset $A\subseteq U$ “doesn’t approach the boundary” between $U$ and $K$, such as the open disc of radius $1$ centered at $(-2,0)$, then it’s fairly easy to see that $i_{*}(A)=A\cup \{(x,y)\mid x>0\}$, and therefore $j^{*}{i}_{*}(A)=\{(x,y)\mid x>0\}$ is the open right half-plane.

On the other hand, consider some open subset $A\subseteq U$ which does approach the boundary, such as

$$A=\{(x,y)\mid x^2+y^2<1\text{and}x<0\}$$

the intersection with $U$ of the open disc of radius $1$ centered at $(0,0)$. A little thought should convince you that in this case, $i_{*}(A)$ is the union of the open right half-plane with the whole open disc of radius $1$ centered at $(0,0)$. Therefore, $j^{*}{i}_{*}(A)$ is the open right half-plane together with the strip $\{(0,y)\mid -1<y<1\}$.

This example suggests that in general, $j^{*}{i}_{*}(A)$ measures how much of the “boundary” between $U$ and $K$ is “adjacent” to $A$. I leave it to some enterprising reader to try to make that precise. Here’s another nice exercise: what can you say about $i^{*}{j}_{*}(B)$ for an open subset $B\subseteq K$?

Let us however go back to our original question of recovering the topology of $X$. Suppose $A\subseteq U$ and $B\subseteq K$ are open such that $A\cup B$ is open in $X$; how does this latter fact manifest as a property of $A$ and $B$? Note first that $(A\cup B)\cap U=A$. Thus, since $i_{*}(A)$ is the largest $V$ such that $V\cap U=A$, we have $A\cup B\subseteq i_{*}(A)$, and therefore $B=j^{*}(A\cup B)\subseteq j^{*}{i}_{*}(A)$. Let me say that again:

$$B\subseteq j^{*}{i}_{*}(A).$$

This is a relationship between $A$ and $B$ which is expressed purely in terms of the topological spaces $U$ and $K$ and the function $j^{*}{i}_{*}:O(U)\to O(K)$, which we have just shown is necessary for $A\cup B$ to be open in $X$.

In fact, it is also sufficient! For suppose this to be true. Since $B$ is open in $K$, there is some open $C\subseteq X$ such that $C\cap K=B$. Given such a $C$, the union $C\cup U$ also has this property, since $U\cap K=\varnothing$. Note that in fact $C\cup U=B\cup U$, and also $B\cup U=j_{*}(B)$, the largest open subset of $X$ whose intersection with $K$ is $B$. (Since $K$, unlike $U$, is not open, there may not be a smallest such, but there is always a largest such.) Now I claim we have

$$A\cup B=j_{*}(B)\cap i_{*}(A)$$

To show this, it suffices to show that the two sides become equal after intersecting with $U$ and with $K$. For the first, we have

$$(j_{*}(B)\cap i_{*}(A))\cap U=j_{*}(B)\cap (i_{*}(A)\cap U)=j_{*}(B)\cap A=A=(A\cup B)\cap U$$

and for the second we have

$$(j_{*}(B)\cap i_{*}(A))\cap K=(j_{*}(B)\cap K)\cap i_{*}(A)=B\cap i_{*}(A)=B=(A\cup B)\cap K$$

using the assumption at the step $B\cap i_{*}(A)=B$.

In conclusion, the topology of $X$ is entirely determined by

• the induced topology of an open subspace $U\subseteq X$,
• the induced topology on its closed complement $K=X\setminus U$, and
• the induced function $j^{*}{i}_{*}:O(U)\to O(K)$.

Specifically, the open subsets of $X$ are those of the form $A\cup B$ — or equivalently, by the above argument, $i_{*}(A)\cap j_{*}(B)$ — where $A\subseteq U$ is open in $U$, $B\subseteq K$ is open in $K$, and $B\subseteq j^{*}{i}_{*}(A)$.

An obvious question to ask now is, suppose given two arbitrary topological spaces $U$ and $K$ and a function $f:O(U)\to O(K)$; what conditions on $f$ ensure that we can define a topology on $X≔U\bigsqcup K$ in this way, which restricts to the given topologies on $U$ and $K$ and induces $f$ as $j^{*}{i}_{*}$? We may start by asking what properties $j^{*}{i}_{*}$ has. Well, it preserves inclusion of open sets (i.e. $A\subseteq A\prime \Rightarrow j^{*}{i}_{*}(A)\subseteq j^{*}{i}_{*}(A\prime )$) and also finite intersections ($j^{*}{i}_{*}(A\cap A\prime )=j^{*}{i}_{*}(A)\cap j^{*}{i}_{*}(A\prime )$), including the empty intersection ($j^{*}{i}_{*}(U)=K$). In other words, it is a finite-limit-preserving functor between posets. Perhaps surprisingly, it turns out that this is also sufficient: any finite-limit-preserving $f:O(U)\to O(K)$ allows us to glue $U$ and $K$ in this way; I’ll leave that as an exercise too.

Okay, that was some fun point-set topology. Now let’s categorify it. Open subsets of $X$ are the same as 0-sheaves on it, i.e. sheaves of truth values, or of subsingleton sets, and the poset $O(X)$ is the (0,1)-topos of 0-sheaves on $X$. So a certain sort of person immediately asks, what about $n$-sheaves for $n>0$?

In other words, suppose we have $X$, $U$, and $K$ as above; what additional data on the toposes $\mathrm{Sh}(U)$ and $\mathrm{Sh}(K)$ of sheaves (of sets, or groupoids, or homotopy types, etc.) allows us to recover the topos $\mathrm{Sh}(X)$? As in the posetal case, we have adjunctions $i_{!}⊣i^{*}⊣i_{*}$ and $j^{*}⊣j_{*}$ relating these toposes, and we may consider the composite $j^{*}{i}_{*}:\mathrm{Sh}(U)\to \mathrm{Sh}(K)$.

The corresponding theorem is then that $\mathrm{Sh}(X)$ is equivalent to the comma category of ${\mathrm{Id}}_{\mathrm{Sh}(K)}$ over $j^{*}{i}_{*}$, i.e. the category of triples $(A,B,\varphi )$ where $A\in Sh(U)$, $B\in \mathrm{Sh}(K)$, and $\varphi :B\to j^{*}{i}^{*}(A)$. This is true for 1-sheaves, $n$-sheaves, $\mathrm{\infty }$-sheaves, etc. Moreover, the condition on a functor $f:\mathrm{Sh}(U)\to \mathrm{Sh}(K)$ ensuring that its comma category is a topos is again precisely that it preserves finite limits. Finally, this all works for arbitrary toposes, not just sheaves on topological spaces. I mentioned in my last post some applications of gluing for non-sheaf toposes (namely, syntactic categories).

One new-looking thing does happen at dimension 1, though, relating to what exactly the equivalence

$$\mathrm{Sh}(X)\simeq ({\mathrm{Id}}_{\mathrm{Sh}(K)}\downarrow j^{*}{i}_{*})$$

looks like. The left-to-right direction is easy: we send $C\in \mathrm{Sh}(X)$ to $(i^{*}C,j^{*}C,\varphi )$ where $\varphi :j^{*}C\to j^{*}{i}_{*}{i}^{*}C$ is $j^{*}$ applied to the unit of the adjunction $i^{*}⊣i_{*}$. But in the other direction, suppose given $(A,B,\varphi )$; how can we reconstruct an object of $\mathrm{Sh}(X)$?

In the case of open subsets, we obtained the corresponding object (an open subset of $X$) as $A\cup B$, but now we no longer have an ambient “set of points” in which to take such a union. However, we also had the equivalent characterization of the open subset of $X$ as $i_{*}(A)\cap j_{*}(B)$, and in the categorified case we do have objects $i_{*}(A)$ and $j_{*}(B)$ of $\mathrm{Sh}(X)$. We might initially try their cartesian product, but this is obviously wrong because it doesn’t incorporate the additional datum $\varphi$. It turns out that the right generalization is actually the pullback of $j_{*}(\varphi )$ and the unit of the adjunction $j^{*}⊣j_{*}$ at $i_{*}(A)$:

$$\begin{array}{ccc}C& \to & j_{*}(B)\\ \downarrow & & {\downarrow }^{j^{*}(\varphi )}\\ i_{*}(A)& \to & j_{*}{j}^{*}{i}_{*}(A)\end{array}$$

In particular, any object $C\in \mathrm{Sh}(X)$ can be recovered from $i^{*}C$ and $j^{*}C$ by this pullback:

$$\begin{array}{ccc}C& \to & j_{*}{j}^{*}C\\ \downarrow & & \downarrow \\ i_{*}{i}^{*}C& \to & j_{*}{j}^{*}{i}_{*}{i}^{*}C\end{array}$$

Now let’s shift perspective a bit, and ask what all this looks like in the internal language of the topos $\mathrm{Sh}(X)$. Inside $\mathrm{Sh}(X)$, the subtoposes $\mathrm{Sh}(U)$ and $\mathrm{Sh}(K)$ are visible through the left-exact idempotent monads $i_{*}{i}^{*}$ and $j_{*}{j}^{*}$, whose corresponding reflective subcategories are equivalent to $\mathrm{Sh}(U)$ and $\mathrm{Sh}(K)$ respectively. In the internal type theory of $\mathrm{Sh}(X)$, $i_{*}{i}^{*}$ and $j_{*}{j}^{*}$ are modalities, which I will denote $I_U$ and $J_U$ respectively. Thus, inside $\mathrm{Sh}(X)$ we can talk about “sheaves on $U$” and “sheaves on $K$” by talking about $I_U$-modal and $J_U$-modal types (or sets).

Moreover, these particular modalities are actually definable in the internal language of $\mathrm{Sh}(X)$. Open subsets $U\subseteq X$ can be identified with subterminal objects of $\mathrm{Sh}(X)$, a.k.a. h-propositions or “truth values” in the internal logic. Thus, $U$ is such a proposition. Now $I_U$ is definable in terms of $U$ by

$$I_U(C)=(U\to C)$$

I’m using type-theorists’ notation here, so $U\to C$ is the exponential $C^U$ in $\mathrm{Sh}(X)$. The other modality $J_U$ is also definable internally, though a bit less simply: it’s the following pushout:

$$\begin{array}{ccc}U\times C& \to & C\\ \downarrow & & \downarrow \\ U& \to & J_U(C)\end{array}.$$

In homotopy-theoretic language, $J_U(C)$ is the join of $C$ and $U$, written $U*C$. And if we identify $\mathrm{Sh}(U)$ and $\mathrm{Sh}(K)$ with their images under $i_{*}$ and $j_{*}$, then the functor $j^{*}{i}_{*}:\mathrm{Sh}(U)\to \mathrm{Sh}(K)$ is just the modality $J_U$ applied to $I_U$-modal types.

Finally, the fact that $\mathrm{Sh}(X)$ is the gluing of $\mathrm{Sh}(U)$ with $\mathrm{Sh}(K)$ means internally that any type $C$ can be recovered from $I_U(C)$, $J_U(C)$, and the induced map $J_U(C)\to J_U(I_U(C))$ as a pullback:

$$\begin{array}{ccc}C& \to & J_U(C)\\ \downarrow & & \downarrow \\ I_U(C)& \to & J_U(I_U(C))\end{array}$$

Now recall that internally, $U$ is a proposition: something which might be true or false. Logically, $I_U(C)=(U\to C)$ has a clear meaning: its elements are ways to construct an element of $C$ under the assumption that $U$ is true.

The logical meaning of $J_U$ is somewhat murkier, but there is one case in which it is crystal clear. Suppose $U$ is decidable, i.e. that it is true internally that “$U$ or not $U$”. If the law of excluded middle holds, then all propositions are decidable — but of course, internally to a topos, the LEM may fail to hold in general. If $U$ is decidable, then we have $U+\neg U=1$, where $\neg U=(U\to 0)$ is its internal complement. It’s a nice exercise to show that under this assumption we have $J_U(C)=(\neg U\to C)$.

In other words, if $U$ is decidable, then the elements of $J_U(C)$ are ways to construct an element of $C$ under the assumption that $U$ is false. In the decidable case, we also have $J_U(I_U(C))=1$, so that $C=I_U(C)\times J_U(C)$ — and this is just the usual way to construct an element of $C$ by case analysis, doing one thing if $U$ is true and another if it is false.

This suggests that we might regard internal gluing as a “generalized sort of case analysis” which applies even to non-decidable propositions. Instead of ordinary case analysis, where we have to do two things:

• assuming $U$, construct an element of $C$; and
• assuming not $U$, construct an element of $C$

in the non-decidable case we have to do three things:

• assuming $U$, construct an element of $C$;
• construct an element of the join $U*C$; and
• check that the two constructions agree in $U*(U\to C)$.

I have no idea whether this sort of generalized case analysis is useful for anything. I kind of suspect it isn’t, since otherwise people would have discovered it, and be using it, and I would have heard about it. But you never know, maybe it has some application. In any case, I find it a neat way to think about gluing.

Let me end with a tantalizing remark (at least, tantalizing to me). People who calculate things in algebraic topology like to work by “localizing” or “completing” their topological spaces at primes, since it makes lots of things simpler. Then they have to try to put this “prime-by-prime” information back together into information about the original space. One important class of tools for this “putting back together” is called fracture theorems. A simple fracture theorem says that if $X$ is a $p$-local space (meaning that all primes other than $p$ are inverted) and some technical conditions hold, then there is a pullback square:

$$\begin{array}{ccc}X& \to & X_p^{\wedge }\\ \downarrow & & \downarrow \\ X_{\mathbb{Q}}& \to & (X_p^{\wedge }{)}_{\mathbb{Q}}\end{array}$$

where $(-{)}_p^{\wedge }$ denotes $p$-completion and $(-{)}_{\mathbb{Q}}$ denotes “rationalization” (inverting all primes). A similar theorem applies to any space $X$ (with technical conditions), yielding a pullback square

$$\begin{array}{ccc}X& \to & \prod _p{X}_{(p)}\\ \downarrow & & \downarrow \\ X_{\mathbb{Q}}& \to & (\prod _p{X}_{(p)}{)}_{\mathbb{Q}}\end{array}$$

where $(-{)}_{(p)}$ denotes localization at $p$.

Clearly, there is a formal resemblance to the pullback square involved in the gluing theorem. At this point I feel like I should be saying something about $\mathrm{Spec}(\mathbb{Z})$. Unfortunately, I don’t know what to say! Maybe some passing expert will enlighten us.

The following paper by Jonathan Heckman of Harvard is either wrong, or trivial, or revolutionary:

Statistical Inference and String Theory

I don't understand it so far but Jonathan claims that one may derive the equations of general relativity – and, in fact, the equations of string theory – from something as general as Bayesian inference by a collective of agents.




It sounds really bizarre because the Bayesian inference seems to be a totally generic framework that may be applied anywhere and that says nothing else about "what the theories should look like" while general relativity and string theory are completely rigid, specific, well-defined theories. How could they be equivalent?




Jonathan considers a collective of agents who are ordered along a \(d\)-dimensional grid. Each of them tries to reconstruct the probabilistic distribution for events that they observe experimentally. Collectively, these distributions define an embedding of a manifold in another manifold and Jonathan rather quickly states that various conditional probabilities we know from the Bayesian inference may be written as the Feynman path integrals with the actions that include \(\sqrt{\det G}\), \(\sqrt{\det h}\), and similar things!

Again, I don't understand it so far but needless to say, a proof that string theory is the same thing as rational thinking – and not just a subset of rational thinking – would be extraordinarily important. ;-) I will keep on reading it.

 

The summer before last, I invited Brendan Fong to Singapore to work with me on my new ‘network theory’ project. He quickly came up with a nice new proof of a result about mathematical chemistry. We blogged about it, and I added it to my book, but then he became a grad student at Oxford and got distracted by other kinds of networks—namely, Bayesian networks.

So, we’ve just now finally written up this result as a self-contained paper:

• John Baez and Brendan Fong, Quantum techniques for studying equilibria in chemical reaction networks.

Check it out and let us know if you spot mistakes or stuff that’s not clear!

The idea, in brief, is to use math from quantum field theory to give a somewhat new proof of the Anderson–Craciun–Kurtz theorem.

This remarkable result says that in many cases, we can start with an equilibrium solution of the ‘rate equation’ which describes the behavior of chemical reactions in a deterministic way in the limit of a large numbers of molecules, and get an equilibrium solution of the ‘master equation’ which describes chemical reactions probabilistically for any number of molecules.

The trick, in our approach, is to start with a chemical reaction network, which is something like this:

and use it to write down a Hamiltonian describing the time evolution of the probability that you have various numbers of each kind of molecule: A, B, C, D, E, … Using ideas from quantum mechanics, we can write this Hamiltonian in terms of annihilation and creation operators—even though our problem involves probability theory, not quantum mechanics! Then we can write down the equilibrium solution as a ‘coherent state’. In quantum mechanics, that’s a quantum state that approximates a classical one as well as possible.

All this is part of a larger plan to take tricks from quantum mechanics and apply them to ‘stochastic mechanics’, simply by working with real numbers representing probabilities instead of complex numbers representing amplitudes!

I should add that Brendan’s work on Bayesian networks is also very cool, and I plan to talk about it here and even work it into the grand network theory project I have in mind. But this may take quite a long time, so for now you should read his paper:

• Brendan Fong, Causal theories: a categorical perspective on Bayesian networks.