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December 31, 2006
Puzzle #11
Posted by John Baez
Happy New Year! Here’s a little puzzle to start the year off…
When was the toothpick invented?
December 26, 2006
This Week’s Finds in Mathematical Physics (Week 243)
Posted by John Baez
In week243 of This Week’s Finds, hear about Claude Shannon, his sidekick Kelly, and how they used information theory to make money at casinos and the stock market. Hear about the new book Fearless Symmetry, which explains fancy number theory to ordinary mortals. Learn about the Dark Ages of our Universe, and how they were ended by the earliest stars. And finally, get a taste of Derek Wise’s work on Cartan geometry, gravity… and hamsters!
December 25, 2006
The Earliest Stars?
Posted by John Baez
You may have read about this in the news, but here are the original articles:
- A. Kashlinsky, R. G. Arendt, J. Mather and S. H. Moseley, New measurements of cosmic infrared background fluctuations from early epochs, to appear in Ap. J. Letters.
- A. Kashlinsky, R. G. Arendt, J. Mather and S. H. Moseley, On the nature of the sources of the cosmic infrared background, to appear in Ap. J. Letters.
Executive summary: using delicate techniques to carefully sift through the infrared background radiation, the authors claim to find radiation not accounted for by previously known sources. Assuming the now-standard ΛCDM cosmology, the sources of this radiation date back to less than 1 billion years after the Big Bang, and were individually much brighter than current-day stars.
December 24, 2006
arXiv Policy Statement?
Posted by John Baez
A while back David Corfield raised some important issues about the academic commons. Here’s a practical question along those lines:
Does the arXiv have an official policy statement someplace where they promise to keep papers there freely available? They should. I haven’t been able to find one! Did I miss it?
Marco Grandis raised this issue in a post to the category theory mailing list.
Puzzle #9
Posted by John Baez
Which famous philosopher is also known as the RaMBaM?
December 21, 2006
Conference: Lie Algebroids and Lie Groupoids in Differential Geometry
Posted by Urs Schreiber
In October 2007 there will be a four-day conference in Sheffield on Lie algebroids and Lie groupoids in Differential Geometry, organized by Kirill Mackenzie and Ieke Moerdijk.
Lectures on Classical Mechanics
Posted by John Baez
In the Spring of 2005 I taught a graduate course on classical mechanics, and Derek Wise took beautifully precise notes. I started with the Lagrangian approach, with a heavy emphasis on action principles, and derived the Hamiltonian approach from that.
Now, as a Christmas present to the world, Blair Smith has converted them into a beautiful typeset PDF document, adding extra material:
- John Baez, Blair Smith and Derek Wise, Lectures on Classical Mechanics. Also available in Postscript format.
Enjoy! And, please report any typos or other errors that you find!
A Little Bit of Geometric Langlands: Relation to Integrable Systems
Posted by Urs Schreiber
As I mentioned in Navigating Geometric Langlands by Analogies, the mathematicians and the string theorists in Hamburg have a a small series of lectures this term, where we try to explain to each other some tiny fraction of what Langlands duality is about.
Last time we had something on the “classical” number-theoretic aspect. I didn’t even try to report on that.
This time, Jörg Teschner spoke about geometric Langlands duality. After briefly mentioning what the main statement is, he concentrated on understanding one aspect of this statement using the language of integrable systems.
Ever since it was found that certain aspects of super Yang-Mills theory are governed by what are called quantum integrable systems, and due to the impact this has on understanding and testing the AdS/CFT duality, string theorists have been interested in integrable systems. That’s one reason why Jörg Teschner decided to emphasize this aspect of the talk.
Roughly, the main point he tried to make could maybe be summarized like this:
The geometric Langlands duality relates (conjecturally) two different derived categories by an equivalence.
But on top of that, the conjecture states that both these derived categories have something like a nice “basis of eigenstates of some operator” (compare my previous entry # on what such a statement would really mean) and that under the equivalence a basis vector on one side is sent to a basis vector on the other.
Now, on one of these two sides, those “basis vectors” (the Hecke eigensheaves) can be understood as coming from common eigenstates of the set of commuting Hamiltonians of some integrable system.
From this point of view of integrable systems, geometric Langlands duality seems to be a statement about when and how an integrable system admits a separation of variables.
If you like integrable systems, that should be interesting.
Common Applications
Posted by David Corfield
I’ve been reading some of Jorg Lemm’s papers in recent days. He’s written a book - Bayesian Field Theory - which I don’t have access to, but he had written a paper of the same name earlier. In it (page 6, note 1) he remarks that:
statistical field theories, which encompass quantum mechanics and quantum field theory in their Euclidean formulation, are technically similar to a nonparametric Bayesian approach.
It is intriguing to see so many constructions of mathematical physics - mean field methods, diffusion models, free energy - find a use in learning theory. But what to make of it? If we think it needs an explanation at all, we might say that perhaps it’s telling us that we only have a limited number of tools, so should expect to use them time and again. If we were washed up on a desert island with just a knife in our pocket, we’d find a host of uses for it, with little in common between them, e.g., opening a clam and sharpening a stick.
David Ruelle favoured this kind of explanation about multiple application in “Is our mathematics natural? The case of equilibrium statistical mechanics.” Bull. Amer. Math. Soc. 19, 259-268 (1988). Our minds have a limited repertoire, which explains why mathematicians keep bumping into the same constructions. Closer to this blog, a similar question is why the deeper reaches of number theory (Langlands programme) and quantum field theory (duality) are so closely related. In Mathematics in the 20th Century, Michael Atiyah’s predictions for the 21st century went thus:
What about the 21st century? I have said the 21st century might be the era of quantum mathematics or, if you like, of infinite dimensional mathematics. What could this mean? Quantum mathematics could mean, if we get that far, ‘understanding properly the analysis, geometry, topology, algebra of various non-linear function spaces’, and by ‘understanding properly’ I mean understanding it in such a way as to get quite rigorous proofs of all the beautiful things the physicists have been speculating about.
Quantization and Cohomology (Week 9)
Posted by John Baez
Here are the final notes from my Fall 2006 course on Quantization and Cohomology:
- Week 9 (Dec. 5) - A glimpse of what’s to come. Phases versus relative phases. Geometric quantization: finding a connection on a U(1) bundle whose curvature is the symplectic 2-form ω on phase space. Why doing this is only possible if ω defines an integral cohomology class - hence the term ‘quantization’.
Last week’s notes are here; the notes from next quarter begin here.
My colleague Apoorva Khare has produced a LaTeX version of the notes for the entire Fall quarter, and Christine Dantas has drawn figures for these notes. The notes still need more polishing, but they’re already very useful:
- John Baez and Apoorva Khare, with figures by Christine Dantas, Course Notes on Quantization and Cohomology, Fall 2006. Also available in Postscript.
December 20, 2006
Excellent Math in Bonn: Opening Colloquium in January
Posted by Urs Schreiber
After decades in which every potential sign of elitism was oficially frowned upon, the last German government had decided that it was quite a pity that every university here was and had to be equal, with none being more equal than the others. They decided it would be lots of fun to have some elite universities, like all those other countries do. So they decided to find them – by decree.
The result is that, as of recently, two Munich universities as well as that in Karlsruhe are officially elite universities™. In addition, a couple of institutes of several other universities, in various fields, have been declared to be clusters of excellence™.
In practice, what this really means is that those places with the new titles will get extra government money from now on. Everybody is excited about that.
Anyway, there is now precisely one “cluster of excellence” concerned with mathematics in Germany, and that’s in Bonn.
Hausdorff Center for Mathematics
As a cluster of excellence, it’s titled
Mathematics: Foundations, Models, Applications.
Luckily, that title is not too restrictive.
I am writing all this because you might be intersted in the cluster’s Opening Colloquium that takes place January 19th and 20th, 2007.
Plenary lectures will be given by
Douglas Arnold (Minneapolis)
Michael Hopkins (Harvard)
Elliott Lieb (Princeton)
László Lovász (Budapest)
Andreu Mas-Colell (Barcelona)
Wendelin Werner (Paris)
So it might be worth a visit.
Research Blogging
Posted by Urs Schreiber
Clifford Johnson ponders the advantages of research blogging in The Blog as a Sharp Tool for Research and now again in Research Blogging.
Related considerations were voiced by Craig Laughton early this year: Exploring the Blogosphere.
I don’t have much to add to that, except for noting that I used blogging in this sense, and almost exclusively in this sense, from the very beginning.
And I’d guess that, long long before my feeble writings, John’s TWFs served a similar purpose.
One big difference is that Clifford Johnson has a non-public blog for his research, which, as he writes #
[…] is also the place where everyone (including me) can say silly things and ask silly questions if we want to, without the whole world watching. That latter is a very important feature, in fact.
For some reason I have always felt like moving private discussions on technical issues out in the open. For me that’s a matter of increasing the reaction rate of research by increasing the reaction surface. And, looking back, it did work for me #.
With the esoteric stuff we are talking about, this seems more important to me than shielding away insights and hiding mistakes. The game here is not Bingo #.
But I am aware that most people feel quite different about this – and quite possibly for good reasons. But I cannot help it. On the other hand, I am fond of having found philosophical support # # from David Corfield.
While I cannot prove it, I think everybody would benefit from seeing more research-related communication done out in the open. The most valuable aspect of many conferences is the conversation one has in between the talks. And this kind of conversation is what I am after.
December 18, 2006
Universal Gerbes
Posted by Urs Schreiber
Jim Stasheff asked me to forward the following question to the -Café audience:
There is a universal principal bundle for any group. Is there also a universal gerbe?
This Week’s Finds in Mathematical Physics (Week 242)
Posted by John Baez
In week242 of This Week’s Finds, see some incredible photos of Saturn’s rings taken by the Cassini spacecraft:
Hear about some of the other exciting space missions NASA may cancel to pay for an expensive plan to send canned primates to Mars. See the Sun in neutrinos. And learn about Jeffrey Morton’s new approach to topological quantum field theory using a double bicategory of cobordisms with corners!
December 15, 2006
Frobenius Algebroids with Invertible Products
Posted by Urs Schreiber
Sometimes, banalities fool us into ignoring important structure. That happens when the banality is the degenerate case of something that is in general more interesting.
One banality is this.
An associative monoidoid (I mean any enriched category, please see the comments below)
with invertible product (this invertibility is the degeneracy)
is naturally a Frobenius monoidoid
only that you would tend to ignore this fact. It’s like ordering Pizza Tonno without Tuna.
December 14, 2006
Higher Categories at the Fields Institute
Posted by John Baez
Excitement is building throughout Toronto as news of this workshop spreads:
- Higher Categories and Their Applications, January 9-13, 2007, part of the Thematic Program in Geometric Applications of Homotopy Theory at the Fields Institute.
Crowds are lining up in front of the Fields Institute, trying to get seats early, wondering when the show will start. To prevent rioting and looting, I have decided to post a tentative schedule including abstracts of some of the talks.