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July 31, 2007
Web Spamming by Academic Publishers
Posted by John Baez
A recent email from Carl Willis mentions a practice that’s been annoying me lately: a particular form of ‘web spamming’ by academic publishers, sometimes called ‘cloaking’. The publishing company gives search engine crawlers access to full-text articles — but when you try to read these articles, typically clicking on a link to a PDF file, you get a ‘doorway page’ demanding a subscription or payment.
Sometimes you’ll even be taken to a page that has nothing to do with the paper you thought you were about to see! That’s what infuriates me the most. I don’t expect free articles from these guys, but it would at least be nice to see basic bibliographical information.
Culprits include Springer, Reed Elsevier, and the Institute of Electrical and Electronic Engineers. The last one seems to have quit — but to see why they did it, check out their powerpoint presentation on this subject, courtesy of Carl Willis.
Higher Gauge Theory and Elliptic Cohomology
Posted by John Baez
After some fun in Greece, I’ve been holed up in Greenwich the last two days preparing my talk for the 2007 Abel Symposium. This is an annual get-together sponsored by the folks who put out the Abel prize, a belated attempt to create something like a Nobel prize for mathematicians.
One of the themes of this year’s symposium is “elliptic objects and quantum field theory”. So, while my true love is higher gauge theory, my talk will emphasize its relation to elliptic cohomology and related areas of math:
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John Baez, Higher Gauge Theory and Elliptic Cohomology.
Abstract: The concept of elliptic object suggests a relation between elliptic cohomology and "higher gauge theory", a generalization of gauge theory describing the parallel transport of strings. In higher gauge theory, we categorify familiar notions from gauge theory and consider "principal 2-bundles" with a given "structure 2-group". These are a slight generalization of nonabelian gerbes. After a quick introduction to these ideas, we focus on the 2-groups associated to any compact simple Lie group . We describe how these 2-groups are built using central extensions of the loop group and how the classifying space for -2-bundles is related to the "string group" familiar in elliptic cohomology. If there is time, we shall also describe a vector 2-bundle canonically associated to any principal 2-bundle, and how this relates to the von Neumann algebra construction of Stolz and Teichner.
July 28, 2007
Algebra 1 versus Algebra 2
Posted by David Corfield
In Delphi, Colin McLarty performed some myth-busting for us. Many of you will have heard of Paul Gordan’s supposed reaction to a result of David Hilbert in the theory of invariants:
This is not Mathematics, it is Theology!
Often this is taken as one of the reactionary old guard standing in the way of the new algebra. However, Colin does a great job explaining how the true story is far more subtle.
Hilbert in 1888 said he found his proof “with the stimulating help of” this very professor Gordan.
Rather than recapping his argument, we may as wait until it appears. Here I want to know more about what happens next. In particular, I’d like to know whether Gian-Carlo Rota’s distinction between Algebra 1 and Algebra 2 holds water. He does this somewhere in English, Chapter III of Indiscrete Thoughts I believe.
Online, all I can find is in Italian. Here Rota picks out key figures in each:
Algebra 1: algebraic geometry and algebraic number theory, represented by Kronecker, Hilbert, Weil, …
Algebra 2: ‘Combinatoria Algebrica’ - algebraic combinatorics, represented by Boole, Capelli, Young, Gordan, Hall, Birkhoff, …
Does this chime with anyone?
July 27, 2007
Arrow-Theoretic Differential Theory
Posted by Urs Schreiber
Using the concept of tangent categories (derived from that of supercategories) I had indicated how to refine my previous discussion of -curvature. Here are more details.
Arrow-theoretic differential theory
Abstract: We propose and study a notion of a tangent -bundle to an arbitrary -category. Despite its simplicity, this notion turns out to be useful, as we shall indicate.
1 Introduction … 1
2 Main results … 2
2.1 Tangent -bundle … 3
2.2 Vector fields and Lie derivatives … 4
2.3 Inner automorphism n-groups … 4
2.4 Curvature and Bianchi Identity for functors … 5
2.4.1 General functors … 5
2.4.2 Parallel transport functors and differential forms … 6
2.5 Sections and covariant derivatives … 6
3 Differential arrow theory … 8
3.1 Tangent categories … 8
3.2 Differentials of functors … 11
4 Parallel transport functors and their curvature … 12
4.1 Principal parallel transport … 12
4.1.1 Trivial G-bundles with connection … 12
Introduction
Various applications of (-)categories in quantum field theory indicate that (-)categories play an important role over and above their more traditional role as mere organizing principles of the mathematical structures used to describe the world: they appear instead themselves as the very models of this world.
For instance there are various indications that thinking of configuration spaces and of physical processes taking place in these as categories, with the configurations forming the objects and the processes the morphisms, is a step of considerably deeper relevance than the tautological construction it arises from seems to indicate.
While evidence for this is visible for the attentive eye in various modern mathematical approaches to aspects of quantum field theory – for instance [FreedQuinn], [Freed] but also [Willerton] – the development of this observation is clearly impeded by the lack of understanding of its formal underpinnings.
If we ought to think of configuration spaces as categories, what does that imply for our formulation of physics involving these configuration spaces? In particular: how do the morphisms, which we introduce when refining traditional spaces from 0-categories to 1-categories, relate to existing concepts that must surely secretly encode the information contained in these morphisms. Like tangent spaces for instance.
Possibly one of the first places where this question was at all realized as such is [Isham]. That this is a piece of work which certainly most physicists currently won’t recognize as physics, while mathematicians might not recognize it as interesting mathematics, we take as further indication for the need of a refined formal analysis of the problem at hand.
Several of the things we shall have to say here may be regarded as an attempt to strictly think the approach indicated in Isham’s work to its end. Our particular goal here is to indicate how we may indeed naturally, generally and usefully relate morphisms in a category to the wider concept of tangency.
For instance his “arrow fields” on categories we identify as categorical tangents to identity functors on categories and find their relation to ordinary vector fields as well as to Lie derivatives, thereby, by the nature of arrow-theory, generalizing the latter concepts to essentially arbitrary categorical contexts.
While there is, for reasons mentioned, no real body of literature yet, which we could point the reader to, on the concrete question we are aiming at, the reader can find information on the way of thinking involved here most notably in the work of John Baez, the spiritus rector of the idea of extracting the appearance of -categories as the right model for the notion of state and process in physics. In particular the text [BaezLauda] as well as the lecture notes [Baez] should serve as good background reading.
The work that our particular developments here have grown out is described in [S1, S2]. Our discussion of the Bianchi identity for -functors should be compared with the similar but different constructions in the world of -fold categories given in [Kock].
July 26, 2007
Homotopy Theory and Higher Categories in Barcelona
Posted by John Baez
In Barcelona there will be a year-long program on homotopy theory and higher categories:
- Homotopy Theory and Higher Categories, at the Centre de Recerca Matemàtica, Barcelona, Catalonia, Spain, 2007-2008 academic year. Organized by Carles Casacuberta (Barcelona), André Joyal (Montreal), Joachim Kock (Barcelona), Amnon Neeman (Canberra) and Frank Neumann (Leicester).
July 25, 2007
Delphic Inspiration
Posted by David Corfield
I’ve returned from the sun of Delphi to the sogginess of England. John has already put up some pictures and a description of the event – Mathematics and Narrative – in his diary. I think the very best part of the meeting was the decision to have each participant be interviewed by another. The suggested length of two to three hours for this process seemed daunting, but it allowed a kind of conversation I’ve never known before. And to have two and a half hours of Barry Mazur’s undivided attention!
When it came to my turn to be interviewed, by my philosopher friend Colin McLarty, I began to see that Alasdair MacIntyre’s notion of a rational tradition of enquiry could be made to do some real work. We get rather used in the humanities to fairly loose schematic descriptions of phenomena, unlike in the hard sciences where predicted entities (such as categorified constructions) had better be found if we are not to lose faith. From the interview, we got the sense that this framework could point us easily to the difficulties other approaches face, and then explain them.
Perhaps we’ll see the Delphi meeting as one of those defining moments in getting a non-relativist practice-oriented philosophy of mathematics off the ground. Elsewhere, I interviewed a third member of this movement, Brendan Larvor, for the fourth edition of The Reasoner.
Question About Representations of Finite Groups
Posted by John Baez
Here’s the first of some questions that have been bugging me. Maybe you can help!
I want to know when we can define the representations of a finite group using not the full force of the complex numbers, but only some subfield, like or . If I knew the answer to this question, it might be important for the groupoidification program, where we’re trying to replace complex vector spaces by groupoids whenever possible.
Suppose is some subfield of the complex numbers. In what follows, ‘representation’ will mean representation on a finite-dimensional complex vector space. Suppose is some group with a representation . Let’s say is defined over if we can find some basis of our vector space such that the matrices corresponding to the linear transformations all have entries lying in .
Question 1. Is there a smallest subfield such that every representation of every finite group is definable over ? If so, what is it?
It’s not hard to see that:
- Every representation of every finite group is definable over when is the field of algebraic numbers.
- Not every representation of every finite group is definable over when . There’s an easy trick to see which ones are.
- Every representation of the symmetric group is definable over when .
- Every representation of the cyclic group is definable over when is the cyclotomic field generated by taking and throwing in a primitive th root of unity.
But what I really want to know is this:
Question 2. Is every representation of every finite group definable over when is the field generated by taking and throwing in all roots of unity? If not, what’s the simplest counterexample?
July 23, 2007
Physical Systems as Topoi, Part III
Posted by Urs Schreiber
Yesterday night I had to interrupt my transcript of Andreas Döring’s talk. Here is the continuation.
July 22, 2007
Physical Systems as Topoi, Part II
Posted by Urs Schreiber
The continuation of my transcript of Andreas Döring’s talk.
Physical Systems as Topoi, Part I
Posted by Urs Schreiber
Here at the -Café, we already had pretty detailed discussions (see A Topos Foundation for Theories of Physics and Topos Theory in the New Scientist) of Andreas Döring and Chris Isham’s recent work
A. Döring, C. Isham
A Topos Foundation for Theories of Physics
quant-ph/0703060.
I am particularly grateful to Squark, for walking us through many of the essential details. For a quick summary, see this comment.
Now today, at Recent Developments in QFT in Leipzig, I was lucky enough to hear Andreas Döring himself give a talk on this work.
Here I’ll reproduce my transcript of what he said.
Making AdS/CFT Precise
Posted by Urs Schreiber
The last session of Recent Developments in QFT in Leipzig was general discussion, which happened to be quite interesting for various reasons.
Deformation Quantization of Surjective Submersions
Posted by Urs Schreiber
I am in Leipzig, attending the last day of Recent Developments in QFT in Leipzig.
First talk this morning was by Stefan Waldman on Deformation quantization of surjective submersions.
What he described is this:
Suppse over some manifold we have a surjective submersion In applications we will want to restrict to the case that is some principal -bundle. But much of the following is independent of that restriction.
Next, suppose that we want to consider (noncommutative) deformations of the algebra of functions on . This is a popular desideratum in many approaches of quantum field theory that try to go beyond the standard model and classical gravity.
Stefan Waldman and his collaborators ask: given such a deformation of base space, what are suitably compatible deformations of the covering space over ?
The motivating idea is this:
There is that wide-spread idea in quantum field theory that theories which go beyond the energy scale currently observable by experimental means will involve noncommutative deformations of the algebra of functions on spacetime. But since physical fields are usually not just functions on spacetime, but either sections of some vector bundles, or connections on these, the question arises in which sense these bundles then have to be deformed, too.
Therefore Waldman is looking for deformations of the algebra of functions on the total space of the bundle which are, in some sense to be determined, compatible with the deformation downstairs.
The first idea is to find deformations such that the pullback extends to an algebra homomorphism of the deformed algebras But they show that in many physically interesting cases, like the Hopf fibration which in physics corresponds to the “Dirac monopole”, there are obstructions for such an extension to exist at all. Hence they reject this idea.
The next idea is to realize just as a bimodule for . But this, too, turns out to be too restrictive.
Finally, they settle for requiring that induces just a one-sided -module structure on