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January 31, 2007
Quantization and Cohomology (Week 12)
Posted by John Baez
This week’s class on Quantization and Cohomology introduced the theme of ‘rigs’ (rings without negatives), foreshadowed last week:
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Week 12 (Jan. 30) - Classical, quantum and statistical mechanics as ‘matrix mechanics’. In
quantum mechanics we use linear algebra over the ring ; in classical mechanics everything is formally the same, but we instead use the rig , where the addition is min and the multiplication is +. As a warmup for bringing statistical mechanics into the picture - and linear algebra over yet another rig - we recall how the dynamics of particles becomes the statics of strings after Wick rotation.
- Supplementary reading: Grigori L. Litvinov, The Maslov dequantization, idempotent and tropical mathematics: a very brief introduction. Also see the longer version here.
January 29, 2007
No Need to Apologise
Posted by David Corfield
Café regular John Armstrong has a blog. It goes by the name of The Unapologetic Mathematician. A subtle allusion to Hardy’s A Mathematician’s Apology, playing cleverly on the two meanings of apology?
CFT in Oberwolfach
Posted by Urs Schreiber
There will be an Arbeitsgemeinschaft (study group) in Oberwolfach, on Algebraic Structures in Conformal Field Theories, April 1 - April 7, 2007.
You can find the program and the application details here.
January 28, 2007
Another Interview
Posted by David Corfield
It’s worth taking a look at an interview Mikio Sato gave to Emmanuel Andronikof in 1990, published in February’s Notices of the American Mathematical Society. Sato is famous for algebraic analysis, D-modules, and the like, about which I know next to nothing. Perhaps if Urs continues to post on Geometric Langlands we’ll hear something about D-modules, as they appear to very relevant. You won’t learn much mathematics from the interview, but it gives a fascinating account of an indirect path to becoming one of the world’s leading mathematicians.
Concerning future directions, this passage caught my eye:
While methods of mathematical physics in quantum field theory have profited various branches of mathematics (topology, braid theory, number theory, geometry), the converse is not necessarily true. Today [remember this is 1990 - DC], mathematical physicists mostly use number theory or algebraic geometry. Mathematical physics is receptive only to higher developed areas of mathematics, some of which are exploited in superstring theory, though not to its full extent. Mathematics has not succeeded in providing a more effective way of computation than perturbation expansions. Of course, there are some primitive methods of computation, like the Monte-Carlo method. All these are kind of brute force computations, not refined mathematics, surely not refined enough for the problems physics is now confronted with, like determining the mass of particles or quarks. All these things are discussed on a very abstract level, not on a quantitative level. So I think that mathematical analysis should be developed much further to match the reality of physics.
Also see Pierre Schapira’s description of Sato’s work in the same edition of the Notices.
January 27, 2007
Peering Through the Veil
Posted by David Corfield
Twice in recent days I have confronted the possibility of experiencing a kind of alienation due to interviews. First, my co-author Darian Leader and I were interviewed by the New Scientist about our book Why Do People Get Ill?. A day or two later we got to see a draft of what was to be selected for publication. Space limitations have meant that statements attributed to one of us are composites of things said by either of us. I don’t think it matters much in terms of the information carried in the interview, but it feels strange to have sentences you never uttered marked as originating from you.
No such chopping in that other recent interview, the one Urs and John gave to Bruce Bartlett about this blog. Even hesitation and laughter have been carefully marked. Here the potential alienation arises from the possibility of being spoken about in a way which clashes with one’s self-image. Of course nothing like this happened, but I would like to take the opportunity to say something about John’s comment about me that when
he’s talking about the philosophy of mathematics, he’s very concerned about the sociology of mathematics, and how people interact, and how you can do mathematics well.
Now ‘sociology’ has a number of uses. On the one hand, it can be taken as a non-normative discipline which seeks to understand and describe how societies operate. Although there may be some or other philosophical stance operating behind the scenes, this activity would seem not to be philosophical as it stands. On the other hand, ‘sociology’ as applied to the study of science and mathematics, as in the ‘sociology of scientific knowledge’, tends to come with a strong dose of social constructivism, and a wish to unmask the resources and techniques of the powerful to represent the way things are. In this context the study of ‘norms’ is largely to understand how the powerful wield certain standards to maintain their position of prominence. We had a discussion about that stance starting back here.
January 26, 2007
Globular Extended QFT of the Charged n-Particle: String on BG
Posted by Urs Schreiber
To begin filling the definition of the charged quantum -particle with life, here I walk through a very simple but still interesting example: “a string on the classifying space of a 2-group ”.
This turns out to be a “globular extened quantum field theory” , which to a “point” of the shape assigns a 2-vector space of states namely the category of modules over the algebra of a certain groupoid – the loop groupoid of the 2-group – and which to a “string” of the shape assigns the 2-linear identity map on this 2-vector space
For the special case that the 2-group is the “string group of a compact, simple and simply connected Lie group” the 2-space of states over the point is the 2-space of -equivariant gerbe modules on , also known as certain D-branes on .
A (2-)state of the string on is a 2-linear 2-map which is nothing but a gerbe module/D-brane over , together with an automorphism
If we close the string by gluing its endpoints by means of a trace, we find that a state of the closed string is a function on connected components of
January 25, 2007
Classical vs Quantum Computation (Week 11)
Posted by John Baez
Today in our course on Classical vs Quantum Computation we covered lots of examples of 2-categories, to show how widespread these gadgets are:
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Week 11 (Jan. 25) - Examples of 2-categories. The 2-category of categories. The fundamental 2-groupoid of a topological space. The 2-category of topological spaces, maps, and homotopies between maps. The 2-category of topological spaces, maps, and homotopies between maps. The 2-category implicit in extended topological quantum field theories, due to Jeffrey Morton. The 2-category implicit in string theory, due to Stolz and Teichner. Monoidal categories as one-object 2-categories. The 2-category of rings,
bimodules and bimodule homomorphisms. Monoidal categories as one-object 2-categories. The 2-category of rings, bimodules and bimodule homomorphisms.
Supplementary reading:- Jeffrey Morton, A double bicategory of cobordisms with corners.
- Stefan Stolz and Peter Teichner, What is an elliptic object? Section 4.2: the bicategory of conformal 0-, 1- and 2-manifolds.
January 24, 2007
Classical vs Quantum Computation (Week 10)
Posted by John Baez
This quarter in our course on Classical vs Quantum Computation, our goal is to repair a gaping hole in the usual application of category theory to computation — especially the lambda-calculus and its quantum generalizations. We want to be able to talk about the process of computation! For this, we need to get serious about 2-categories…
- Week 10 (Jan. 18) - Categorifying the concept of ‘category’ to get the concept of ‘2-category’ - in detail.
The Globular Extended QFT of the Charged n-Particle: Definition
Posted by Urs Schreiber
After thinking about it for a while (A B C D E F G H I J) it seems that I am finally at a point where I can venture to state a comprehensive formal definition of the structure whose working title was the charged quantum -particle.
The following definition is taken from the beginning of
The Globular Extended QFT of the String propagating on the Classifying Space of a strict 2-Group
which develops one of simplest interesting examples in more detail (to be discussed in a followup post).
The two definitions, discussed in detail below, roughly go like this:
Definition 1. A charged -particle is a setup internal to .
Definition 2. The quantization of a charged -particle is the -functor on obtained by pull-pushing through the correspondence
Cocycle Category
Posted by Urs Schreiber
Here is another guest post by Bruce Bartlett.
Luckily, Bruce is still at Fields in Toronto, attending the Thematic Program on Geometric Applications of Homotopy Theory.
Here he reports on something very interesting that is intimately related to our discussion of anafunctors.
Quantization and Cohomology (Week 11)
Posted by John Baez
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Week 11 (Jan. 23) - Action as a functor from a category of “configurations” and “paths” to the real numbers (viewed as a one-object category). Three things physicists do with this functor: find its critical points, find its minima, and integrate its exponential. The analogy between the (classical) principle of least action and the (quantum) principle of path integration. The underlying analogy between the real numbers equipped the operations min and +, and the complex numbers with operations + and ×.
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Supplementary reading: The Hamilton-Jacobi equation.
- Homework on another way to see Action as a functor.
- answers by Jeffrey Morton.
- answers by Toby Bartels.
- answers by Miguel Carrión Álvarez.
- answers by Derek Wise.
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Supplementary reading: The Hamilton-Jacobi equation.