The n-Category Café

Faster-than-light neutrinos? Boring… let’s see something really revolutionary.

Edward Nelson, a math professor at Princeton, is writing a book called Elements in which he claims to prove the inconsistency of Peano arithmetic.

It’s a long shot, but I can’t resist saying a bit about it.

guest post by John Huerta

Hi. Since this is my first ever post to the n-Café, let me introduce myself: I’m John Huerta, former student of John Baez, and now a postdoc at Australian National University. I was hired by Peter Bouwknegt, a string theorist who, like me, divides his time between the Departments of Mathematics and Theoretical Physics. In the six weeks that I’ve been here, I’ve already had a chance to see some of Australia’s amazing wildlife. I even got to feed kangaroos and wallabies! Here I am, offering some food to a wallaby:

I hope to see a lot more of Australia while I’m here! More seriously, I plan to do some work on T-duality and generalized geometry, some fascinating areas of mathematical physics on which Peter is an expert.

That’s enough about me. The real reason I am writing this post is to tell you about my first solo paper, which I just posted to the arXiv:

• John Huerta, Division Algebras and Supersymmetry III.

Abstract. Recent work applying higher gauge theory to the superstring has indicated the presence of ‘higher symmetry’. Infinitesimally, this is realized by a ‘Lie 2-superalgebra’ extending the Poincaré superalgebra in precisely the dimensions where the classical superstring makes sense: 3, 4, 6 and 10. In the previous paper in this series, we constructed this Lie 2-superalgebra using the normed division algebras. In this paper, we use an elegant geometric technique to integrate this Lie 2-superalgebra to a ‘Lie 2-supergroup’ extending the Poincaré supergroup in the same dimensions.

Briefly, a ‘Lie 2-superalgebra’ is a two-term chain complex with a bracket like a Lie superalgebra, but satisfying the Jacobi identity only up to chain homotopy. Simple examples of Lie 2-superalgebras arise from 3-cocycles on Lie superalgebras, and it is in this way that we constructed the Lie 2-superalgebra above. Because this 3-cocycle is supported on a nilpotent subalgebra, our geometric technique applies, and we obtain a Lie 2-supergroup integrating the Lie 2-superalgebra in the guise of a smooth 3-cocycle on the Poincaré supergroup.

I would love your comments on this paper. They would really help me to improve it! Below the fold, I’ll tell you what the paper is really about.

This October we have the next QVEST meeting

• Quarterly Seminar on Topology and Geometry

  7th of October, 2011

  Utrecht University

  seminar website

The speakers are

• Laurent Bartholdi (Göttingen)

  Growth and Poisson boundaries of groups

  Abstract: Let $G$ be a finitely generated group. A rich interplay between algebra and geometry arises by viewing G as a metric space, or as a metric measured space. I will describe two invariants of finitely generated groups, namely growth and Poisson boundary, and explain by new examples that their relationship is deep, but still mysterious.

  Its growth function, $\gamma(n)$, counts the number of group elements that can be written as a product of at most $n$ generators. This function depends on the choice of generators, but only mildly: say $\gamma$ is equivalent to $\delta$ if $\gamma(n) \leq \delta(C n) \leq \delta(C^2 n)$ for some positive $C$; then the equivalence class of $\gamma$ is independent of the choice of generators.

  For example, the growth of $\mathbb{Z}^d$ is asymptotic to $n^d$, while the growth of a free group is asymptotic to $2^n$. There are groups whose growth function is known to lie strictly between polynomials and exponentials; I will describe the first examples for which the asymptotic growth is known. I will also describe an example of a group of exponential growth, whose Poisson boundary is trivial for all finitely-supported random walks. Perhaps surprisingly, both examples come from the same general construction, permutational wreath products.

• Chenchang Zhu (Göttingen)

  Higher extensions of Lie algebroids, integration of Courant algebroids and string Lie 2-algebras

  Abstract: Recently, many efforts have been made to integrate a Courant algebroid, namely to find a global object associated to a Courant algebroid (For example, a global object corresponds to a Lie algebra is a Lie group). One of the reasons is probably that the standard Courant algebroid serves as the generalized tangent bundle of a generalized complex manifold of Hichin and Gualtieri. Thus the integration will help to understand the global symmetry of such manifolds. Our idea is that we first view an exact Courant algebroid as an extension of the tangent bundle by its coadjoint representation (up to homotopy) a la Abad-Crainic, then we perform the integration by the usual method of integration of an extension. We find that such higher extensions of Lie algebroids also include the example of string Lie 2-algebras.

• Chris Rogers (Göttingen)

  Higher Symplectic Geometry and Geometric Quantization

  Higher analogues of algebraic and geometric structures studied in symplectic geometry naturally arise on manifolds equipped with a closed non-degenerate form of degree greater than or equal to 2. In this talk, I will first explain how such a manifold gives an L-∞ algebra of “Hamiltonian” differential forms, just as a symplectic manifold gives a Poisson algebra of functions. I will then describe how to prequantize these manifolds and, within this context, sketch the relationship between the $L_\infty$ algebra of Hamiltonian forms and the $L_\infty$ structure on a Courant algebroid. Finally, I’ll discuss generalizations of real polarizations, and describe how twisted vector bundles play the role of the “quantum states” in higher geometric quantization.

If you would like to attend but have questions on anything, please contact me by email.

There is much that is odd about motivic measure if it is judged by measure theory in the sense of twentieth century analysis […] The first peculiarity is that the measure is not real-valued.

—Thomas Hales, What is motivic measure?, Bulletin of the AMS 42 (2005), 119–135.

This post isn’t about motivic measure, though you should definitely take a look at Hales’s excellent article (especially to find out what the second peculiarity is). This post will, however, share something of the spirit of motivic measure, including a flexible attitude towards where measure takes its values.

Suppose that we have some set, and a collection of “subsets” that we want to be able to “measure”. I’ll keep this very vague and general for the moment, though when I make it precise it genuinely will be quite general. The word “measure” isn’t used here with its standard meaning: we’re just assigning a “quantity” to each of our sets in some plausible way.

The crucial point is that whatever “quantity” means, it needn’t mean “real number”. And all I ask of a “measure” $\phi$ is that it satisfies the inclusion-exclusion principle: $$\phi(A_1 \cup \cdots \cup A_n) = \sum_i \phi(A_i) - \sum_{i \lt j} \phi(A_i \cap A_j) + \cdots$$ whenever $n \geq 0$ and $A_1, \ldots, A_n$ are sets for which this makes sense.

What is the universal way to measure a collection of sets?

Strange. No sooner had I set down ‘my friend Colin McLarty’ in the previous post, than I find him repaying the compliment here. Being an erudite man, however, he prefers to do so in Latin:

There is no category theory today without equality. And I don’t think there ever will be. Amicus <David> Corfield sed magis amica veritas. (p. 39)

This is a variant of Amicus Plato sed magis amica veritas (Plato is my friend, but truth is a better friend).

No detail is given there of the need for equality, but in his paper ‘There is no ontology here’, Chap 14 of The Philosophy of Mathematical Practice (preprint), Colin writes

Gelfand and Manin overstate an important insight when they call isomorphism ‘useless’ compared to equivalence:

“Contrary to expectations [isomorphism of categories] appears to be more or less useless, the main reason being that neither of the requirements $G F = 1_C$ and $F G = 1_D$ is realistic. When we apply two natural constructions to an object, the most we can ask for is to get a new object which is canonically isomorphic to the old one; it would be too much to hope for the new object to be identical to the old one.” (Gelfand and Manin, 1996, p. 71)

This is actually not true even in Gelfand and Manin’s book. Their central construction is the derived category $D(A)$ of any Abelian category $A$. Given $A$, they define $D(A)$ up to a unique isomorphism (1996, §III.2). They use the uniqueness up to isomorphism repeatedly. The notion of isomorphic categories remains central. Yet for many purposes equivalence is enough.

Gelfand and Manin’s book is Methods of Homological Algebra.

Can it really be that the derived category construction is usefully defined up to isomorphism?

I’m just back from the Belgian city of Ghent, where I heard from Catarina Dutilh Novaes that earlier this year there was quite a storm on the FOM (Foundations of Mathematics) mailing list over comments by Voevodsky suggesting that he didn’t understand Gödel’s incompleteness results. I gave up reading FOM several years ago after the tone of the attacks on my friend Colin McLarty, who was defending category theoretic foundations, reached a particular low. I hear that generally it’s a friendlier place now.

You can read about the Voevodsky case at Caterina’s blog – M-Phi – here and the two follow-up posts. She sums up in the third of these posts:

So it would seem that the main conclusion to be drawn from these debates, and as pointed out by Steve Awodey in one of his messages to FOM, is that the consistency of PA is really a minor, secondary issue within a much broader and much more ambitious new approach to the foundations of mathematics.

Yes, better to spend your time learning about Univalent Foundations, I’m sure. And a very good place to start is Mike Shulman’s series of six posts on Homotopy Type Theory listed here.

All over the world, 2013 will be a special year for the Mathematics of Planet Earth. At Banff they’re inviting mathematicians to organize workshops on this theme, and the deadline is September 30th. I want to apply.

Since I’m starting to apply category-theoretic ideas to complex networked systems found in mathematical biology, ecology and climate physics, I’ll probably do an application on “network theory”. If you’re interested, let me know.

For more details, including more about what I mean by “network theory”, go here.

I’ve started using Google+ to share fun tidbits of information about math and physics — the kind of stuff I used to put near the start of each issue of This Week’s Finds. It’s quick and easy.

There are lots of cool people on Google+, sharing all sorts of information. It’s become one of my main sources of news.

Currently Google+ works by invitation only. If I know you and like you, I’ll be glad to send you an invitation. (Please only ask if we’ve had some pleasant encounters in cyberspace, and not many unpleasant ones.)

And if you want, I can also add you to my “Mathematicians”, “Physicists” or “Azimuth” circles. If I do that — and only if I do that — you’ll see certain more technical items that I don’t want to inflict on the uninterested masses.

Here are some samples of the stuff I’m talking about…