Combinatorial physics

This is to announce a new academic discipline, combinatorial physics. This week I have been at a workshop on the subject (more on this later), and there are several conferences scheduled (one in France, one just approved at the Galileo Galilei Instite, one coming up at AIMS), as well as a journal, the Annales de l’Institut Henri Poincaré D, subtitled “Combinatorics, Physics and their Interactions”, which will start publishing next year.

What’s it all about? I wondered as I travelled to Cardiff for the meeting. There seem to be lots of potential connections. For one thing, several of the competing theories of quantum gravity stress discreteness, and some (such as loop quantum gravity and causal set theory) are explicitly founded on discrete structures: the philosophy is that the universe is discrete at some very small scale. In addition, Hopf bifurcations and catastrophe theory show how, even in a continuous universe, discrete effects can arise naturally from continuous causes. These are things about which I have almost no detailed knowledge.

When I arrived, I found people talking about things which were very familiar to me (permutation patterns, alternating sign matrices, species) as well as others which were not so familiar but still on my radar (lattice paths, non-crossing matchings). However, when the conference began, it was clear that the largest part of it was really just statistical mechanics in disguise; a somewhat wasted opportunity? I think I only heard the words “Tutte polynomial” once, in Mireille Bousquet-Mélou’s talk.

Like any conference, there was good and not-so-good; and specific to this one, some of the talks by physicists drove me crazy. But there were a few excellent talks. Stephen Tate talked about the interaction of species and statistical mechanics; I had heard him speak for an hour on this at Queen Mary, but even in a twenty minute talk he was beautifully clear and has some nice results. Mireille Bousquet-Mélou gave, as she always does, a lovely talk, on counting forests in planar maps; and Mark Dukes introduced us to the world of webs, a concept coming from QCD but purely combinatorial in nature, linking graphs, posets and permutations, with lots of good problems.

But for me the best talk was by Jessica Striker. Maybe I am biased; she endeared herself to me by starting off with one of my papers with Dima Fon-Der-Flaass, and actually naming an object which we didn’t bother to give a name to (the “toggle group”), but I feel that her talk deserves a post to itself, which it will have when I get round to it.

For the record, the AIHPD gives a list of specific subject areas which they cover:

  • Combinatorics of renormalization
  • Combinatorics of cluster, virial and related expansions
  • Discrete geometry and combinatorics of quantum gravity
  • Graph polynomials and statistical-mechanics models
  • Topological graph polynomials and quantum field theory
  • Physical applications of combinatorial Hopf algebras, matroids, combinatorial species, and other combinatorial structures
  • Exact solutions of statistical-mechanical models
  • Combinatorics and algebra of integrable systems
  • Computational complexity and its relation with statistical physics
  • Computational/algorithmic aspects of combinatorial physics
  • Interactions of combinatorial physics with topology, geometry, probability theory, or computer science

About Peter Cameron

I count all the things that need to be counted.
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2 Responses to Combinatorial physics

  1. Gordon Royle says:

    The list of topics has a significant overlap with “the mathematics of Dominic Welsh”!

    But please elaborate on what aspects of the talks by physicists drove you most crazy – excessive assumptions on audience knowledge of (strange) terminology?

    • Both of those, but what really drove me crazy was a speaker who put everything in the language of category theory (presumably to make it rigorous), then took [n] to be the set {1,…,n}, and later said “Let [n] and [m] be disjoint”.

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