14 thoughts on “Open Thread

  1. Curiously enough, I would characterize my Algebra prof as an uptight nerd and my Real Analysis prof as a dirty hippy (well, actually he’s cleaned up a bit, but he has photos on his webpage in which he is quite clearly a dirty hippy and he still wears the tie-die shirts from time to time).

  2. So. What are people working on at the moment?

    I’m currently dabbling in an eclectic mix of infinitary combinatorics and operator algebras. Not areas with much intersection at the moment (barring the occasional exception like Akemann and Weaver’s counterexample to Naimark’s problem), but I’m working on some things which might lead to a bit more. Probably not, but even if they don’t I’ll have learned a whole bunch of interesting mathematics.

  3. My undergrad real analysis professor (whose own research was then on multi-dimensional real spaces, R^{\alpha} where \alpha is any positive real number, not necessarily an integer) used sometimes to begin lectures with 5 minutes of silent meditation, in order for us to get in the right mood for theorem-proving. So, I’m not sure about this generalization. However, a famous category theorist once told me that he became a category theorist (in the 1970s) because most of the analysts he met at the time were very very macho, and only the category theorists left their egos at home. I know at least two other successful PhDs in analysis who switched careers afterwards (out of Pure Math entirely) because they could not stand the macho culture.

  4. Apropos of nothing:

    When I was a student I took a course in optimisation. In order to make it more entertaining the lecturer regularly took a short break at the lecture midpoint in order to tell us about how to deal with bears. Keep food outside your tent ‘n’ all that. His educational method was successful. 20 years later I can still remember much of his advice, and now that I no longer live in the UK there’s a chance that some of his advice might even be useful. Unfortunately I don’t remember anything about the simplex method.

  5. Uptight twits is what they are…convergence this and convergence that.
    We Algebraists know better than to rush all about some infinite dimensional Hilbert space. We know how to stop a smell the patchouli.

    For David:

    I am working on Clifford Algebras over Markov Chains, and I am ashamed…deeply ashamed. Probability, and PDE’s; two things I swore I would never touch. Well, one out of two ain’t bad. Still, not too far from computing cohomology of complex Grassmannians….at least that is what I tell myself…

  6. Hmm. Ok, I know what a Clifford algebra is, and I know what a Markov chain is, but I confess I haven’t a clue what a Clifford algebra over a Markov chain might be. What are they? (And where do they crop up?)

  7. Take your Markov chain with N states, and impose your favorite distance squared metric on the states of the chain. Use this to embed in everyone’s favorite manifold – R^N. Use your defined inner product and duality to impose an exterior product. Now hopefully you are only some short steps away from defining your favorite algebraic invarients on the the chain (perhaps by the route of embedding in the appropriate Grassmannian for example).

    This would be useful for collaborative filtering or automatic classification for a start. Probably more applicable as theoretical tool.

  8. I started reading this on Clifford algebras and Markov chains: http://www.siue.edu/~sstaple/index_files/clfgrph1114.pdf

    One thing that strikes me is that it uses the Berezin integral. This is well known to physicists as it’s used to compute things about fermions in Quantum Field Theory. In fact, it looks a bit like it gives a nice interpretation of those integrals in fermionic QFT in terms of self-avoiding walks. You can already interpret QFT in terms of random walks with an alternative probability theory – ie. complex probability amplitudes. Seems like there is a connection.

    Are you able to say anything more about this michael?

  9. Thanks for directing me to that! The paper was new to me.

    vis a vis “interpretation of those integrals in fermionic QFT in terms of self-avoiding walks.”, either you are reading more into the paper than is there, or I am reading less. The jist of the paper is augmenting regular matrix multiplication techniques for the adjacency matrix for a graph, or the transition matrix for a Markov chain (that one would use to compute nifty things like probaility that starting in state i the chain is in state j after k steps [for Markov chains] or the number of Hamiltonian cycles in a graph [for an adjacency matrix]) by insisting on a Clifford product on the elements instead of just the normal field multiplication. Alternativly, one could think of it as a “Clifford action” analogous to a group action piggybacking on the matrix multiplication.

    It is actually a very elegant idea that gives a very slick derivation of some of these enumerables. A nice club to have in the bag so to speak.

    I am not a physicist, nor do I play one on tv, so I have no exposure to the fermionic QFT stuff to enable me to make an informed judgment, but I would be willing to bet that your intuition is corrrect. In fact, it seems so plausable that it is hard to believe that no one has done it. There is a paper in that idea for someone – extant or otherwise.

  10. I may be reading more into the paper than is there – but that’s probably a good thing!

    Quick summary of QFT: you typically spend your day having to integrate ‘amplitudes’ (which are a complex valued analogue of probabilities) over the space of paths from A to B. This is an infinite dimensional space but the catch is that there isn’t a suitable measure defined on it. So you typically approach it through dicing up the paths into ‘walks’ with a finite number of steps and the space of paths with n steps is finite dimensional. You then look at the limit as the number of steps goes to infinity. Turns out that in practice the integrand is often of the form exp(-x.Ax) (or approximately so) for some matrix A and so the answer we want is proportional to 1/(sqrt det A). The theory is very similar to the theory of Brownian motion and for very simply QFTs it is basically identical to Brownian motion apart from a factor of i(=sqrt(-1)) that appears.

    That’s for bosonic particles. When we come to fermions, physicists do something very weird and which can seem very unmotivated. Suddenly they decide that all of the variables in the integration are anticommuting. The integral is replaced with the Berezin integral and that 1/(sqrt det A) becomes det A modulo some factors. Miraculously the theory seems to work. So my hunch is that we can interpret this as something like the integral over all self-avoiding walks, or something related. This is also motivated by the fact that identical fermions are particles that don’t like to sit in the same state as each other so in a sense it doesn’t seem too weird that their paths might be self-avoiding.

    Hmmm…the analogy is too clear. I’d almost put money on the Markov chain Clifford algebra stuff having been derived from statistical field theory in the first place.

  11. Considering that the very first reference of the paper is “Fermionic stochastic calculus in Dirac-Fock space” I am inclined to agree with you :)

    It may be though that there is a circle of ideas going on here that has not yet been put in a coherent unified state. It would be interesting to work on, but since it is outside my immediate knowledge, I would have to learn too much stuff to do anything quickly enough. That is what Ars Mathematica is for. It is like a math research RSS feed!

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