Georgia Institute of Technology College of Sciences

This talk is intended to be a cocktail of many things. I will start with standard random matrices (called GUE in the slang) and formal computations which leads one to the main problem of counting planar diagrams. This was done by physicists, though the main computation of generating functions for such planar diagrams go through an analytic tools. Here I will change the topic to analysis, and get through with the help of Chebyshev polynomials and how these can be used to solve a minimization problem and then from there to compute several generating functions of planar diagrams. Then I will talk about tridiagonalization which is a main tool in matrix analysis and point out an interesting potential view on this subject.

Markov bases have been developed in algebraic statistics for exact goodness-of-fit testing. They connect all elements in a fiber (given by the sufficient statistics) and allow building a Markov chain to approximate the distribution of a test statistic by its posterior distribution. However, finding a Markov basis is often computationally intractable. In addition, the number of Markov steps required for converging to the stationary distribution depends on the connectivity of the sampling space.In this joint work with Caroline Uhler and Sarah Cepeda, we compare different test statistics and study the combinatorial structure of the finite lattice Ising model. We propose a new method for exact goodness-of-fit testing. Our technique avoids computing a Markov basis but builds a Markov chain consisting only of simple moves (i.e. swaps of two interior sites). These simple moves might not be sufficient to create a connected Markov chain. We prove that when a bounded change in the sufficient statistics is allowed, the resulting Markov chain is connected. The proposed algorithm not only overcomes the computational burden of finding a Markov basis, but it might also lead to a better connectivity of the sampling space and hence a faster convergence.

The quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff, says that a typical Hamiltonian system of n degrees of freedom on a typical energy surface has a dense orbit. This question is wide open. In this talk I will explain a recent result by V. Kaloshin and myself which can be seen as a weak form of the quasi-ergodic hypothesis. We prove that a dense set of perturbations of integrable Hamiltonian systems of two and a half degrees of freedom possess orbits which accumulate in sets of positive measure. In particular, they accumulate in prescribed sets of KAM tori.

We will see how a result in von Neumann algebras (a theory developed by von Neumann to give themathematical framework for quantum physics) gave rise, rather serendipitously, to an elementary but very usefulinvariant in the theory of ordinary knots in threel space. Then we'll look at some subsequent developments of the theory, and talk about a thorny problem which remains open.