Georgia Institute of Technology College of Sciences

We are going to discuss a generalization of the classical relation between Jacobi matrices and orthogonal polynomials to the case of difference operators on lattices. More precisely, the difference operators in question reflect the interaction of nearest neighbors on the lattice Z^2. It should be stressed that the generalization is not obvious and straightforward since, unlike the classical case of Jacobi matrices, it is not clear whether the eigenvalue problem for a difference equation on Z^2 has a solution and, especially, whether the entries of an eigenvector can be chosen to be polynomials in the spectral variable. In order to overcome the above-mentioned problem, we construct difference operators on Z^2 using multiple orthogonal polynomials. In our case, it turns out that the existence of a polynomial solution to the eigenvalue problem can be guaranteed if the coefficients of the difference operators satisfy a certain discrete zero curvature condition. In turn, this means that there is a discrete integrable system behind the scene and the discrete integrable system can be thought of as a generalization of what is known as the discrete time Toda equation, which appeared for the first time as the Frobenius identity for the elements of the Pade table.

Catalan numbers arise in many enumerative contexts as the counting sequence of combinatorial structures. We consider natural local moves on some realizations of the Catalan sequence and derive estimates of the mixing time of the corresponding Markov chains. We present a new O(n^2 log n) bound on the mixing time for the random transposition chain on Dyck paths, and raise several open problems, including the optimality of the above bound. (Joint work with Prasad Tetali and Damir Yelliusizov.)

The question of global regularity vs. finite time blow-up remains open for many fluid equations. Even in the cases where global regularity is known, solutions may develop small scales as time progresses. In this talk, I will first discuss an active scalar equation which is an interpolation between the 2D Euler equation and the surface quasi-geostrophic equation. We study the patch dynamics for this equation in the half-plane, and prove that the solutions can develop a finite-time singularity. I will also discuss a passive transport equation whose solutions are known to have global regularity, and our goal is to study how well a given initial density can be mixed if the incompressible flow satisfies some physically relevant quantitative constraints. This talk is based on joint works with A. Kiselev, L. Ryzhik and A. Zlatos.

We discuss a dual version of a problem about perfect matchings in cubic graphs posed by Lovász and Plummer. The dual version is formulated as follows: "Every triangulation of an orientable surface has exponentially many groundstates"; we consider groundstates of the antiferromagnetic Ising Model. According to physicist, the dual formulation holds. In this talk, I plan to show a counterexample to the dual formulation (**), a method to count groundstates which gives a better bound (for the original problem) on the class of Klee-graphs, the complexity of the related problems and if time allows, some open problems. (**): After that physicists came up with an explanation to such an unexpected behaviour!! We are able to construct triangulations where their explanation fails again. I plan to show you this too. (This is joint work with Marcos Kiwi)