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.

We will start with a description of geometric and measure-theoretic objects associated to certain convex functions in R^n. These objects include a quasi-distance and a Borel measure in R^n which render a space of homogeneous type (i.e. a doubling quasi-metric space) associated to such convex functions. We will illustrate how real-analysis techniques in this quasi-metric space can be applied to the regularity theory of convex solutions u to the Monge-Ampere equation det D^2u =f as well as solutions v of the linearized Monge-Ampere equation L_u(v)=g. Finally, we will discuss recent developments regarding the existence of Sobolev and Poincare inequalities on these Monge-Ampere quasi-metric spaces and mention some of their applications.

We discuss asymptotics of multiple orthogonal polynomials with respect to Nikishin systems generated by two measures (\sigma_1, \sigma_2) with unbounded supports (supp(\sigma_1) \subset \mathbb{R}_+, supp(\sigma_2) \subset \mathbb{R}_-); moreover, the second measure \sigma_2 is discrete. We focus on deriving the strong and weak asymptotic for a special system of multiple OP from this class with respect to two Pollaczek type weights on \mathbb{R}_+. The weak asymptotic for these polynomials can be obtained by means of solution of an equilibrium problem. For the strong asymptotic we use the matrix Riemann-Hilbert approach.