Georgia Institute of Technology College of Sciences

It is well known that the Horn inequalities characterize the relationship of eigenvalues of Hermitian matrices A, B, and A+B. At the same time, similar inequalities characterize the relationship of the sizes of the Jordan models of a nilpotent matrix, of its restriction to an invariant subspace, and of its compression to the orthogonal complement. In this talk, we provide a direct, intersection theoretic, argument that the Jordan models of an operator of class C_0 (such operator can be thought of as the infinite dimensional generalization of matrices, that is an operator will be annihilated by an H-infinity function), of its restriction to an invariant subspace, and of its compression to the orthogonal complement, satisfy a multiplicative form of the Horn inequalities, where ‘inequality’ is replaced by ‘divisibility’. When one of these inequalities is saturated, we show that there exists a splitting of the operator into quasidirect summands which induces similar splittings for the restriction of the operator to the given invariant subspace and its compression to the orthogonal complement. Our approach also explains why the same combinatorics solves the eigenvalue and the Jordan form problems. This talk is based on the joint work with H. Bercovici.

The Riemann mapping theorem says that every simply connected proper plane domain can be conformally mapped to the unit disk. I will discuss the computational complexity of constructing a conformal map from the disk to an n-gon and show that it is linear in n, with a constant that depends only on the desired accuracy. As one might expect, the proof uses ideas from complex analysis, quasiconformal mappings and numerical analysis, but I will focus mostly on the surprising roles played by computational planar geometry and 3-dimensional hyperbolic geometry. If time permits, I will discuss how this conformal mapping algorithm implies new results in discrete geometry, e.g., every simple polygon can be meshed in linear time using quadrilaterals with all angles \leq 120 degrees and all new angles \geq 60 degrees (small angles in the original polygon must remain).

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.