Georgia Institute of Technology College of Sciences

We will prove a recent version of the weighted Carleson Embedding Theorem for vector-valued function spaces with matrix weights. Time permitting, we will discuss the applications of this theorem to estimates on well-localized operators. This result relies heavily on the work of Kelly Bickel and Brett Wick and is joint with Sergei Treil.

In the recent past multiple orthogonal polynomials have attracted great attention. They appear in simultaneous rational approximation, simultaneous quadrature rules, number theory, and more recently in the study of certain random matrix models. These are sequences of polynomials which share orthogonality conditions with respect to a system of measures. A central role in the development of this theory is played by the so called Nikishin systems of measures for which many results of the standard

I will discuss two different Lax systems for the Painleve II equation. One is of size 2\times 2 and was first studied by Flaschka and Newell in 1980. The other is of size 4\times 4, and was introduced by Delvaux, Kuijlaars, and Zhang in 2010. Both of these objects appear in problems in random matrix theory and closely related fields. I will describe how they are related, and discuss the applications of this relation to random matrix theory.

The conventional point of view is that the Lagrangian is a scalar object, which through the Euler-Lagrange equations provides us with one single equation. However, there is a key integrability property of certain discrete systems called multidimensional consistency, which implies that we are dealing with infinite hierarchies of compatible equations. Wanting this property to be reflected in the Lagrangian formulation, we arrive naturally at the construction of Lagrangian multiforms, i.e., Lagrangians which are

The main result to be discussed will be the boundedness from $L^\infty \times L^\infty$ into $BMO$ of bilinear pseudodifferential operators with symbols in a range of bilinear H\"ormander classes of critical order. Such boundedness property is achieved by means of new continuity results for bilinear operators with symbols in certain classes and a new pointwise inequality relating bilinear operators and maximal functions. The role played by these estimates within the general theory will be addressed.

Some interesting applications of extremal trigonometric polynomials to the problem of stability of solutions to the nonlinear autonomous discrete dynamic systems will be considered. These are joint results with A.Khamitova, A.Korenovskyi, A.Solyanik and A.Stokolos

I will give a series of elementary lectures presenting basic regularity theory of second order HJB equations. I will introduce the notion of viscosity solution and I will discuss basic techniques, including probabilistic techniques and representation formulas. Regularity results will be discussed in three cases: degenerate elliptic/parabolic, weakly nondegenerate, and uniformly elliptic/parabolic.

Bernstein's inequality connecting the norms of a (trigonometric) polynomial with the norm of its derivative is 100 years old. The talk will discuss some recent developments concerning Bernstein's inequality: inequalities with doubling weights, inequalities on general compact subsets of the real line or on a system of Jordan curves. The beautiful Szego-Schaake–van der Corput generalization will also be mentioned along with some of its recent variants.

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

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.