Georgia Institute of Technology College of Sciences

I will state two different inequalities which are poorly understood, even utterly mysterious. They are stated purely in terms of conditional expectations over dyadic intervals. Motivations and connections carry one into the area of Furstenberg's multilinear ergodic averages.

I will survey recent results about the convergence of the Wash-Fourier series near L1. Joint work with Michael Lacey.

In this talk I will present recent work, in collaboration with J.Dolbeault, G. Tarantello and A. Tertikas,about the symmetry properties of extremal functions for (interpolation)functional inequalities playing an important rolein the study of long time behavior of evolution diffusion equations.Optimal constants are rarely known,in fact one can write them explicitely only when the extremals enjoymaximal symmetry. This is why the knowledge of the parameters' regionswhere symmetry is achieved is of big importance.

Let C[-1, 1] be the space of continuous functions on [-1, 1], and denote by \Delta^2 the set of convex functions f \in C[-1, 1]. Also, let E_n(f) and En^{(2)}_n(f) denote the degrees of best unconstrained and convex approximation of f \in \Delta^2 by algebraic polynomials of degree < n, respectively. Clearly, E_n(f) \le E^{(2)}_n (f), and Lorentz and Zeller proved that the opposite inequality E^{(2)}_n(f) \le CE_n(f) is invalid even with the constant C = C(f) which depends on the function f \in \Delta^2.

The Seiberg-Witten equations, introduced by Edward Witten in 1994, are a first-order semilinear geometric PDE that have led to manyimportant developments in low-dimensional topology. In this talk,we study these equations on cylindrical 4-manifolds with boundary, which we supplement with (Lagrangian) boundary conditions that have a natural Morse-Floer theoretic interpretation.

This will be an expository talk on the study of orthogonal polynomials on the real line and on the unit circle. Topics include recurrence relations, recurrence coefficients and simple examples. The talk will conclude with applications of orthogonal polynomials to other areas of research.

A separated sequence of real numbers is called a Polya sequence if the only entire functions of zero type which are bounded on this sequence are the constants. The Polya-Levinson problem asks for a description of all Polya sequences. In this talk, I will present some points of the recently obtained solution. The approach is based on the use of Toeplitz operators and de Branges spaces of entire functions. I will also present some partial results about the related Beurling gap problem.

Consider self-adjoint operators $A, B, C : \mathcal{H} \to \mathcal{H}$ on a finite-dimensional Hilbert space such that $A + B + C = 0$. Let $\{\lambda_j (A)\}$, $\{\lambda_j (B)\}$, and $\{\lambda_j (C)\}$ be sequences of eigenvalues of $A, B$, and $C$ counting multiplicity, arranged in decreasing order. In 1962, A. Horn conjectured that the relations of $\{\lambda_j (A)\}$,$\{\lambda_j (B)\}$, and $\{\lambda_j (C)\}$ can be characterized by a set of inequalities defined inductively.

The Schur-Agler class is a subclass of the bounded analytic functions on the polydisk with close ties to operator theory. We shall describe our recent investigations into the properties of rational inner functions in this class. Non-minimality of transfer function realization, necessary and sufficient conditions for membership (in special cases), and low degree examples are among the topics we will discuss.

When Calderón studied his commutators, in connection with the Cauchy integral on Lipschitz curves, he ran into the bilinear Hilbert transform by dropping an average in his first commutator. He posed the question whether this new operator satisfied any L^p estimates. Lacey and Thiele showed a wide range of L^p estimates in two papers from 1997 and 1999. By dropping two averages in the second Calderón commutator one bumps into the trilinear Hilbert transform. Finding L^p estimates for this operator is still an open question.