Georgia Institute of Technology College of Sciences

This will be a survey talk on the ongoing classification problem for Carleson and reverse Carleson measures for the de Branges-Rovnyak spaces. We will relate these problems to some recent work of Lacey and Wick on the boundedness of the Cauchy transform operator.

I will introduce the cluster value problem, and its relation to the Corona problem, in the setting of Banach algebras of analytic functions on unit balls. Then I will present a reduction of the cluster value problem in separable Banach spaces, for the algebras $A_u$ and $H^{\infty}$, to those spaces that are $\ell_1$ sums of a sequence of finite dimensional spaces. This is joint work with William B. Johnson.

This talk discusses exponential frames and Riesz sequences in L^2 over a set of finite measure. (Roughly speaking, Frames and Riesz sequences are over complete bases and under complete bases, respectively). Intuitively, one would assume that the frequencies of an exponential frame can not be too sparse, while those of an exponential Riesz sequence can not be too dense. This intuition was confirmed in a very general theorem of Landau, which holds for all bounded sets of positive measure. Landau's proof involved a deep study of the eigenvalues of compositions of

Complex analysis in several variables is very different from the one variable theory. Hence it is natural to expect that operator theory on Bergman spaces of pseudoconvex domains in $\mathbb{C}^n$ will be different from the one on the Bergman space on the unit disk. In this talk I will present several results that highlight this difference about compactness of Hankel operators. This is joint work with Mehmet Celik and Zeljko Cuckovic.

In 1980, T. M.

For dimensions n greater than or equal to 3, and integers N greater than 1, there is a distribution of points P in a unit cube [0,1]^{n}, of cardinality N, for which the discrepancy function D_N associated with P has an optimal Exponential Orlicz norm. In particular the same distribution will have optimal L^p norms, for 1 < p < \infty. The collection P is a random digit shift of the examples of W.L. Chen and M. Skriganov.

On the Hardy space, by means of an elegant and ingenious argument, Widom showed that the spectrum of a bounded Toeplitz operator is always connected and Douglas showed that the essential spectrum of a bounded Toeplitz operator is also connected. On the Bergman space, in 1979, G. McDonald and the C. Sundberg showed that the essential spectrum of a Toeplitz operator with bounded harmonic symbol is connected if the symbol is either real

In this talk, we will characterize the compact operators on Bergman spaces of the ball and polydisc. The main result we will discuss shows that an operator on the Bergman space is compact if and only if its Berezin transform vanishes on the boundary and additionally this operator belongs to the Toeplitz algebra. We additionally will comment about how to extend these results to bounded symmetric domains, and for "Bergman-type" function spaces.

A recent conjecture in harmonic analysis that was exploredin the past 20 years was the A_2 conjecture, that is the sharp bound onthe A_p weight characteristic of a Calderon-Zygmund singular integraloperator on weighted L_p space. The non-sharp bound had been knownsince the 1970's, but interest in the sharpness was spurred recentlyby connections to quasiconformal mappings and PDE. Finally solved infull by Hytonen, the proof is complex, intricate and lengthy. A new "simple" approach using local mean oscillation and positive operatorbounds was published by Lerner.

We will study one and two weight inequalities for several different operators from harmonic analysis, with an emphasis on vector-valued operators. A large portion of current research in the area of one weight inequalities is devoted to estimating a given operators' norm in terms of a weight's A_p characteristic; we consider some related problems and the extension of several results to the vector-valued setting. In the two weight setting we consider some of the difficulties of characterizing a two weight inequality through Sawyer-type testing