Georgia Institute of Technology College of Sciences

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.

Orthonormal bases (ONB) are used throughout mathematics and its applications. However, in many settings such bases are not easy to come by. For example, it is known that even the union of as few as two intervals may not admit an ONB of exponentials. In cases where there is no ONB, the next best option is a Riesz basis (i.e. the image of an ONB under a bounded invertible operator). In this talk I will discuss the following question: Does every finite union of rectangles in R^d, with edges parallel to the axes, admit a Riesz basis

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.

In this talk, we will discuss a T1 theorem for band operators (operators with finitely many diagonals) in the setting of matrix A_2 weights. This work is motivated by interest in the currently open A_2 conjecture for matrix weights and generalizes a scalar-valued theorem due to Nazarov-Treil-Volberg, which played a key role in the proof of the scalar A_2 conjecture for dyadic shifts and related operators. This is joint work with Brett Wick.

There is a long standing asymptotic relationship in several areas of analysis, between polynomials and entire functions of exponential type. Many extremal problems for polynomials of degree n turn into analogous extremal problems for entire functions of exponential type, as the degree n approaches infinity. We discuss some of the old such as Bernstein's constant on approximation of |x|, and recent work on Plancherel-Polya and Nikolskii inequalities.

When does the spectrum of an operator determine the operator uniquely?-This question and its many versions have been studied extensively in the field of inverse spectral theory for differential operators. Several notable mathematicians have worked in this area. Among others, there are important contributions by Borg, Levinson, Hochstadt, Liebermann; and more recently by Simon, Gesztezy, del Rio and Horvath, which have further fueled these studies by relating the completeness problems of families of functions to the inverse

We discuss bi-parameter Calderon-Zygmund singular integrals from the point of view of modern probabilistic and dyadic techniques. In particular, we discuss their structure and boundedness via dyadic model operators. In connection to this we demonstrate, via new examples, the delicacy of the problem of finding a completely satisfactory product T1 theorem. Time permitting related non-homogeneous bi-parameter results may be mentioned.