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Series: Analysis Seminar

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).

Series: Analysis Seminar

The sharp A2 weighted bound for martingale transforms can be proved by a new elementary method. With additional work, it can be extended to the euclidean setting. Other generalizations should be possible.

Series: Analysis Seminar

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.

Series: Analysis Seminar

This talk concerns a theory of "multiparameter singularintegrals." The Calderon-Zygmund theory of singular integrals is a welldeveloped and general theory of singular integrals--in it, singularintegrals are associated to an underlying family of "balls" B(x,r) on theambient space. We talk about generalizations where these balls depend onmore than one "radius" parameter B(x,r_1,r_2,\ldots, r_k). Thesegeneralizations contain the classical "product theory" of singularintegrals as well as the well-studied "flag kernels," but also include moregeneral examples. Depending on the assumptions one places on the balls,different aspects of the Calderon-Zygmund theory generalize.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

Series: Analysis Seminar

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
spectral problems of the Schr ̈odinger operator. In this talk, we will discuss the role played by the Toeplitz
kernel approach in answering some of these questions, as described by Makarov and Poltoratski. We will
also describe some new results using this approach. This is joint work with Mishko Mitkovski.

Series: Analysis Seminar

We discuss asymptotics of multiple orthogonal
polynomials with respect to Nikishin systems generated by two
measures (\sigma_1, \sigma_2) with unbounded supports
(supp(\sigma_1) \subset \mathbb{R}_+,
supp(\sigma_2) \subset \mathbb{R}_-); moreover, the second
measure \sigma_2 is discrete. We focus on deriving the strong and weak
asymptotic for a special system of multiple OP from this class with respect
to two Pollaczek type
weights on \mathbb{R}_+. The weak asymptotic for these
polynomials can be obtained by means of solution of an equilibrium problem.
For
the strong asymptotic we use the matrix Riemann-Hilbert approach.

Series: Analysis Seminar

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.