Seminars and Colloquia by Series

Wednesday, February 18, 2015 - 14:05 , Location: Skiles 005 , Chris Bishop , SUNY Stony Brook , , Organizer: Michael Lacey
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).
Wednesday, January 21, 2015 - 14:00 , Location: Skiles 005 , Michael Lacey , Georgia Tech , Organizer: Michael Lacey
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. 
Wednesday, January 14, 2015 - 14:00 , Location: Skiles 005 , Maxim Derevyagin , University of Mississippi at Oxford , , Organizer: Doron Lubinsky
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.
Wednesday, November 19, 2014 - 14:00 , Location: Skiles 006 , Brian Street , University of Wisconsin, Madison , Organizer: Michael Lacey
 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.
Wednesday, November 12, 2014 - 14:00 , Location: Skiles 005 , Diego Maldonado , Kansas State University , Organizer:
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.
Wednesday, October 22, 2014 - 14:00 , Location: Skiles 005 , Kelly Bickel , Bucknell University , Organizer:
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.
Wednesday, October 15, 2014 - 14:00 , Location: Skiles 005 , Jingbo Xia , SUNY - Buffalo , Organizer:
Wednesday, October 1, 2014 - 14:00 , Location: Skiles 005 , Rishika Rupum , Texas A&M , Organizer:
  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.
Monday, June 2, 2014 - 14:05 , Location: Skiles 005 , Alexander Aptekarev , Keldysh Institute, Russia , Organizer: Jeff Geronimo
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.
Wednesday, April 16, 2014 - 14:00 , Location: Skiles 005 , Henri Martikainen , Georgia Tech , Organizer:
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.