## Seminars and Colloquia by Series

Thursday, April 2, 2015 - 11:05 , Location: rm 005 , Karl Liechty , DePaul University , Organizer: Jeff Geronimo

Karl Liechty is the
winner of the 2015 Szego prize in orthogonal polynomials and special functions.

I will discuss two different Lax systems for the Painleve II equation. One is of size 2\times 2 and was first studied by Flaschka and Newell in 1980. The other is of size 4\times 4, and was introduced by Delvaux, Kuijlaars, and Zhang in 2010. Both of these objects appear in problems in random matrix theory and closely related fields. I will describe how they are related, and discuss the applications of this relation to random matrix theory.
Wednesday, April 1, 2015 - 14:00 , Location: Skiles 005 , Virginia Naibo , Kansas State University , Organizer: Brett Wick
The main result to be discussed will be the boundedness from $L^\infty \times L^\infty$ into $BMO$ of bilinear pseudodifferential operators with symbols in a range of bilinear H\"ormander classes of critical order. Such boundedness property is achieved by means of new continuity results for bilinear operators with symbols in certain classes and a new pointwise inequality relating bilinear operators and maximal functions. The role played by  these estimates within the general theory will be addressed.
Wednesday, March 25, 2015 - 14:00 , Location: Skiles 005 , Chris Schwanke , University of Mississippi , , Organizer: Michael Lacey
In this talk, we demonstrate how to use convexity to identify specific operations on Archimedean vector lattices that are defined abstractly through functional calculus with more concretely defined operations. Using functional calculus, we then introduce functional completions of Archimedean vector lattices with respect to continuous, real-valued functions on R^n that are positively homogeneous. Given an Archimedean vector lattice E and a continuous, positively homogeneous function h on R^n, the functional completion of E with respect to h is the smallest Archimedean vector lattice in which one is able to use functional calculus with respect to h. It will also be shown that vector lattice homomorphisms and positive linear maps can often be extended to such completions. Combining all of the aforementioned concepts, we characterize Archimedean complex vector lattices in terms of functional completions. As an application, we construct the Fremlin tensor product for Archimedean complex vector lattices.
Wednesday, March 11, 2015 - 14:05 , Location: Skiles 005 , Jordan Greenblat , UCLA , Organizer: Michael Lacey
In the course of their work on the Unique Games Conjecture, Harrow, Kolla, and Schulman proved that the spherical maximal averaging operator on the hypercube satisfies an L^2 bound independent of dimension, published in 2013. Later, Krause extended the bound to all L^p with p > 1 and, together with Kolla, we extended the dimension-free bounds to arbitrary finite cliques. I will discuss the dimension-independence proofs for clique powers/hypercubes, focusing on spectral and operator semigroup theory. Finally, I will demonstrate examples of graphs whose Cartesian powers' maximal bounds behave poorly and present the current state and future directions of the project of identifying analogous asymptotics from a graph's basic structure.
Wednesday, March 4, 2015 - 14:05 , Location: Skiles 005 , Dmitriy Dmitrishin , Odessa National Polytechnic University , Organizer: Jeff Geronimo
Some interesting applications of extremal trigonometric polynomials to the problem of stability of solutions to the nonlinear autonomous discrete dynamic systems will be considered. These are joint results with A.Khamitova, A.Korenovskyi, A.Solyanik and A.Stokolos
Wednesday, February 25, 2015 - 14:00 , Location: Skiles 005 , Wing Li , Georgia Institute of Technology , Organizer: Brett Wick
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 is an operator will be annihilated by an H-infinity function), of its restriction to an invariant subspace, and of its compression to the orthogonal complement, satisfy a multiplicative form of the Horn inequalities, where ‘inequality’ is replaced by ‘divisibility’. When one of these inequalities is saturated, we show that there exists a splitting of the operator into quasidirect summands which induces similar splittings for the restriction of the operator to the given invariant subspace and its compression to the orthogonal complement. Our approach also explains why the same combinatorics solves the eigenvalue and the Jordan form problems. This talk is based on the joint work with H. Bercovici.
Thursday, February 19, 2015 - 15:30 , Location: Skiles 006 , Chris Bishop , SUNY Stony Brook , Organizer:
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, 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.