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Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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

Series: Analysis Seminar

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.

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

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.