Seminars and Colloquia by Series

Reaching L^1 via Extrapolation Theory

Series
Analysis Seminar
Time
Wednesday, April 8, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Carlos DomingoUniversity of Barcelona
The classical Rubio de Francia extrapolation allows you to obtain strong-type estimates for weights in A_p (and every p>1) if you can show that it holds for some p_0>1. However, the endpoint p=1 has to be treated separately. In this talk we will explain how to deduce weak-type (1,1) estimates for A_1 weights if we have a certain restricted weak-type inequality at some level p_0>1. We will then show how this approach can be applied to the Bochner-Riesz operator at the critical index and Fourier multipliers.

A new variational principle for integrable systems

Series
Analysis Seminar
Time
Tuesday, April 7, 2015 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Sarah LobbUniversity of Sidney
The conventional point of view is that the Lagrangian is a scalar object, which through the Euler-Lagrange equations provides us with one single equation. However, there is a key integrability property of certain discrete systems called multidimensional consistency, which implies that we are dealing with infinite hierarchies of compatible equations. Wanting this property to be reflected in the Lagrangian formulation, we arrive naturally at the construction of Lagrangian multiforms, i.e., Lagrangians which are the components of a form and satisfy a closure relation. Then we can propose a new variational principle for discrete integrable systems which brings in the geometry of the space of independent variables, and from this principle derive any equation in the hierarchy.

Two Lax systems for the Painleve II equation

Series
Analysis Seminar
Time
Thursday, April 2, 2015 - 11:05 for 1 hour (actually 50 minutes)
Location
rm 005
Speaker
Karl LiechtyDePaul University

Please Note: 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.

On an endpoint mapping property for certain bilinear pseudodifferential operators

Series
Analysis Seminar
Time
Wednesday, April 1, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Virginia NaiboKansas State University
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.

Functional Completions and Complex Vector Lattices

Series
Analysis Seminar
Time
Wednesday, March 25, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chris SchwankeUniversity of Mississippi
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.

Maximal Bounds on Cartesian Powers of Finite Graphs

Series
Analysis Seminar
Time
Wednesday, March 11, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jordan GreenblatUCLA
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.

Fejer-Riesz type argument in non-linear dynamics

Series
Analysis Seminar
Time
Wednesday, March 4, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dmitriy DmitrishinOdessa National Polytechnic University
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

Intersection theory and the Horn inequalities for invariant subspaces

Series
Analysis Seminar
Time
Wednesday, February 25, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Wing LiGeorgia Institute of Technology
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.

Conformal mapping and optimal meshes

Series
Analysis Seminar
Time
Thursday, February 19, 2015 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Chris BishopSUNY Stony Brook
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).

Conformal mapping and optimal meshes

Series
Analysis Seminar
Time
Wednesday, February 18, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chris BishopSUNY Stony Brook
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).

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