Random matrices and integrable systems working seminar

Courant Institute

Spring 2008

Tuesdays, 4:00--5:00pm, WWH room 1013.
For any questions or remarks please email Irina Nenciu.

Tue 02/19 Gerald Teschl
University of Vienna
Stability of the periodic Toda lattice under short range perturbations
Abstract: We consider the stability of the periodic Toda lattice under a short range perturbation. We prove that the perturbed lattice asymptotically approaches a modulated lattice.
More precisely, let g be the genus of the hyperelliptic curve associated with the unperturbed solution. We show that, apart from the phenomenon of the solitons traveling on the quasi-periodic background, the n/t-pane contains g+2 areas where the perturbed solution is close to a finite-gap solution in the same isospectral torus. In between there are g+1 regions where the perturbed solution is asymptotically close to a modulated lattice which undergoes a continuous phase transition (in the Jacobian variety) and which interpolates between these isospectral solutions. In the special case of the free lattice (g=0) the isospectral torus consists of just one point and we recover the classical result.
Both the solutions in the isospectral torus and the phase transition are explicitly characterized in terms of Abelian integrals on the underlying hyperelliptic curve.
Our method relies on the equivalence of the inverse spectral problem to a matrix Riemann-Hilbert problem defined on the hyperelliptic curve and generalizes the so-called nonlinear stationary-phase-steepest-descent method by Deift and Zhou for Riemann-Hilbert problem deformations to Riemann surfaces.
Tue 02/26 Peter Albers
C0--rigidity of Poisson brackets - work by Entov - Polterovich
Special time: 3:45--5:00pm Abstract: I will report on very recent work by Entov-Polterovich on C0--rigidity of Poisson brackets. Their results are easy to state and very surprising. In their proof Entov-Polterovich crucially use the theory of geodesics of the Hofer metric on the group of Hamiltonian diffeomorphisms.
The aim of the talk is to give a gentle introduction to Hofer's geometry on the group of Hamiltonian diffeomorphisms and to explain the input of symplectic geometry into the proof of Entov-Polterovich.
Tue 03/04 No seminar No seminar
Tue 03/11 Manuel Dominguez de la Iglesia
Universidad de Sevilla
Differential properties of some families of matrix valued orthogonal polynomials and applications
Abstract: In the last years new families of matrix valued orthogonal polynomials (MOP) on the real line have been found satisfying second order differential equations with coefficients independent of the degree of the polynomial. We give an overview of the techniques that have led to these examples. These families are among those that are likely to play in the case of matrix orthogonality the role of the classical families of Hermite, Laguerre and Jacobi in the case of scalar orthogonality.
Apart from this richness, several new and certainly interesting phenomena, absent in the scalar theory, have appeared. For instance, a fixed family of MOP can be common eigenfunctions of several linearly independent second order differential operators, a fixed second order differential operator can have infinitely many different families of MOP as eigenfunctions, or that there exist families of MOP satisfying odd order differential equations.
We also show applications that these families have in quasi-birth-and-death processes or time-and-band limiting problems.
Tue 03/18 Spring break No seminar
03/17--03/19 DIMACS Workshop on Random Matrices
DIMACS Center, CoRE Building, Rutgers University
(see also the following directions to the DIMACS Center)
Tue 03/25 No seminar No seminar
Tue 04/01 No seminar No seminar
Tue 04/08 Kirill Vaninsky
Michigan State University
Poisson formalism for integrable systems and Riemann surfaces
Abstract: Until now Poisson brackets for integrable systems and associated spectral curves were considered as separate objects. In this talk we explain how Poisson brackets can be obtained from the spectral curves. Novel approach allows to recover known solutions of the Yang-Baxter equation in elementary way.
Tue 04/015 No seminar No seminar
Tue 04/22 Felix Krahmer
The role of Chebyshev polynomials for optimizing exponentially accurate Sigma-Delta quantization schemes
(Joint work with Percy Deift and Sinan Gunturk)
Abstract: Gunturk (2003) showed that one can achieve exponential accuracy for the Analog-Digital conversion method of Sigma-Delta modulation by using sparse convolution filters. We address the question, what maximum error decay rate one can achieve using such a construction.
We show that for a family of filters that is optimal in this sense, their non-zero entries must be distributed as the zeros of a Chebyshev polynomial of the second kind. The proof uses a nonlinear matrix identity for the zeros of Chebyshev polynomials.
The talk will not assume any signal processing background.