# Spring 2008

Tuesdays, 4:00--5:00pm, WWH room 1013.

For any questions or remarks please email Irina Nenciu.

February | ||
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Tue 02/19 | Gerald TeschlUniversity 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 AlbersCIMS |
C^{0}--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
C^{0}--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. | |

March | ||

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 |

April | ||

Tue 04/01 | No seminar |
No seminar |

Tue 04/08 | Kirill VaninskyMichigan 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 KrahmerCIMS |
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. | ||