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Analysis-Geometry
Seminar
Meets
Fridays 2-3 pm in 509 Lake Hall
We
generally go to lunch beforehand.
Please join us at 457 Lake Hall at 12:30 pm.
This seminar features talks in the fields of
partial differential equations, functional analysis, differential geometry and
topology, and mathematical physics.
The organizers are: Maxim Braverman,
Bob McOwen, Mikhail Shubin, Peter
Topalov
Upcoming Talks:
·
September 18,
2009
Speaker: A. Gamburd (UC
Title: Semi-product and spectral gap
bounds
Abstract: We
present a new approach to establishing expansion and spectral gap bounds for finitely
generated subgroups of linear groups based on using tools from additive combinatorics, in particular sum-product estimates.
·
October 2, 2009
Speaker: John Gonzalez (Northeastern University)
Title: Solutions of the Nonlinear
Schrodinger Equation with Prescribed Asymptotics at
Infinity
Abstract: We prove
local existence and uniqueness of solutions for the one-dimensional nonlinear
Schrodinger (NLS) equations in classes of smooth functions that admit an
asymptotic expansion at infinity in decreasing powers of x. We
show that an asymptotic solution differs from a genuine solution by a Schwartz
class function which solves a generalized version of the NLS equation. The latter equation is solved by discretization methods.
·
October 9, 2009
Speaker: Peter Topalov (Northeastern
University)
Title: On Liouville
integrability of h-projectively
equivalent Kahler metrics
Abstract: Under a non-degeneracy condition we classify the compact connected Kahler manifolds admitting pairs of h-projectively equivalent metrics. Any such manifold is biholomorphically equivalent to CP^n and has integrable geodesic flow.
·
October 16, 2009
Speaker: Evgeny Lakshtanov (
Title: Scattering by obstacles with impedance boundary conditions
Abstract: We consider acoustic scattering by obstacles with general resistive and reactive impedance boundary conditions. We will touch the following topics: existence of the solution, poles of the S-matrix and their possible multiplicity, different kinds of acoustic resonators and their relationship, resonance regimes of scattering by obstacles coated in a springy material; the question when S-matrix has kernel (null-space) and its physical meaning, connectivity with poles.
·
October 30, 2009 (NEW
DATE!!!)
Speaker: Robert McOwen (Northeastern
University)
Title: Differentiability of Solutions to Elliptic Equations via Dynamical
Systems
Abstract: We
consider a second-order elliptic equation in divergence form and seek mild
conditions on the coefficients that imply that weak solutions are Lipschitz continuous or differentiable. We assume the
coefficients have modulus of continuity satisfying the square Dini condition; the additional conditions that we obtain
are fairly sharp, as examples show. Our method involves the study of
asymptotic properties of solutions to a dynamical system that is derived
from the coefficients of the elliptic equation.
·
November 6, 2009
Speaker: H.-T. Yau (
Title: Eigenvalue statistics of random matrices - a dynamical approach
Abstract: TBA
·
November 13, 2009
Emma Previato
(Boston University)
Title: Algebraic solutions of the dispersionless KP
hierarchy
Abstract: The Kadomtsev-Petviashvili (KP) equation is an integrable non-linear PDE, whereas its dispersionless
limit (dKP) develops shock waves. However, geometric
features of dKP have emerged, in particular,exact solutions have been shown to take the
form of tau functions and Schwarzian derivatives,
with a dispersionful limit satisfying Hirota's Bilinear Difference Equations.
In this talk, an overview of such techniques will be given, to highlight their
non-algebraic nature, and will be followed by the construction (joint work with
S. Matsutani) of a class of solutions which are
algebraic functions on Riemann surfaces and can be expressed in terms of theta
functions.
Boris Vainberg (UNC at
Title: Small diameter asymptotics for the wave
problems in networks of thin fibers and applications to optical delay lines
Abstract: Problems in 3D web-like domains for the Helmholtz
equation will be reduced to a corresponding 1D problem on the limiting graph.
The gluing conditions at vertices of the graph are expressed in terms of some
scattering matrices. Periodic networks will be considered. Reflection,
transmission and the delay will be discussed for waves propagating through a
finite slab of periodic medium. The results are joint
with Prof. S. Molchanov.
·
November 20, 2009
Speaker: Mark Kelbert (
Title: New proof of
Abstract: This
talk addresses the issue of the proof of the entropy power inequality, an
important tool in the analysis of Gaussian channels of information
transmission, proposed by