Analysis and PDE Seminar

Analysis and PDE Seminar-Master's Talk

TITLE:  Upper Bounds for the Eigenvalue Counting Function of the Laplacian and Schrodinger Operators
 
ABSTRACT:  TheWeyl asymptotics provide information about the asymptotic distribution of the eigenvalues of the Laplacian in a bounded domain. Polya conjectured that the Weyl asymptotics actually provide an upper bound for the eigenvalue counting function. In the late 70's, Cwickel, Rosenbljum and Lieb independently (and with di erent techniques) proved an upper bound on the number of negative eigenvalues of a Schrodinger operator on Rd. I will give a motivated presentation of Lieb's technique, which uses path integrals and the Feynman- Kac formula. Lieb also used this bound to get a bound proportional to Polya's conjecture.
Date:
-
Location:
745 Patterson Office Tower
Event Series:

Analysis and PDE Seminar

TITLE: Traveltime Tomography with local data
 
ABSTRACT: We study the boundary rigidity problem with partial data consisting of determining locally the Riemannian metric of a Riemannian manifold with boundary from the distance function measured at pairs of points near a xed point on the boundary. We show that one can recover a conformal factor near a strictly convex point where we have the information, uniquely and in a stable way. In
particular, this implies that we can determine locally the isotropic sound speed of a medium by measuring the travel times of waves joining points close to a convex point on the boundary.  The local results lead to a global lens rigidity uniqueness and stability result assuming that the manifold is foliated by strictly convex hypersurfaces.
 
Date:
-
Location:
745 Patterson Office Tower
Event Series:

Analysis and PDE Seminar

TITLE:  Eigenvalue statistics for random Schrodinger operators
 
ABSTRACT: Random Schrodinger operators model the propagation of noninteracting electrons in disordered media. The study of random Schrodinger operators combines the spectral theory of self-adjoint operators and probability theory. I'll review the basic spectral properties of these operators such as the deterministic spectrum and Anderson localization. Estimates for the eigenvalues of the corresponding Schrodinger operators restricted to nite regions, such as the Wegner and Minami estimates, will be presented. These estimates will be used to characterize the local eigenvalue statistics and level spacing statistics for various models and energy regimes.
 
Date:
-
Location:
745 Patterson Office Tower
Event Series:

Analysis and PDE Seminar

TITLE:  Boundary Korn Inequality and Neumann Problems in Homogenization of Elliptic Systems of Elasticity
 
ABSTRACT:  This talk concerns with a family of elliptic systems of linear elasticity with rapidly oscillating periodic coecients, arising in the theory of homogenization.  We establish uniform optimal regularity estimates for solutions of Neumann problems in a bounded Lipschitz domain with L2 boundary data. The proof relies on a boundary Korn inequality for solutions of systems of linear elasticity
and uses a large-scale Rellich estimate. This is a joint work with Jun Geng and Liang Song.
 
Date:
-
Location:
745 Patterson Office Tower
Event Series:

Analysis and PDE Seminar

TITLE:  The Lp Carleman estimate and a partial data inverse problem
 
ABSTRACT:  In this talk I'll describe an inverse problem, explain what a Carleman estimate
is, and talk about what the two have to do with each other. Then I'll discuss a
particular (Lp) Carleman estimate, and use the context of the inverse problem
to talk about why one might want to prove it. Time permitting, I'll discuss the
proof. This is recent work with L. Tzou.
1
Date:
-
Location:
745 Patterson Office Tower
Event Series:

Analysis and PDE Seminar--Qualifying Exam

Title: Nachman's Inverse Conductivity Result

Abstract: In his 1996 paper, Nachman addressed the inverse conductivity problem in two dimensions.  If the electrical conductivity γ ∈ W2, p(ℝ2)​ and is bounded below, then it is uniquely determined by its corresponding Dirichlet-to-Neumann operator Λγ​.  The proof of this result is constructive, involving the scattering problem for the Schrodinger equation at zero energy.  In this talk, we will give an overview of Nachman's approach.

Date:
-
Location:
745 Patterson Office Tower
Event Series:

Analysis and PDE Seminar

TITLE:  A hybrid inverse source problem for radiative transport

ABSTRACT:  The radiative transport equation (RTE) is a model for light propagation inside a scattering medium. A classic inverse problem for the RTE is as follows.  Suppose we have an object where light propagation is modeled by the RTE, which contains a source of light. Given the ability to measure light intensity on the boundary, can we recover the light source exactly? In this talk I will give a brief introduction to the RTE and its inverse source problem, and discuss recent work on improving the stability of the problem using so-called hybrid methods. This is joint work with John Schotland and Guillaume Bal.

Date:
-
Location:
745 Patterson Office Tower
Event Series:
Subscribe to Analysis and PDE Seminar