Abstracts for some talks (Spring Semester 2008)
=================================================================
Tuesday, February 26, 4:00 p.m., VV B139
Sergey Borodachov (Georgia Institute of Technology)
Optimal cubature formulas for integration along a ball, which use information along concentric spheres
Abstract: I plan to talk on optimal cubature formulas, which recover
the integral of a function along a d-dimensional ball from its mean
values along n concentric (d-1)-dimensional spheres inside the ball. We
look for the best radii of the node spheres and the best weights.
Optimality of the cubature formula is understood in the sense of
minimal worst case error over certain classes of differentiable
functions. In particular, we consider the class of functions, which
vanish on the boundary of the ball and are solutions to the Poisson
equation with the right hand side bounded by 1 in the L_1-norm.
This problem is an extension of the classical Kolmogorov-Nikol'skii
problem about optimal quadratures.
(Joint work with Professor Vladislav Babenko from Dnepropetrovsk
University, Ukraine)
=================================================================
Thursday, February 28, 2:25 p.m., VV B115
Artem Zvavitch (Kent State University)
Brunn-Minkowski type inequalities for Gaussian Measure
In this talk we will present a joint work with Richard Gardner.
We will discuss the Brunn-Minkowski type inequalities
for Gaussian Measure in $R^n$. The best-known of these
are Ehrhard's inequality, and the weaker
logarithmic concavity inequality.
We obtain some results concerning other inequalities of this type,
as well as a best-possible dual Gaussian Brunn-Minkowski inequality
(where the Minkowski sum is replaced by radial sum).
=================================================================
Tuesday, April 8, 4:00 p.m., VV B139
Song-Ying Li (UC-Irvine)
Title: Complex Monge-Ampere Operators in Complex Analysis and
Pseudo-Hermitian CR Manifolds.
Abstract: In this talk, I will talk about the nature and
applications of the
complex Monge-Amp\`ere operators in complex analysis.
I also connect the Monge-Amp\`ere operator to
pseudo-Hermitian structures, pseudo Ricci curvature, pseudo scalar
curvatures of CR manifolds. Finally, I will provide some applications of
Monge-Amp\`ere operators for characterizations of balls.
==================================================================
Tuesday, April 15, 4:00 p.m., VV B139
Liz Vivas (University of Michigan)
Fatou-Bieberbach Domains
Abstract:
We present an example of a Fatou-Bieberbach domain attracted to a fixed point
of an automorphism of C^2 tangent to the identity along a degenerate
characteristic direction.
==================================================================
Tuesday, April 22, 4:00 p.m., VV B139
Allan Greenleaf (University of Rochester)
Cloaking for the Conductivity and Helmholtz Equations
Abstract: I will describe how to cloak a region in $R^3$ from observation,
i.e., not just make its contents invisible,
but make undetectable the fact that anything is being hidden.
The basic construction works for several apparently different
types of
waves. I will focus on the cases of either electrostatic measurements
(where the observations correspond to boundary values of solutions
of the conductivity equation), and polarized electromagnetic waves,
acoustic waves and quantum mechanical matter waves
(boundary values of solutions of the Helmholtz equation).
The basic tool is analysis on certain singular Riemannian manifolds.
(Joint work with Yaroslav Kurylev, Matti Lassas and Gunther Uhlmann.)
================================================================
Monday, May 5, 2:25 p.m., VV B329
Michael Frazier (University of Tennessee)
Estimates for Green's Functions of Schrödinger Operators
Abstract: If T is a bounded linear operator on L^2 (\mu) with norm less
than one, then I-T has an inverse given by a Neumann series. Suppose T is
represented by integration against a symmetric kernel K(x,y). Under the
condition that the reciprocal of K is a quasimetric, we obtain an
exponential lower bound for the kernel of the inverse of I-T. Under an
appropriate smallness condition on T, we obtain an upper bound of the same
type.
These results were motivated by the inhomogeneous, time-independent
Schrodinger equation. We obtain estimates for the Green's function of the
Schrodinger operator for a very general class of domains. Examples
include the potential -c|x|^{-2} in n dimensions for n>2.
Our methods also apply to operators with fractional potential replacing
the Laplacian. These operators relate to alpha-stable Levy processes in
the same way that the Laplacian relates to Brownian motion.
=================================================================
Tuesday, May 6, 4:00 p.m., VV B139
Lillian Pierce (Princeton University)
TITLE: Discrete analogues in harmonic analysis
ABSTRACT:
Recently there has been increasing interest in discrete analogues of
classical operators in harmonic analysis. A few discrete analogues
can be handled immediately by direct comparison with the classical
continuous case, but many others present significant difficulties
unique to the discrete setting. This talk will describe a menagerie
of new results for discrete operators, including twisted discrete
singular Radon transforms (in both the translation invariant and
quasi-translation invariant settings), discrete analogues of
fractional integral operators along lower dimensional quadratic
surfaces, and a discrete analogue of fractional integration on the
Heisenberg group. Although these are genuinely analytic results, key
aspects of the methods come from number theory; this talk will
highlight the roles played by theta functions, Waring's problem, the
Hypothesis K* of Hardy and Littlewood, and the circle method.