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.