Heat Kernels of a Class of Degenerate Elliptic Operators
Speaker:Prof. Der-Chen Chang
Department of Mathematics and Statistics and
Department of Computer Science
Georgetown University
Date & Time:19 Jul 2012 (Thursday) 15:00
Venue:J214
Organized by:Department of Mathematics

Abstract

In this talk, we first discuss the geometry induced by a class of second-order ubelliptic operators. This class contains degenerate elliptic and hypoelliptic operators (such as the Grushin operator and the Baouendi-Goulaouic operator). Given any two points in the space, the number of geodesics and the lengths of those geodesics are calculated. We find modified complex action functions and show that the critical values of these functions will recover the lengths of the corresponding geodesics. We also find the volume elements by solving transport equations. Then heat kernels for these operators are obtained. Finally we link these heat kernels to sharp estimates for Kohn Laplacian on a family of pseudo-convex hypersurfaces.

Biography

Prof. Chang is currently Professor of Department of Mathematics and Statistics and Department of Computer Science of Georgetown University. He obtained his Ph.D. in Princeton University, and is a former student of E.M. Stein. Prof. Chang is studying Fourier analysis and several complex variables. His publications include Heat Kernels for Elliptic and Sub-elliptic Operators: Methods and Techniques, Sub-Riemannian Geometry: General Theory and Examples, Geometric Analysis on the Heisenberg Group and Its Generalizations, Geometric Mechanics on Riemannian Manifolds and Its Applications to Partial Differential Equations and Laguerre Calculus on the Heisenberg Group etc.