Colloquium

Spectral dimensions of Laplacians defined by

measures with overlaps and some applications

 Dr. Sze-Man Ngai

Georgia Southern University and Hunan Normal University

 Abstract: The spectral asymptotics of a Laplacian has been shown to be closely related to heat kernel estimates, which under suitable conditions determine whether wave propagates with finite or infinite speed. We observe that some self-similar measures defined by finite or infinite iterated function systems with overlaps are ``essentially of finite type", which allows us to extract useful measure-theoretic properties of iterates of the measure. We describe methods to obtain, under this condition, a closed formula for the spectral dimension of the Laplacian. We will mention some applications to heat kernel estimates, wave propagation speed, and computing the Lq-spectrum of the measure. Part of this work is joint with Qingsong Gu, Jiaxin Hu, Wei Tang, and Yuanyuan Xie.

Bio: Sze-Man Ngai received his Ph.D. from University of Pittsburgh in 1995. After receiving his Ph.D., he held research and teaching positions at The Chinese University of Hong Kong, Cornell University, and Georgia Institute of Technology. He joined the Department of Mathematical Sciences of Georgia Southern University in 2000. His current research interests include the multifractal formalism of fractal measures, self-affine tiles, and fractal differential equations.