Speaker: Xianzhe Dai, University of California, Santa Barbara

Time: 20:00-21:00 pm, October 20, 2022, GMT+8

Venu: Zoom Meeting ID: 891 8743 0260 Password: 654321

Abstract:

The classical Weyl's law describes the asymptotic behavior of eigenvalues of the Laplace Beltrami operator in terms of the geometry of the underlying space. Namely, the growth order is given by (half of) the dimension and the limit by the volume. The study has a long history and is important in mathematics and physics.  In a very recent joint work with S. Honda, J. Pan and G. Wei, we find two surprising types of  Weyl's laws for some compact Ricci limit spaces. The first type could have power growth of any order (bigger than one). The other one has an order corrected by logarithm as some fractals even though the space is 2-dimensional. Moreover, the limits in both types can be written in terms of the singular sets of null capacities, instead of the regular sets.  These are the first examples with such features for Ricci limit spaces. Our results depend crucially on analyzing and developing important properties of the examples constructed by J. Pan and G. Wei (GAFA 2022).

Biography:

Xianzhe Dai is a professor at the Department of Mathematics at the University of California, Santa Barbara. He specializes in Differential Geometry and Geometric Analysis, in particular the Atiyah-Singer index theory.  He obtained his PhD in 1989 from Stony Brook University. After a C. L. E Moore Instructorship at MIT, he went to the University of Southern California before moving to UCSB. He was a Sloan Research Fellow from 1993 to 1995.

Source: SRMC