Mathematics Colloquium

Upcoming Colloquium

Fundamental Gap Estimate for Convex Domains
Dr. Guofang Wei, UC Santa Barbara

February 10, 2023
12:00pm-1:00pm

Image
Guofang Wei

This talk will be held via Zoom.

Join 2/10 Zoom
Meeting ID: 910 8240 4817
Passcode: 073879

Abstract

The fundamental (or mass) gap refers to the difference between the first two eigenvalues of the Laplacian or more generally for Schrödinger operators. It is a very interesting quantity both in mathematics and physics as the eigenvalues are possible allowed energy values in quantum physics. In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap conjecture that difference of first two eigenvalues of the Laplacian with Dirichlet boundary condition on convex domain with diameter D in the Euclidean space is greater than or equal to 3π2/D2. In several joint works with X. Dai, Z. He, S. Seto, L. Wang (in various subsets) the estimate is generalized, showing the same lower bound holds for convex domains in the unit sphere. In sharp contrast, in recent joint work with T. Bourni, J. Clutterbuck, X. Nguyen, A. Stancu and V. Wheeler (a group of women mathematicians), we prove that there is no lower bound at all for the fundamental gap of convex domains in hyperbolic space in terms of the diameter. Very recently, jointed with X. Nguyen, A. Stancu, we show that even for horoconvex (which is much stronger than convex) domains in the hyperbolic space, the product of their fundamental gap with the square of their diameter has no positive lower bound. All necessary background information will be introduced in the talk.

Bio Sketch

Guofang Wei is a mathematician in the field of differential geometry and geometric analysis. She is a professor at the University of California, Santa Barbara. Professor Wei earned a doctorate in mathematics from the State University of New York at Stony Brook in 1989, under the supervision of Detlef Gromoll. In 2013, she became a fellow of the American Mathematical Society, for "contributions to global Riemannian geometry and its relation with Ricci curvature."

The research of Professor Wei has been concentrated on global Riemannian geometry--the interaction of curvature with the underlying geometry and topology which includes the study of the fundamental groups, comparison geometry, manifolds with integral curvature bounds, spaces with weak curvature bounds, the eigenvalue of Laplacian, and more.