# Mathematics Colloquium

## Upcoming Colloquium

**Fundamental Gap Estimate for Convex Domains**

Dr. Guofang Wei, UC Santa Barbara

February 10, 2023

12:00pm-1:00pm

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 3π^{2}/D^{2}. 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.

The Mathematics Colloquium is a unique opportunity for students to learn about new developments in mathematics and what mathematics and statisticians do after they graduate. Hosted by the Department of Mathematics and Statistics at California State University, Long Beach, the weekly meetings invite guests from universities, research laboratories, and industry to present and discuss current topics in mathematics. All students are encouraged to attend*.*

## Schedule

The schedule for Spring 2023 will be posted shortly after the start of the semester.

Dr. Guofang Wei, UC Santa Barbara

### 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 3π^{2}/D^{2}. 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.

## Previous Colloquia

The Mathematics Colloquium Archive has the Colloquia from previous semesters.