Colloquium - Guofang Wei

Ohio State Garden of Constants
March 20, 2025
3:00PM - 4:00PM
EA0160

Date Range
2025-03-20 15:00:00 2025-03-20 16:00:00 Colloquium - Guofang Wei Guofang Wei  University of California, Santa BarbaraTitleFundamental Gap and Log-concavity of First EigenfunctionAbstractThe fundamental (or mass) gap refers to the difference between the first two eigenvalues of the Laplacian or more generally for Schr\"{o}dinger operators. It is a very interesting quantity both in mathematics and physics as the eigenvalues are possible allowed energy values in quantum physics. The log-concavity of the first eigenfunction plays an important role in the fundamental gap estimate. For convex domains in the Euclidean space, Brascamp-Lieb proved the log-concavity of the first Dirichelt eigenfunction back in the 70s.  Since then there have been many works in this direction. We will survey many recent fantastic results for convex domains in Euclidean spaces, spheres, hyperbolic spaces and surfaces with positive curvature with Dirichlet boundary conditions, starting with the breakthrough of Andrews-Clutterbuck.  Then we will present a very recent work on horoconvex domains in the hyperbolic space. This last result is joint with Ling Xiao. All necessary background information will be introduced in the talk.    EA0160 America/New_York public

Guofang Wei  
University of California, Santa Barbara

Title
Fundamental Gap and Log-concavity of First Eigenfunction

Abstract
The fundamental (or mass) gap refers to the difference between the first two eigenvalues of the Laplacian or more generally for Schr\"{o}dinger operators. It is a very interesting quantity both in mathematics and physics as the eigenvalues are possible allowed energy values in quantum physics. The log-concavity of the first eigenfunction plays an important role in the fundamental gap estimate. For convex domains in the Euclidean space, Brascamp-Lieb proved the log-concavity of the first Dirichelt eigenfunction back in the 70s.  Since then there have been many works in this direction. We will survey many recent fantastic results for convex domains in Euclidean spaces, spheres, hyperbolic spaces and surfaces with positive curvature with Dirichlet boundary conditions, starting with the breakthrough of Andrews-Clutterbuck.  Then we will present a very recent work on horoconvex domains in the hyperbolic space. This last result is joint with Ling Xiao. All necessary background information will be introduced in the talk.  

 

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