Topics in Spectral Geometry
Student No.:50
Time:【Schedule updated】Mon/Wed 10:10-12:00
Instructor:Richard Schoen  [Stanford University]
Place:Conference Room 4, Floor 2, Jin Chun Yuan West Building
Starting Date:2013-3-18
Ending Date:2013-5-22



【Schedule updated】Lecture on April 8 will be changed to April 9 at 13:00-14:50. Lectures on April 15 and 17 will be canceled. 



We will begin with some of the classical work on eigenvalues forRiemannian manifolds including the basic upper and lower bounds, the Lichnerowicz theorem, the heat kernel, and connections with isoperimetric quantities such as the Cheeger constant. We will describe a recent method by B. Andrews and J. Clutterbuck using the heat equation to obtain a sharp bound on the fundamental gap and a sharp lower bound for Riemannian metrics. We will then discuss work from recent papers on the determination of metrics of fixed area on surfaces which maximize the first eigenvalue.



Basic Riemannian geometry, linear elliptic and parabolic PDE



A reference for the classical material is chapters III and IV of the book "Lectures on Differential Geometry" by R. Schoen and S. T. Yau. References for the other material will be posted.