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Math 257A-topics on differential geometry
Spring, 2011


Course Description

Parabolic PDE, besides its connection with physics, probability, applied mathematics, has been a very effective tools in solving problems in geometric analysis. In this course we plan to cover several important old and recent progresses in geometric evolution equations. The topics include Weyl's asymptotics on eigenvalues via heat kernel, Andrews-Clutterbuck's recent parabolic approach on the fundamental gap conjecture, new parabolic proof on Li-Yau-Zhong-Yang's eigenvalue estimate, dimension estimates of the spaces of holomorphic functions of polynomial growth, etc. The underline theme is various important estimates on the gradient and continuity modulus of solutions to the parabolic equations. The nonlinear geometric PDEs such as the mean curvature flow and Ricci flow will also be introduced. Even though we shall start with the maximum principle and other basics on parabolic PDE, we will cover the most recent development and the very current technology of the subject.

There will be no final exam.

The complete course schedule will be available.


Instructors

Name Office E-mail Phone Office Hours
Ni, Lei AP&M 5250 lni@math.ucsd.edu 534-2704 MWF 11:00-11:50am


Course Time and Location

Section Instructor Time Place
A00 Ni MWF 10:00-10:50 am APM 5829


Texts

Text: (1) Linear and quasilinear equations of parabolic type, O.A. Ladyzenskaya, V. A. Solonnikov and N. N. Uralceva; (not required)

(2) Partial differential equations of parabolic type, A. Friedman; (not required)

(3) Hamilton's Ricci flow, B. Chow, P. Lu and L. Ni. (not required)


Exams

There will be no exam.

Handout

Tauberian Theorem


Schedule

The course schedule will be available.


Grades

Grades will be based on the following percentages.
Homework 100%

Last modified: Wed July 29, 14:49:08 PST 2009