Geometric Measure Theory and Its Applications
October - December 2012, Tuesday 13:00-15:00 Prof. G-Q Chen.
Geometric Measure Theory has contributed greatly to the development of the calculus of variations, partial differential equations, and geometric analysis, and has important applications to differential geometry, stochastic analysis, dynamical systems, differential topology, mathematical physics, among others. This course is an introduction to some facets of the theory and its applications. The course starts with a brief review on basic measure theory and is followed by the topics including Hausdorff measures, area/co-area formulae, Sobolev functions, BV functions, sets of finite perimeter, divergence-measure fields, Gauss-Green theorems and normal traces, *differentiability and approximation, *varifolds and currents, connections with and applications to nonlinear PDEs (the topics with * are optional, depending on the course development).
Basic real analysis or the equivalent is the only essential prerequisite. However, some familiarity with basic measure theory, functional analysis, and differential geometry is desirable.