Math
191b
Section 1 

Topics
in Analysis and Geometry 

Winter 201415  

Instructor:
Vlad Markovic, 364 Sloan, 6263954350, markovic @ caltech
dot edu 



Description:
This
will
be a seminar style course in which various topics (and a
number of open problems that vary in difficulty) will be discussed. We will start with the Theory of Harmonic Mappings between manifolds and its applications in other fields, like Teichmuller Theory and Topology. In particular, I will explain the Heath Flow Equation which is used to construct harmonic mappings. We will discuss another time evolution equation called the Euler Equation that governs the motion of an ideal fluid. The very basics of this theory will be presented and we will derive the known invariants of the ideal fluid flow. In the second part of the course I will talk about the dynamics of geodesic flows on hyperbolic manifolds, and present some new and old theorems about counting various dynamically constructed objects that are significant in topology like closed geodesics and surfaces inside a higher dimensional manifold. We will also talk about similar problems in higher rank symmetric spaces and explain this perspective. 






