Math 211a: Topics in Differential Geometry and Analysis

This semester, Math 211a is going to be about the Atiyah-Singer index theorem. This is a remarkable result at the interface of analysis, topology, and differential geometry, that has been used in fields as diverse as algebraic geometry, number theory, and mathematical physics.

My plan is to start at the beginning, and to follow the beautiful book of John Roe, called `Elliptic Operators, Topology, and Asymptotic Methods' (2nd edition). It is available in the bookstore, and is well worth owning a copy. (If you want to use a copy from the library, make sure you use the second edition, not the first; they're pretty different.) The prerequisites for the course are the basic first-year courses in Analysis, Topology and Algebra, as well as a working knowledge of manifolds (110a; 110b would be helpful as well). Although Roe's book is pretty much self-contained, I will develop additional material as required.

The index theorem is a theorem about the solutions to certain differential equations, defined by elliptic differential operators. Roe's book gives the basic analytical properties of these operators, for a special (but fundamental) case of Dirac operators (this will be explained at some length). If you took Dmitry's course last year, you should have seen some of this material in a more general setting; some special properties of Dirac operators permit a more direct treatment. The proof of the index theorem is via the so-called heat equation method, which combines some basic functional analysis and PDE theory, with some nice algebraic manipulations due to Getzler. We will study applications and assorted other topics in the area, as time permits.

The class is scheduled for MWTh, 10-11, in room 226. I would like to keep this schedule, unless there is mass opposition. In any event, the first class will be Thursday, Aug. 29, 10 am, room 226.