M 392C Geometry/Topology/Physics (54370)

 

Dan Freed

W 4pm – 7pm

RLM 9.166

 

Course Description: The Dirac operator is a first-order linear elliptic differential operator, originally introduced in Lorentz signature, but with many incarnations and applications in Riemannian geometry, differential topology, and beyond. On a compact manifold, or family of compact manifolds, a Dirac operator has many topological and geometric invariants. The most basic is the Fredholm index, which only depends on the kernel of the operator. The topological formula for the index is the Atiyah-Singer index theorem. Geometric invariants, such as the Atiyah-Patodi-Singer eta-invariant and determinant line bundle, are constructed from the entire spectrum of the Dirac operator, and there are geometric versions of the index theorem which pertain. These invariants have many contemporary applications in geometry and physics.

Students in the course will give many of the lectures; the instructor will provide materials and coaching.