Syllabus
The goal is to describe the Atiyah—Singer index theorem for
(generalized) Dirac type operators on compact manifold by introducing
all the ingredients from geometry and analysis and derive some important
examples and applications.
The index theorem is a fundamental result going back to the sixties
bridging between analysis and geometry.
Particular cases of the index formula are the Hirzebruch signature
formula, the Gauss-Bonnet formula and Riemann-Roch-Grothendieck.
There are nowadays many proofs of the formula. I’ll talk about the one
based on the heat equation associated to a Dirac operator and the
Getzler rescaling.
It is the most direct and analytical proof requiring the least amount of
tools from algebraic topology and might be interesting also for Ph.D
students in functional analysis. Furthermore it is very flexible. It has
been already generalized to a large number of more complicated geometric
structures.
Program
1. GEOMETRIC PRELIMINARIES: vector bundles and connections,
characteristic classes, some Riemannian geometry, Spin structures and
Clifford modules, differential operators and Dirac type operators.
2. ANALYSIS: basics of Fredholm operators, Generalized Laplacians,
Sobolev spaces and elliptic operators, Hodge theory, the heat equation
associated to a Dirac type operator and the MacKean-Singer formula.
3. Limits of the supertrace of the heat kernel, Getzler rescaling.
4. APPLICATIONS: Hirzebruch signature formula, Chern-Gauss-Bonnet
formula, Riemann-Roch-Hirzebruch.
CONTEMPORARY INDEX THEORY: towards the end of the course Prof. Paolo
Piazza from Rome, La Sapienza will talk about contemporary index theory
and applications.
website: https://paoloanto.wordpress.com
email: pantonini@gmail.com
