Atiyah-Singer Index Theorem
Atiyah-Singer index theorem is a landmark of mathematics of the 20th century. It is regarded as a bridge between two broad branches of mathematics: topology and analysis. It also summarizes several important classical theorems, such as the Riemann-Roch theorem in algebraic geometry and the Gauss-Bonnet-Chern theorem in differential geometry. The development of the index theory was also intertwined with the development of physics, especially quantum field theory. The Atiyah-
Singer index theorem is a manifestation of the unity of mathematics and the close companionship between math and physics.
This course will be a mixture of topology, geometry and analysis. I will start with basic functional analytic properties of Fredholm operators such as the homotopy invariance of the Fredholm index. Then I will move to the discussion of topological K-theory, including the Bott periodicity theorem and the definition of topological index of elliptic operators. Instead of the original K-theory proof of the index theorem by Atiyah and Singer, I will present the analytic proof using heat kernels for the most elegant case of Dirac operators.
The main references are 1. Booss-Bleecker, Topology and Analysis, 2. Berline-Getzler-Vergne, Heat kernels and Dirac operators
1. absolute necessary ones: basic differential geometry on smooth manifolds, basic algebraic topology (homoloy groups), basic functional analysis (Hilbert spaces) 2. preferrable ones: PDE (elliptic equation, Sobolev spaces), more advanced algebraic topology (homotopy groups, characteristic classes of vector bundles)