Title: MATH 666 Atiyah-Singer Index Theorem
Prerequisites: Topology I and Differential Geometry I, or instructor's approval.
Course Description: Atiyah-Singer index theorem is one of the most important landmarks 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 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 mathematicsand the close companionship between math and physics. In this course I
will focus on the most important case of the index theorem, i.e., the case of Dirac operators. I will follow part of the book of Berline-Getzler-Vergne (Heat kernels
and Dirac operators) which includes an elegant proof of the index theorem using heat equation.
Average time dedicated per week (estimate): 4 hours