## Index Theory

#### A brief history of the subject:

The index theorem of Atiyah and Singer is a milestone of modern mathematics. This result computes a global invariant , namely the index ( = dimention kernel-dimension cokernel ) of an elliptic differential operator on a compact manifold, in terms of the local data defined by the operator and the manifold . Special cases of this theorem include results such as the Gauss-Bonnet-Chern theorem , Hirzebruch's signature theorem, the Riemann-Roch-Hirzebruch theorem and identification of A-genus of a spin manifold as the index of the Dirac operator.

The goal of this course is to present the " heat equation proof" of the index theorem for generalized Dirac operators ( so far the most important elliptic operators ).

**Marking Policy: ** Student are expected to prepare and present reports on selected topics of current
interest . I will choose these topics in consultation with each student .

#### References:

[1] Elliptic operators, topology and asymptotic methods, by J. Roe, published by Longman, 1988.

[2] Heat Kernels and Dirac operators , by N. Berline, E. Getzler and M. Vergne ,Springer, 1992.