Notes on the Atiyah-Singer Index Theorem

This is arguably one of the deepest and most beautiful results in modern geometry, and in my view is a must know for any geometer/topologist. It has to do with elliptic partial differential operators on a compact manifold, namely those operators P with the property that dim ker P, dim coker P < ∞. In general these integers are very difficult to compute without some very precise information about P. Remarkably, their difference, called the index of P , is a " soft " quantity in the sense that its determination can be carried out relying only on topological tools. You should compare this with the following elementary situation. Suppose we are given a linear operator A : C m → C n. From this information alone we cannot compute the dimension of its kernel or of its cokernel. We can however compute their difference which, according to the rank-nullity theorem for n × m matrices must be dim ker A − dim coker A = m − n. Michael Atiyah and Isadore Singer have shown in the 60's that the index of an elliptic operator is determined by certain cohomology classes on the background manifold. These cohomology classes are in turn topological invariants of the vector bundles on which the differential operator acts and the homotopy class of the principal symbol of the operator. Moreover, they proved that in order to understand the index problem for an arbitrary elliptic operator it suffices to understand the index problem for a very special class of first order elliptic operators, namely the Dirac type elliptic operators. Amazingly, most elliptic operators which are relevant in geometry are of Dirac type. The index theorem for these operators contains as special cases a few celebrated results: the Gauss-Bonnet theorem, the Hirzebruch signature theorem, the Riemann-Roch-Hirzebruch theorem. In this course we will be concerned only with the index problem for the Dirac type elliptic operators. We will adopt an analytic approach to the index problem based on the heat equation on a manifold and Ezra Getzler's rescaling trick. Prerequisites: Working knowledge of smooth manifolds, and algebraic topology (especially cohomology). Some familiarity with basic notions of functional analysis: Hilbert spaces, bounded linear operators, L 2-spaces. Part II. The statement and some basic applications of the index theorem, [19]. About the class There will be 3-4 homeworks containing routine exercises which involve the basic notions introduced during the …