Aspects of index theory

Abstract

Index theory is the theory of linear maps with finite dimensional nullspaces and cokernels. In particular, the index of such a map is the dimension of its nullspace minus that of its cokernel. We survey some aspects of this theory in this thesis. In particular, the first chapter talks about general index theory, the second is a survey of the relations between the analytic and topological aspects of the index. The third talks about the index of pseudodifferential operators and the forth deals with differential complexes. The fifth chapter surveys the Chern-Weil Theorem, thereby building enough material to state the Atiyah-Singer Index Theorem in Chapter 6.