Index theory is an operator algebraic approach in the study of the geometric and topological properties of a manifold (or more general geometric objects), by means of some analytical invariants produced by the solutions of elliptic differential operators on the manifold. When the manifold admits a group action, the index theory of the elliptic operators, which commute with the group action, provides ingredients in tackling problems in representation and number theory. In this talk, I shall introduce the ideas of index theory in connection with representation theory and then speak about several indices of invariant elliptic operators, together with aspects in other areas. The talk involves my work on $L^2-index$ formula and joint work with Bai-Ling Wang on orbifold index.
The talk is intended to be introductory and accessible to graduate students.