Index Theory of Non-compact $G$-manifolds

TL;DR

This paper reviews the Atiyah-Singer index theorem, its extension to transversally elliptic operators, and recent progress in generalizing the index theorem to non-compact G-manifolds, highlighting its mathematical significance and developments.

Contribution

It surveys classical and recent generalizations of the index theorem, focusing on non-compact G-manifolds and transversally elliptic operators, advancing understanding in geometric analysis.

Findings

Review of classical Atiyah-Singer index theorem

Discussion of transversally elliptic operators

Overview of recent generalizations to non-compact manifolds

Abstract

The index theorem, discovered by Atiyah and Singer in 1963, is one of most important results in the twentieth century mathematics. It found numerous applications in analysis, geometry and physics. Since it was discovered numerous attempts to generalize it were made, see for example [5, 3, 4, 16, 12] to mention a few; some of these generalizations gave rise to new very productive areas of mathematics. In this lectures we first review the classical Atiyah-Singer index theorem and its generalization to so called transversally elliptic operators (Atiyah, 1974) due to Atiyah and Singer. Then we discuss the recent developments aimed at generalization of the index theorem for transversally elliptic operators to non-compact manifolds, [24, 10].