Quid of applications of the Atiyah-Singer index theorem

par Mathis Alleysson

mercredi 17 sept. 2025, 18:00 → 20:00 Europe/Paris

129 Room Picard (IMT 1R2)

In the early 1960s, Atiyah and Singer published their celebrated index theorem, introducing a powerful framework in which the index of an elliptic operator can be expressed using purely topological data. Their work also built a connection between two previously separate areas of mathematics: the analytic theory of elliptic differential equations and the topology of manifolds. Numerous generalizations and related results have arisen from this theorem, making it one of the foundational starting points of index theory.
The story of this theorem begins with the observation that the analytic index of an elliptic operator is invariant under continuous deformations. This invariance suggested that the index had a topological origin. In this talk, the focus will be on the topological index. Our goal is to clarify how this index is defined (making use of connections and characteristic classes) and how it can be computed in practice. If time allows, we will also explore how classical results such as the Gauss-Bonnet and Riemann-Roch theorems appear as consequences of such computations.