Colloquium

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Organizer(s) ABE Noriyuki, IWAKI Kohei, KAWAZUMI Nariya (chair), KOIKE Yuta
URL https://www.ms.u-tokyo.ac.jp/seminar/colloquium/index_e.html

2008/05/23

16:30-17:30   Room #123 (Graduate School of Math. Sci. Bldg.)
Jean-Michel Bismut (Univ. Paris-Sud, Orsay)
Functional integration and index theory

[ Abstract ]
The heat equation proof of the Atiyah-Singer index theorem involves a local `fantastic cancellation' mechanism, which has long been unexplained conceptually.

In this lecture, I will show how the supersymmetric formalism introduced by physicists has ultimately led to a new understanding of this cancellation mechanism. Ideas of Witten and Atiyah relating the index theorem to the localization formulas of Duistermaat-Heckman in equivariant cohomology have ultimately led to a renewed understanding of the cancellation mechanism as being of geometric nature (albeit in infinite dimensions). The key fact is that when interpreting the heat equation method for the proof of the index theorem, integrals of measures on the loop space of the given manifold, which one obtains via Ito stochastic calculus, should be properly interpreted as integrals of differential forms on the loop space.

I will then explain how this new understanding of the local index theorem has naturally led to a better understanding of spectral invariants, and often to the proof of certain key properties.