談話会・数理科学講演会

過去の記録 ~07/21次回の予定今後の予定 07/22~

担当者 足助太郎,寺田至,長谷川立,宮本安人(委員長)
セミナーURL https://www.ms.u-tokyo.ac.jp/seminar/colloquium/index.html

2008年05月23日(金)

16:30-17:30   数理科学研究科棟(駒場) 123号室
お茶&Coffee&お菓子: 16:00~16:30 (コモンルーム)
Jean-Michel Bismut 氏 (Univ. Paris-Sud, Orsay)
Functional integration and index theory

[ 講演概要 ]
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.