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Integration by localisation in finite and infinite dimensions

Wednesday, 17 March, 2021 - 16:00

Joint SISSA-ICTP Colloquia
Integration by localisation in finite and infinite dimensions
Wednesday 17 March 2021 - 16:00
Anton Alekseev - Université de Genève, Switzerland

Integral calculus is an art. One of the most surprising techniques in the calculation of multi-dimensional integrals is called localisation. In a typical example of localisation, an integral is presented as a sum of a finite number of simple contributions associated to fixed points of an action of a compact group (e.g. a circle) on the integration domain.

Localisation was discovered by Duistermaat and Heckman in their study of symplectic geometry of coadjoint orbits. They showed that certain oscillatory integrals can be computed exactly by taking the first two terms of their stationary phase expansion. Berline and Vergne, and Atiyah and Bott, gave a conceptual explanation of this phenomenon in terms of equivariant cohomology.

In this talk, we will start with some simple low dimensional examples, and then we will consider an infinite dimensional example of coadjoint orbits of the Virasoro algebra. Elements of Virasoro coadjoint orbits can be thought of as Schroedinger operators on the circle. Recently, Stanford and Witten considered formal Duistermaat-Heckman localisation formulas for the corresponding orbital integrals. If time permits, we will discuss possible mathematical interpretations of these formulas.

Based on a joint work with S. Shatashvili.