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Joint SISSA-ICTP Colloquia | Integration by localisation in finite and infinite dimensions

11 March 2021
A seminar by Anton Alekseev, Université de Genève, Switzerland

On Wednesday 17 March at 16:00 , SISSA-ICTP Colloquia are back with a talk by Anton Alexeev, professor of Mathematics at the University of Geneva and winner of the Medal of the Erwin Schrödinger Institute for Mathematics and Physics for the year 2020.

Abstract

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.)

This Colloquium will be held via Zoom webinar.

Pre-registration is required here
After registering, you will receive a confirmation email containing information about joining the webinar.