MS Seminar (Mathematics - String Theory)

Speaker: Raphael Ponge (Seoul National University)
Title: Noncommutative Geometry and the Hirzebruch-Riemann-Roch Formula
Date: Wed, Aug 31, 2011, 15:30 - 17:00
Place: Seminar Room A
Related File: 497.pdf
Abstract: One aim of noncommutative geometry (in the sense of Alain Connes) is to translate the classical tools of differential geometry in the Hilbert space formalism of quantum mechanics. It is then expected that noncommutative geometry will enable us to deal with a variety of problems whose "noncommutative nature" prevents us from using the usual tools of differential geometry (e.g., quantum space-time, diffeomorphism-invariant geometry).

The Hirzebruch-Riemann-Roch formula gives a geometric expression for the holomorphic Euler characteristic of a compact complex manifold in terms of the integral of characteristic forms. It can be seen as a special case of the local index formula of Atiyah-Singer, which ultimately holds in the framework of noncommutative geometry.

In this talk, it is planned to report on on-going projects about using noncommutative geometry to study the following geometric settings:

- Biholomorphism-invariant geometry of (strongly pseudoconvex) complex domains, i.e., the geometry in the presence of the action of an arbitrary group of biholomorphisms of such a domain.

- Contactomorphism-invariant geometry of contact manifolds, i.e., the geometry in the presence of the action of an arbitrary group of contactomorphisms of a given contact manifold.

The main goal is to reformulate the Hirzebruch-Riemann-Roch formula and the Atiyah-Singer index formula in these settings.