Series: Mathematics Colloquium

Date: Thursday, December 12, 2002

Time: 4:30 - 5:30 PM

Place: 102 McAllister Building

Host: Ping Xu

Refreshments: 4:00 - 4:30 PM, in 212 McAllister

Speaker: Boris Tsygan, Northwestern

Title: Some Problems in Noncommutative Differential Geometry

Abstract:

Many geometric objects related to a smooth manifold can be defined in
terms of the algebra A of functions on this manifold.  Very often
these objects can be defined in a manner that makes sense for any
algebra A, commutative or not.  Studying associative algebras by such
methods of geometric origin is the subject of noncommutative geometry.
The algebras in question are often themselves of geometric provenance,
usually algebras of operators of certain kind acting on some geometric
object.  Far from being just an extension of classical methods,
noncommutative geometry exhibits striking new phenomena; namely, it
possesses very unusual symmetries.  First of all, in words of Alain
Connes, "Noncommutative spaces evolve with time", i.e. many of them
have a one-parameter group of automorphisms, as was discovered in the
modular theory of factors of type III.  Even more unusually, the very
set of noncommutative calculi possesses a large symmetry group. The
nature of this group depends on the model of calculus.  For one model,
it is a group related to the Galois group of the field of rational
numbers.  For another, it is a renormalization group of physical
origin.  Since Alain Connes' central idea is to find a unified
symmetry group combining the two, a systematic comparison of the two
models of noncommutative calculus seems to be a reasonable course of
action.  We give a survey of recent works on the subject, due to
Connes-Moscovici and to Tamarkin and myself.