Education, Master Class, Topics 2003/2004

Master Class 2003/ 2004

Noncommutative Geometry

Noncommutative Geometry: N.P. Landsman, I. Moerdijk
Period: October 2003 - June 2004

Noncommutative geometry is a recent development in pure mathematics, which combines and to some extent unifies such diverse areas of mathematics as algebra, topology, geometry, and functional analysis in a powerful way. Here is the guiding principle. Algebraic topology and differential geometry provide tools such as topological K-theory, de Rham cohomology, differential calculus on manifolds, and index theory, which can be reformulated purely algebraically in terms of the commutative ring of continuous or smooth functions on the underlying space. Motivated mainly by problems in quantum mechanics and in the theory of foliated manifolds, these tools have successfully been extended to certain non-commutative algebras. This development has led to "non-commutative geometry", a subject which has now acquired a life of its own.

In the noncommutative setting, the first three tools listed above are replaced by the K-theory of C*-algebras, cyclic cohomology, and spectral triples, respectively, which in combination lead to deep noncommutative analogues of the Atiyah-Singer index theorems. These techniques are ultimately based on the theory of operator algebras and on homological algebra, whereas many interesting examples in noncommutative geometry come from Lie groupoids and their associated Lie algebroids and convolution algebras. This program has mainly been formulated and developed by Alain Connes, and anyone interested in this course should first have a look at his amazing book Noncommutative Geometry from 1994.
WHAT DO WE OFFER? The aim of the master class is to bring students to a level at which they understand the techniques and ideas in Connes' book as well in more recent research literature. This goal will be achieved through a program involving 6 one-semester lecture courses and one intense full-year seminar, at which the students themselves are the main speakers.
WHOM ARE WE LOOKING FOR? Applicants should have completed at least three years of undergraduate studies in pure mathematics or theoretical and mathematical physics. It is particularly important that they have finished first courses in algebra, functional analysis and Hilbert spaces, topology, and differential geometry.
Courses We offer two semesters of three courses and a seminar each. Most of these will take place at the University of Utrecht, some activities might take place at the University of Amsterdam (which can be reached from Utrecht by a train journey of less than 30 min.). Some courses will be accompanied by exercise classes.