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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.
Brochure
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