CONNES' NONCOMMUTATIVE GEOMETRY

**Coming talk :**

**
It is yet unclear whether/when the seminar will start up again during fall
2004 !!!
**

** 11.12.03** : Erik Alfsen talked about Clifford algebras (II).

** 26.11.03** : Erik Alfsen talked about Clifford algebras (I).

**06.11.03 ** : Per Holm gave an introduction to Clifford
algebras.

**22.10.03 ** : Nadia Larsen talked about Morita equivalence and
C*-algebras.

**22.05.03** : Sergey Neshveyev talked
about Fredholm operators on C*-modules and

the non-commutative
Atiyah-Janichindex theorem (cf. 4.1-4.4 in ENG).

**08.05.03** : John Rognes talked about topological K-theory and
Bott
periodicity (cf. 3.2.3.3 and 3.6 in ENG).

** 03.04.03** : Erik B. continued to talk about
K-theory for C*-algebras. (cf. 3.1-3.2 in ENG).

** 27.03.03** : Kjetil Røysland talked about
K-theory for C*-algebras from a module point of view (cf. 3.1 in ENG).

** 20.03.03** : Erik B. continued to talk about C*-modules (cf. 2.5
in ENG).

** 13.03.03** : Erik B. talked about C*-modules (cf. 2.5 in ENG).

** 27.02.03** : John Rognes talked about vector bundles, topological
K-theory and algebraic K-theory.

**20.02.03** : Kjetil Røysland talked about vector bundles
and the Serre-Swann theorem (cf. 2.1-2.3 in ENG).

**13.02.03** : Erik B. finished his review of some fundamentals
apects of C*-algebra theory
(cf. 1.3 and 1A in ENG).

**06.02.03** : Erik B. continued his review of C*-algebra theory
(cf. 1.2, 1.3 and 1A in ENG).

**30.01.2003 :** For the ease of participants not familiar with
C*-algebras, Erik Bédos began the seminar with a review of
some of their fundamental aspects, emphasizing the commutative case (cf.
1A, 1.1 and 1.2 in the book "ENG").

Connes' noncommutative geometry deals with the unification of mathematics under the aegis of the quantum apparatus, that is, the theory of operators and C*-algebras. In recent years it has been a rich topic of research with discoveries leading to an increasing number of applications in mathematics and theoretical physics.

The recent book "Elements of Noncommutative Geometry" (ENG) by J.M. Gracia-Bondía, J. Várilly and H. Fuigeroa (Birkhäuser, 2001, 685 p. ISBN 0-8176-4124-6) is meant to be a gentle introduction to the language and techniques of noncommutative geometry, and covers some of the core topics of this subject (e.g. Fredholm operators on C*-modules, noncommutative integral and differential calculi, spectral triples).

The aim of this working seminar is to give the opportunity to its participants to learn more about noncommutative geometry by lecturing for each other from this book. It is divided in four parts ( called "topology", "calculus and linear algebra", "geometry" and "trends") and it seems realistic that one should use about one semester on each part to be able to absorb thoroughly the presented material. This semester (Spring 2003) will therefore be devoted to part I, that is, to foundational topics in noncommutative topology : spaces and vector bundles from a "noncommutative" point of view, some aspects of K-theory, Fredholm operators on C*-modules. This part of the book requires some basic knowledge in topology and in functional analysis. Some familiarity with the basics of C*-algebras will also be helpful. However, some of the basics results are reviewed in the first chapter while some others are dealt with in an appendix.

The theme for each seminar session will be announced by e-mail. Those who would like to be included in the e-mail list for the seminar should send an e-mail to Erik (bedos@math.uio.no).

The seminar fits nicely within the Institute's Strategic University Program SUPREMA "New contexts in geometry and arithmetic" (2003-2006). One of the goal of this program is to study and understand the diverse constituents that naturally assembly into Connes' recent work on the Riemann hypothesis, and the seminar will hopefully have achieved a part of this goal at the end of 2004.

Erik Bédos and Kjetil Røysland (organizers)

List of Previous talks.