The talks for the whole conference have been videotaped and together with the transparencies will be available soon. After a weekend break, the conference moved to the Institut Henri Poincare (IHP) at the heart of Paris on Monday. The talks on Monday were given by Dirk Kreimer ( Diffeomorphism invariance, locality and the residue: a physicist’s harvest of a friend’s work) ; Ali Chamseddine ( The little key to uncover the hidden noncommutative structure of space-time); Gianni Landi (Quantum Groups and Quantum Spaces are Noncommutative Geometries); Michel Dubois Violette (Moduli spaces for regular algebras) and myself (Hopf cyclic cohomology and noncommutative geometry: some new thoughts and a tribute to Alain).

On Tuesday the theme of the conference moved to von Neumann algebras and was a tribute to Alain Connes’ immense legacy in the subject. The speakers were Anthony Wassermann (Non-commutative geometry and conformal field theory); Vaughan Jones (Operations on planar algebras and subfactors) ; Dietmar Bisch (Free product of planar algebras and inclusions of subfactors); Sorin Popa (Rigidity phenomena in von Neumann algebras of group actions); Stefan Vaes (Explicit computations of all bifinite Connes’ correspondences for certain II_1 factors); and Dimitri Shlyakhtenko (Free entropy dimension, L^2 derivations and stochastic calculus).

Wednesday’s talks were focused on the Baum-Connes conjecture, topology and C*-algebras. They were given by Vincent Lafforgue (Strengthening property (T)); Gennadi Kasparov (A K-theoretic index formula for transversally elliptic operators), Nigel Higson (The Baum-Connes conjecture and the Mackey analogy), Marc Rieffel (A new look at ‘Matrix algebras converge to the sphere’); Guoliang Yu (Higher index theory of elliptic operators and noncommutative geometry); and Paul Baum (Noncommutative algebraic geometry and the representation theory of p-adic groups).

The talks on Thursday were given by Joachim Cuntz (C*-algebras associated with the ax+b-semigroup over N); Uffe Haggerup (Connes’ classification of injective factors seen from a new perspective); Dan Voiculescu (Free analysis: relativistic quantum opportunities); Alain Connes (Thermodynamics of endomotives and the zeros of zeta); and Henri Moscovici (Spectral geometry of noncommutative spaces).

The last lecture on Thursday was delivered by Henri Moscovici , a long time collaborator and friend of Alain. Henri gave a nice survey of some of the developments in Connes’ work in noncommutative geometry directly inspired by index theory, and the transverse geometry of foliations. As we know one of the offsprings has been the discovery of cyclic cohomology in 1981 and Hopf cyclic cohomology in 1998.

The conference on Thursday ended with a reception at Jussieu and the dedication of a birthday gift to Alain, a telescope to celebrate his far sighted visionary work!

There were four talks in the last day of the conference on Friday, all by former students of Alain.

Alain Valette talked about “Proper isometric actions on Hilbert and Banach spaces” a piece of geometric group theory which is inspired by the Baum-Connes conjecture. Then George Skandalis talked about “Holonomy groupoid and C*-algebra of a foliation” and showed that many aspects of the interaction between noncommutative geometry and foliation theory generalize to the set up of singular foliations.

At the end of the talk of Georges the whole audience gave him and the other organizers a big ovation for the splendid organizing job.

The talk of Marc Rosso was on “quantum groups and algebraic combinatorics”. The last talk of the conference was given by Jean-Benoit Bost who talked on “Diophantine approximation and noncommutative geometry”. He sketched his conceptual geometric approach to Diophantine approximation as a theory of characteristic numbers in the context of Arakelov geometry and some intriguing analogies with noncommutative geometry and elliptic theory on noncommutative spaces.

The talks were a perfect testimony of the health of the subject of noncommutative geometry with so many interactions with other fields including operator algebras, physics, analysis, topology and number theory which is the most recent open frontier. The conference was a marvelous tribute to the breath and depth of Alain Connes’ contributions to mathematics.