Non Commutative Geometry
Conference in honor of Alain
Paris, 2007, March 29th to April 6th
Specifically designed and by
now well-equipped to handle exceptional spaces, whose coordinates do
not commute but which do occur naturally in both mathematics and
physics, Noncommutative Geometry undergoes a period of vigorous growth.
It has been the source lately of many interesting new developments in
several different fields of both core mathematics and mathematical
The goal of this meeting is twofold: to highlight the most significant
recent advances, some of which will be evoked below, and to identify and prefigure new emerging
- The many occurrences in the last several years of
remarkable connections with Number Theory,
such as quantum statistical-mechanical systems with spontaneous
symmetry breaking of arithmetic nature, the spectral realization of the
zeros of L-functions,
modular Hecke algebras and their Hopf symmetries, have been recently
unified under a common geometric framework based on the noncommutative
space of Q-lattices
modulo the commensurability relation.
- In the realm of Theoretical Physics,
the renormalization in perturbative QFT was given a spectacular
mathematical foundation, in terms of a Hopf algebra associated to
Feynman graphs and of the Birkhoff factorization of loops in the
prounipotent complex Lie group underlying this Hopf algebra.
Furthermore, the renormalization group itself was found to be a (1-parameter) subgroup of a
universal group of symmetries of a motivic
nature that governs the structure of the divergences in renormalizable
quantum field theories.
In another development, the noncommutative geometrization of the
standard model was recently enhanced by a `spectral action principle'
which allows the complete recovery of the standard model Lagrangian.
- The traditional connections with Geometry and Topology have
been dramatically reinforced by a series of recent major advances
towards the elucidation of the Baum-Connes and Novikov conjectures, in
which operator K-theory and
cyclic cohomology play a decisive role.
A local index formula in the framework of spectral geometry of
noncommutative spaces was proved, and was successfully applied to
the analysis of the transverse geometry of foliations. In the process,
new Hopf algebras of `quantum symmetries' have been discovered, and the
solution to the transverse index problem turned out to hinge on the
development of a cyclic cohomological apparatus specific to Hopf
algebras. On the analytic side, the theory of locally compact quantum
groups has made significant strides.
- The more traditional, actually historical, interaction with
the Ergodic Theory,
involving group actions on measure spaces, has recently witnessed
spectacular progress, which led to the proof of the first rigidity and
superrigidity results in the general context of operator algebras.
In a related direction, that of Probability Theory,
a series of powerful new tools were invented, based on free analogues
of the classical probabilistical notions, and they were instrumental in
the solution of longstanding problems.