Non Commutative Geometry
Conference in honor of Alain Connes

Paris, 2007, March 29th to April 6th

Objectives


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 physics.

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 directions.

  • 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.