KMS States and Non-Commutative Geometry

There have recently been new results on extending various aspects of the theory of KMS states from its original context of von Neumann algebras to C*-algebras and understanding the more complex structure that arises in the C*-setting (see for example, Laca-Neshveyev, arXiv:math/0304435 ‘KMS states of quasi-free dynamics on Pimsner algebras’). KMS states lead naturally to type III von Neumann algebras. Noncommutative Geometry has recently been proposed to have application to situations where type III algebras arise. There are currently two points of view, that of Alain Connes and Henri Moscovici (arXiv:math/0609703 ‘Type III and spectral triples’), and that of Carey-Phillips-Rennie (arXiv:0801.4605 ‘Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras’). These developments seem to be relevant to Bost-Connes systems, the Connes-Marcolli program in number theory, proposed applications of semifinite noncommutative geometry to invariants of C*-algebras, Hopf algebras and quantum groups, and KMS theory in NCG generally (see for example Laca-Larsen-Neshveyev, arXiv:0710.3452 ‘On Bost-Connes type systems for number fields’ arXiv:math/0609316 ‘Hecke algebras of semidirect products and the finite part of the Connes-Marcolli C*-algebra’).

These developments represent a very broad range of topics, some quite old and now possibly unfamiliar to many younger researchers, some very recent and as yet not widely known. Thus there is a need for a topical workshop exploring aspects of these recent developments and setting the scene for future research. There was a very successful meeting at the Hausdorff Institute for Mathematics in Bonn in July, 2008 on related topics that made very clear the great potential for rapid progress. The workshop itself is intended for about 40 people, including a number of graduate students and post-doctoral fellows.