Geometry and Quantum Theory (GQT)


The GQT colloquium takes place on the final two days, June 7 and 8, and consists of nine research talks. These are attended by Ph.D. students, members of the GQT staff, and visitors from abroad. We aim at organizing research talks which feature topics covered during the school, though some of the talks are unrelated.

The program is as follows:

Thursday 7 June

9:00-10:00: arrival
10:00-11:00: Lennart Meier (Utrecht)
11:30 -12:30: Arne Smeets (Nijmegen)
14:00-15:00: Vito Zenobi (Göttingen)
15:30-16:30: Mario Garcia Fernandez (ICMAT Madrid)
19:00: dinner

Friday 8 June
9:00-10:00: Niek de Kleijn (Copenhagen), Deformation Quantization and the Algebraic Index Theorem
10:15 – 11:15: Thomas Grimm (ITF, Utrecht)
11:30 – 12:30: Jens Kaad (Syddansk Universitet Odense)
14:00-15:00: Claire Debord (Université Clermont-Auvergne)
15:30 – 16:30: Ralph Klaasse (ULB)



Niek de Kleijn (Copenhagen)

Deformation Quantization and the Algebraic Index Theorem

There are many roads that lead to the index theorem. In this talk I will walk the algebraic road. This roads starts with the work of Fedosov and was fully realized by Nest–Tsygan in the 90’s. The main idea comes form the fact that pseudo-differential operators, or at least their symbol algebra, may be obtained through deformation quantization. Since the index of a pseudo-differential operator depends only on its (principal) symbol one may try to formulate the index theorem (and its proof) using merely the symbol algebra. This leads to the algebraic version of the index theorem phrased in terms of cyclic homology and obtained for deformation quantizations of more general manifolds than cotangent bundles. In this talk I will give a history of the theorem. This will lead us through the topics of deformation quantization and formal geometry and will have us ending up in recent mathematical research.