Last week there was a meeting on spectral geometry and index theory at the Universidad de Los Andes in Bogotá, Colombia, organized by Alexander Cardona and Jean Carlos Cortissoz. The Encuentro “Geometría Espectral y Teoría del Índice” consisted of four afternoons of lectures and research talks, and this light but effective setup made it a very pleasant meeting. Even more because of the beautiful surroundings and interesting city.
The lectures were organized in three series, and were aimed at an undergraduate/graduate audience. Steven Rosenberg (Boston University) lectured about the Atiyah-Singer index theorem, Alexander Cardona about Index theorem for deformation algebras and I lectured on the Connes-Moscovici index theorem in noncommutative geometry. In his lectures, Cardona gave an overview of Fedosov’s deformation quantization of symplectic manifolds, followed by Fedosov’s index theorem, connecting this with the b+B-cocycle constructed by Connes, Flato and Sternheimer. The lectures of Rosenberg and myself followed a similar pattern: starting with the minimum but required preliminaries, we arrived at the statement of the two index theorems. Then, after briefly sketching their proofs, we discussed some applications, notably in the computation of the dimension of moduli spaces (both in the commutative and noncommutative case) and to quantum groups.
Besides the lectures, there were three very interesting research talks by Leonardo Cano (Universität Bonn) on “Spectral deformations of the Laplacian on manifolds”, Monika Winklmeier (Universidad de Los Andes) “On the spectrum of the Klein-Gordon Operator” and by Andrés Vargas (Universität Bonn) on “Geometry and pinching of spin manifolds”. The week ended with the talk by Steven Rosenberg in the mathematics colloquium, on index theorems on loop spaces.
In conclusion, I think that with the many bright master students and the newly started Ph.D. program, the Universidad de Los Andes – and in particular the math department – has still a lot more to offer in the near future!