The development of quantum physics shows that non-commutativity is a basic feature of nature, giving rise e.g. to the uncertainty principle. The mathematical theory of non-commutative geometry complements this insight and explains, for instance, certain quantization phenomena. It involves the study of operator algebras, symmetries, cyclic homology, and the investigation of isomorphism conjectures. Many problems here are of functional analytic nature. The integration of category theory and homological algebra with functional analysis will be a fundamental new approach to understand algebraic, smooth, and analytic properties of the underlying spaces. This shall be applied to classify von Neumann algebras, which are a basic building block in the modeling of quantum theories. New topological tools for the study of singular spaces will allow to understand topological quantum field theories and to grasp the topological meaning.of their moduli space. Geometrically, isomorphism conjectures are strongly related to geometric rigidity, in particular in the context of geometric group theory, where we thrive for a quasi-isometry classification of discrete groups and their representations.
At the same time, the new rigidification beyond homological phenomena will allow for applications of non-commutative geometry in representation theory and number theory, e.g. for spectral descriptions of zeros of L-functions. The number theoretic investigations will be completed by the study of the asymptotic behavior of Gauss sums and Fourier coefficients of automorphic functions on the one hand, and via arithmetic geometry by the study of rational points on higher dimensional varieties. This shall be fruitfully complemented by numerical experiments, involving also fundamental questions about symbolic calculation and automatic theorem proving.
The center will consist of three research groups in the areas "Non-commutative geometry", "Algebraic geometry and topology", and "Symmetries in number theory". The complementary expertise of the mentors and the members of the research groups will stimulate an intense exchange on these different themes and their applications, in particular their applications in quantum physics. The center will also be part of the thriving research environment given by the GK 535 and its planned successor and will participate in SPP 1154 and the EU Network EU-NCG. Moreover, its research program will be a key element in the framework of a new collaborate research center (SFB) to be established until 2013. Its PhD students will be enrolled in the graduate school GAUSS and profit from its structured course program.
The Center with its unique combination of complementary branches from mathematics and quantum physics thus has the potential of becoming an internationally leading research institution, continuing the fruitful exchange between mathematicians and physicists for which Göttingen is well known.