Project C: Non-commutative geometry and probability

In this part of our research program, geometry, groups and their actions are studied with methods related to analysis. Actions are studied using concepts from functional analysis, non-commutative geometry, ergodic theory or probability. Groups are studied using their representations on Hilbert spaces and by associating algebras of operators. Differential and algebraic geometry is generalized to the noncommutative setting using generalized "algebras of functions". Dynamical systems and processes are studied from the point of view of probabilistic or ergodic properties. Many of the projects in this research area have close connections to problems and structures studied in number theory or in topology and thus to the research areas A and B.
One of the starting points for projects C1, C2, C3 is the fact that one can associate algebras with geometric objects, groups, group actions or groupoids. These can be algebras of functions or convolution algebras or a mixture of both. They are usually C*-algebras but in some cases also locally convex algebras and encode important properties of the original objects. On the other hand, the study of the algebras often allows to perform the investigation in a framework which includes generalized versions of geometry, groups etc., like non-commutative geometry or quantum groups.
Project C1 studies actions on spaces constructed algebraically, such as adele spaces or group duals. The ergodic properties of such actions and the algebras associated with them have intriguing structures. For instance, the algebra constructed from an affine action on the space of finite adeles over a global field can also be described using generators labeled by the elements of the ring of integers in the field. An important role in this project is also played by groupoids and duality theory. There are close connections to number theory and to algebra and thus to research area A.
In projects C2 and C3, generalized homology/cohomology theories such as K-theory and its bivariant forms, but also cyclic homology and dimension theory play a leading role. These projects have many points in common with the topology projects in area B (K-theory, isomorphism conjectures, dimension functions). Important objects of study here are group algebras or crossed products. An important aspect is the problem of determining the K-theory of such algebras (Baum-Connes conjecture). This particular problem is parallel to part of the program in B6.
Project C4 is related to C1, C2, C3 in that it studies quantum field theory in the setting of non-commutative geometry. It uses concepts from this theory such as spectral triples and studies quantum field theories defined on a non-commutative version of space-time. These can often be mapped to matrix models with particular properties. The main example is a candidate for a non-perturbative renormalization of an interacting quantum field theory in four dimensions. The project aims at completing the constructive renormalization proof, which requires hard analysis and combinatorics, but also number theory through iterated integrals which evaluate to polylogarithms and zeta functions. The use of combinatorics, of random matrices and of an iteratively generated ring of functions establishes close links to the probability projects C5 and C6. The spectral triple underlying the above quantum field theory is based on supersymmetric quantum mechanics. This connects to topological quantum field theory, which plays a role in area B and can also be linked to nets of von Neumann algebras on the circle, as well as to elliptic cohomology.
Markov chains are an excellent tool from probability theory to investigate the geometric structure of state spaces. They are the common theme of the projects in C5. In particular, we will focus on such situations where the state space of the random walk is itself subject to a random choice. Such situations arise naturally in the theory of random walks in random environments, random walks on random trees, or as well in the study of Markov chain Monte Carlo methods for random measures. This project is, of course, intimately related with the other probability project C6. However, via its structures it is connected to projects focusing on the theory of buildings (part A), while its methods are also used in and partially derived from differential geometry (part B).
C6 is an interdisciplinary project between pure and applied mathematics. The topic is to study certain aspects of sequences of random functions with a focus on the spectrum of random matrices of increasing dimension with stochastically dependent entries and the convergence and stationary regime of iterated function systems (IFS). High dimensional random matrices are connected to free probability and hence to von Neumann algebras as well as to number theory, but also get much attention from theoretical physics. We will try to attack the problems from the ergodic point of view as well as from a probabilistic perspective. Concerning IFS, the goal is to understand their asymptotic behaviour under appropriate contraction conditions on the chosen functions. The project is related to C5 by the use of probabilistic methods, but also to C4, via a combinatorial approach to random matrix theory and to C1 by its connections to von Neumann algebras and ergodic theory.


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