Topological combinatorics and amenability

• Applicant: Professor Dr. Friedrich Martin Schneider
• Subject Area: Mathematics
• Term: from 2015 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 283169120

Amenability ranges among the most powerful and well-recognized concepts in modern mathematics -- drawing exciting connections between group theory, functional analysis, probability theory, topological dynamics, ergodic theory, and combinatorics. Recent developments in Ramsey theory revealed the necessity of a better understanding of amenability for non-locally compact topological groups. The aim of the proposed project is to investigate amenability for topological groups and dynamical systems by means of asymptotic combinatorial properties of the underlying uniform spaces with regard to the respective group actions. In other words, I want to study topological groups and dynamical systems via approximating them by finite combinatorial objects, such as bipartite graphs and simplicial complexes. This approach shall be utilized to establish general topological versions of classical theorems for locally compact groups due to Følner, Tarski, Cohen, and Grigorchuk. From this, I expect both new insight into those classical results (for instance in terms of alternative proofs) and interesting applications to areas where local compactness is a rare property (for example operator theory or infinite combinatorics). Furthermore, I aim at finding a suitable generalization of the Ornstein-Weiss theorem for arbitrary amenable topological groups. This is supposed to serve as a basis for defining dynamical invariants (e.g. topological entropy) for continuous actions of amenable topological groups, which in turn would provide a first step towards a general ergodic theory for amenable topological groups. Since the Department of Mathematics at the University of Auckland encompasses outstanding expertise in general topology, functional analysis, group theory, and combinatorics, the intended research fellowship at Auckland would significantly advance the proposed project and lay the foundations for promising long-term collaborations concerning combinatorial methods in topological group theory.

• DFG Programme: Research Fellowships
• International Connection: New Zealand
• Host: Professor Warren Moors, Ph.D.