Eversince, the study of symmetry in mathematics and mathematical physics has been fundamental
to a thourough understanding of most of the fundamental notions. Group theory in all its forms
is the theory of symmetry and thus an indispensible tool in many of the basic theoretical sciences.
The study of infinite symmetry groups is especially challenging, since most of the tools from the
sophisticated theory of finite groups break down and new global methods of study have to be found.
In that respect, the interaction of group theory and the study of group rings with methods from ring
theory, probability, Riemannian geometry, functional analyis, and the theory of dynamical systems
has been extremely fruitful in a variety of situations. In this proposal, I want to extend this line of
approach and introduce novel approaches to longstanding and fundamental problems.
There are four main interacting themes that I want to pursue:
(i) Groups and their study using ergodic theory of group actions
(ii) Approximation theorems for totally disconnected groups
(iii) Kaplansky’s Direct Finiteness Conjecture and p-adic analysis
(iv) Kervaire-Laudenbach Conjecture and topological methods in combinatorial group theory
The theory of `2-homology and `2-torsion of groups has provided a fruitful context to study global
properties of infinite groups. The relationship of these homological invariants with ergodic theory
of group actions will be part of the content of Part (i). In Part (ii) we seek for generalizations of
`2-methods to a context of locally compact groups and study the asymptotic invariants of sequences
of lattices (or more generally invariant random subgroups). Part (iii) tries to lay the foundation of a padic
analogue of the `2-theory, where we study novel aspects of p-adic functional analysis which help
to clarify the approximation properties of (Z/pZ)-Betti numbers. Finally, in Part (iv), we try to attack
various longstanding combinatorial problems in group theory with tools from algebraic topology and
p-local homotopy theory.
Fields of science
Call for proposal
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