Objective Eversince, the study of symmetry in mathematics and mathematical physics has been fundamentalto a thourough understanding of most of the fundamental notions. Group theory in all its formsis 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 thesophisticated 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 ringtheory, probability, Riemannian geometry, functional analyis, and the theory of dynamical systemshas been extremely fruitful in a variety of situations. In this proposal, I want to extend this line ofapproach 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 theoryThe theory of `2-homology and `2-torsion of groups has provided a fruitful context to study globalproperties of infinite groups. The relationship of these homological invariants with ergodic theoryof 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 sequencesof lattices (or more generally invariant random subgroups). Part (iii) tries to lay the foundation of a padicanalogue of the `2-theory, where we study novel aspects of p-adic functional analysis which helpto clarify the approximation properties of (Z/pZ)-Betti numbers. Finally, in Part (iv), we try to attackvarious longstanding combinatorial problems in group theory with tools from algebraic topology andp-local homotopy theory. Fields of science natural sciencesmathematicspure mathematicsalgebralinear algebranatural sciencesmathematicsapplied mathematicsmathematical physicsnatural sciencesmathematicsapplied mathematicsdynamical systemsnatural sciencesmathematicspure mathematicstopologyalgebraic topologynatural sciencesmathematicspure mathematicsmathematical analysisfunctional analysisoperator algebra Keywords group theory functional analysis ergodic theory algebraic topology Programme(s) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Topic(s) ERC-CoG-2015 - ERC Consolidator Grant Call for proposal ERC-2015-CoG See other projects for this call Funding Scheme ERC-COG - Consolidator Grant Host institution TECHNISCHE UNIVERSITAET DRESDEN Net EU contribution € 2 000 000,00 Address HELMHOLTZSTRASSE 10 01069 Dresden Germany See on map Region Sachsen Dresden Dresden, Kreisfreie Stadt Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 2 000 000,00 Beneficiaries (1) Sort alphabetically Sort by Net EU contribution Expand all Collapse all TECHNISCHE UNIVERSITAET DRESDEN Germany Net EU contribution € 2 000 000,00 Address HELMHOLTZSTRASSE 10 01069 Dresden See on map Region Sachsen Dresden Dresden, Kreisfreie Stadt Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 2 000 000,00