EPSRC Reference: 
EP/N033787/1 
Title: 
Classifying spaces for proper actions and almostflat manifolds 
Principal Investigator: 
Petrosyan, Dr N 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Sch of Mathematical Sciences 
Organisation: 
University of Southampton 
Scheme: 
First Grant  Revised 2009 
Starts: 
01 December 2016 
Ends: 
30 September 2018 
Value (£): 
99,083

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
In this research, we will combine techniques from Geometric Group Theory, Topololgy, and Geometry to work on two objectives.
In the last twenty years, nonpositively curved spaces and groups have been at the forefront of Geometric Group Theory and Topology. Their importance is underlined by I. Agol's breakthrough solution of the Virtual Haken Conjecture of Thurston using the machinery of nonpositively curved cube complexes developed by D. Wise. Also, in the last decade, the BaumConnes and the FarrellJones Conjectures have been verified for many (nonpositively curved) classes of groups, paving the way for computations in algebraic K and Ltheories via their classifying spaces. These conjectures connect many different fields of mathematics and have far reaching applications in Topology, Analysis, and Algebra. The time is therefore right to investigate (finiteness) properties of such groups and to construct models for classifying spaces for proper actions with geometric properties that are suitable for computations. Our first objective is to construct such models for classifying spaces of proper actions for some important classes of groups such as Coxeter groups and the outer automorphism group of rightangled Artin groups, and to investigate Brown's conjecture.
Our second objective is on almostflat manifolds. These manifolds are a generalisation of flat manifolds introduced by M. Gromov. They occur naturally in the study of Riemannian manifolds with negative sectional curvature and play a key role in the study of collapsing manifolds with uniformly bounded sectional curvature. The characteristic properties of these manifolds that we will investigate such as Spin structures and cobordisms play an integral part in modern manifold theory. Spin structures have many applications in Quantum Field Theory and in Mathematical Physics. In particular, the existence of a Spin structure on a smooth orientable manifold allows one to define spinor fields and a Dirac operator which can be thought of as the square root of the Laplacian. Dirac operator is essential in describing the behaviour of fermions in Particle Physics. It is also an important invariant in Pure Mathematics arising in AtiyahSinger Index Theorem, Connes's Noncommutative Differential Geometry, the SchrodingerLichnerowicz formula, Kostant's cubic Dirac operator, and many other areas. The methods by which we propose to study almostflat manifolds arise from the interactions between Geometry/Topology and Group Theory. This is largely due to the fact that the topology of these manifolds is completely classified by their fundamental groups.

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Organisation Website: 
http://www.soton.ac.uk 