Classifying spaces for proper actions and almost-flat manifolds

In this research, we will combine techniques from Geometric Group Theory, Topololgy, and Geometry to work on two objectives.

In the last twenty years, non-positively 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 non-positively curved cube complexes developed by D. Wise. Also, in the last decade, the Baum-Connes and the Farrell-Jones Conjectures have been verified for many (non-positively curved) classes of groups, paving the way for computations in algebraic K- and L-theories 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 right-angled Artin groups, and to investigate Brown's conjecture.

Our second objective is on almost-flat 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 Atiyah-Singer Index Theorem, Connes's Noncommutative Differential Geometry, the Schrodinger-Lichnerowicz formula, Kostant's cubic Dirac operator, and many other areas. The methods by which we propose to study almost-flat 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.

The proposed research lies in the field of Pure Mathematics and therefore it will mostly have academic impact. However, because of its proximity to areas such as Applied Combinatorial Group Theory and Applied Algebraic Topology, Crystallography, Physics and String Theory, there is a long-term impact on more applied scientific fields.

There are rapidly growing applications of non-positively curved groups and spaces in Biology, Robotics, and Computer Science. On the other hand, (almost-)crystallographic groups, (almost-)flat manifolds and orbifolds are fundamental objects in Crystallography, Particle Physics, and String Theory. In the proposed research we will address some fundamental questions regarding these groups and spaces such as constructing computationally good models for classifying spaces of groups for proper actions and investigating the existence of Spin structures on almost-flat manifolds. From the first part, there are foreseeable applications to other more applied branches of Mathematics via the Isomorphism Conjectures. In the second, we study the existence of Spin structures on almost-flat manifolds. 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. Therefore, there are potential benefits of the proposed research to physicists, biologists, and computer scientists.

As most of the envisioned impact of the grant proposal will be long-term in nature, we will focus on the initial societal impact. The results of the proposed research will be made available in preprint form on open-access repositories such as Math arXiv. They will then be published in leading international mathematical journals. The PDRA and I will communicate our work to wide range of mathematical audiences at national and international conferences and seminars. We will seek to deliver lectures to broader public audiences.

A key feature of the grant proposal is the funding of a postdoctoral position (PDRA) to work on the problems outlined in the research proposal. This will give the PDRA the unique opportunity to learn the underlying theory and the methods of the proposed research project outlined in the Case for Support. He or she will undoubtedly grow and become an established researcher in the fields of Geometry Group Theory and Topology acquiring new knowledge and techniques by conducting research, writing scientific articles and developing communication skills by giving research talks and presentations.