Geometrical Methods in Algebra and Topology

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The PI proposes to work on a series of projects that incorporate techniques from algebra, geometry, and topology, with a view to applications in all three of these fields. More precisely, the PI proposes to work on the following four circles of ideas:(1) Answering various classical questions in topology (such as the Novikov conjecture, the Borel conjecture, computation of the algebraic K-theory) under suitable geometric hypotheses on the spaces.(2) Working on problems related to the quasi-isometric classification of finitely generated groups.(3) Exploring the boundary between the space of CAT(-1) metrics and the space of negatively curved Riemannian metrics on a fixed manifold.(4) Studying the topology (cohomology, bounded cohomology, submanifolds) of locally symmetric spaces.These projects are intended to exhibit the interdependence of methods and results between the three fields of algebra (primarily group theory), geometry (negatively curved spaces) and topology (both geometric and algebraic).Groups are sets equipped with a "multiplication", and classically arose as symmetries of underlying geometric objects. Geometry is the study of properties of spaces equipped with some additional structure, usually of a metric nature (i.e. allowing one to measure distances in the space). In the field of topology, one forgets about the underlying distance, and only retain the notion of points in the space being "close together". While these three fields have always had some overlap, research from the last twenty years have led to an increasingly fertile exchange of ideas and applications amongst them. The PI's research is intended to further develop the links between these three fields.