Objective Moduli spaces are spaces which describe all mathematical objects of some type. This proposal concerns the study of certain moduli spaces via techniques from homotopy theory, from several different points of view. The main moduli spaces in which we are interested are moduli spaces of manifolds, or equivalently classifying spaces of diffeomorphism groups of manifolds. We are also interested in spaces of positive scalar curvature metrics on smooth manifolds, which we study by relating them to moduli spaces of smooth manifolds.The study of moduli spaces of manifolds via homotopy theory has seen a great deal of development in the last 20 years, the breakthrough result being Madsen and Weiss' calculation of the stable homology of moduli spaces of surfaces. More recently, Galatius and I have established analogous results for manifolds of higher dimension.A main goal of this proposal is to study the homology of moduli spaces from a multiplicative point of view. This leads to higher-order forms of the phenomenon of homological stability in which the failure of ordinary homological stability is itself stable. Remarkably, our methods developed to handle moduli spaces of manifolds are sufficiently general to yield deep new results when applied to other moduli spaces in algebra and topology, such as moduli spaces of modules (equivalently, classifying spaces of general linear groups) or moduli spaces of graphs (equivalently, classifying spaces of automorphism groups of free groups). In each case our methods give new information about their homology outside of the traditional stable range.Other goals of this proposal are to form new connections between spaces of Riemannian metrics of positive scalar curvature and infinite loop spaces, and to investigate the structure of tautological subrings of the cohomology of moduli spaces of manifolds, especially in relation to the tautological rings of moduli spaces of Riemann surfaces studied in algebraic geometry. Fields of science natural sciencesmathematicspure mathematicstopologynatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsalgebraalgebraic geometry Programme(s) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Topic(s) ERC-2017-STG - ERC Starting Grant Call for proposal ERC-2017-STG See other projects for this call Funding Scheme ERC-STG - Starting Grant Host institution THE CHANCELLOR MASTERS AND SCHOLARS OF THE UNIVERSITY OF CAMBRIDGE Net EU contribution € 974 526,00 Address TRINITY LANE THE OLD SCHOOLS CB2 1TN Cambridge United Kingdom See on map Region East of England East Anglia Cambridgeshire CC 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 € 974 526,00 Beneficiaries (1) Sort alphabetically Sort by Net EU contribution Expand all Collapse all THE CHANCELLOR MASTERS AND SCHOLARS OF THE UNIVERSITY OF CAMBRIDGE United Kingdom Net EU contribution € 974 526,00 Address TRINITY LANE THE OLD SCHOOLS CB2 1TN Cambridge See on map Region East of England East Anglia Cambridgeshire CC 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 € 974 526,00
