Details of Grant 

EPSRC Reference:	EP/R004730/2
Title:	Optimal transport and geometric analysis
Principal Investigator:	Mondino, Professor A
Other Investigators:
Researcher Co-Investigators:
Project Partners:	Abdus Salam ICTP SISSA - ISAS University of Toronto
Department:	Mathematical Institute
Organisation:	University of Oxford
Scheme:	First Grant - Revised 2009
Starts:	01 January 2020	Ends:	31 March 2020	Value (£):	16,699
EPSRC Research Topic Classifications:	Algebra & Geometry Mathematical Analysis
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:
Summary on Grant Application Form
The subject of study of differential geometry are smooth manifolds, which correspond to smooth curved objects of finite dimension. In modern differential geometry, it is becoming more and more common to consider sequences (or flows) of smooth manifolds. Typically the limits of such sequences (or flows) are non smooth anymore. It is then useful to isolate a natural class of non smooth objects which generalize the classical notion of smooth manifold, and which is closed under the process of taking limits. If the sequence of manifolds satisfy a lower bound on the sectional curvatures, a natural class of non-smooth objects which is closed under (Gromov-Hausdorff) convergence is given by special metric spaces known as Alexandrov spaces; if instead the sequence of manifolds satisfy a lower bound on the Ricci curvatures, a natural class of non-smooth objects, closed under (measured Gromov-Hausdorff) convergence, is given by special metric measure spaces (i.e. metric spaces endowed with a reference volume measure) known as RCD(K,N) spaces. These are a 'Riemannian' refinement of the so called CD(K,N) spaces of Lott-Sturm-Villani, which are metric measure spaces with Ricci curvature bounded below by K and dimension bounded above by N in a synthetic sense via optimal transport. In the proposed project we aim to understand in more detail the structure, the analytic and the geometric properties of RCD(K,N) spaces. The new results will have an impact also on the classical world of smooth manifolds satisfying curvature bounds.
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Organisation Website:	http://www.ox.ac.uk