EPSRC Reference: 
EP/R004730/2 
Title: 
Optimal transport and geometric analysis 
Principal Investigator: 
Mondino, Dr A 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

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 nonsmooth objects which is closed under (GromovHausdorff) 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 nonsmooth objects, closed under (measured GromovHausdorff) 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 LottSturmVillani, 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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Summary 

Date Materialised 


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Further Information: 

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