# Optimal transport and geometric analysis

Lead Research Organisation:
University of Oxford

Department Name: Mathematical Institute

### Abstract

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.

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.

### Planned Impact

The main beneficiaries of the proposed research will be:

-Research mathematicians working in the field of non-smooth spaces satisfying lower Ricci curvature bounds in a synthetic sense; they will profit directly from the new techniques developed in the proposed research.

-The proposal aims to have an impact on a broader group of mathematicians, in particular, researchers working on geometric analysis, differential geometry, metric geometry, optimal transportation, functional inequalities.

-On a longer time scale advanced techniques in geometric analysis and optimal transportation will have an impact on applied sciences, for example economics, physics and mathematical biology.

-Graduate students from the relevant areas will profit directly from interactions with the PI and his collaborators.

In order to reach these beneficiaries, the PI will do the following:

-the work will be presented at key international conferences, such as the Rolf Nevalinna Colloquium to be held at ETH-Zurich in June 2017, at specialized workshops like the ones in MFO Oberwolfach, and at several departmental seminars in UK universities and overseas.

-The PI will organize an international workshop on geometric analysis and optimal transport with applications, with up to 20 invited speakers, at the University of Warwick.

-To ensure timely availability of the project's results to other academics working in the area, the PI will post these on various preprint servers, like arXiv and cvgmt.

-The work will be published in high quality general journals like Inventiones Mathematicae, Journal of the European Mathematical Society, Proceedings of the London Mathematical Society, and specialized ones like Journal of Differential Geometry, Calculus of Variations and Partial Differential Equations, Journal of Functional Analysis.

-The PI will collaborate on various sections of the project with Dr. Fabio Cavalletti (SISSA-Trieste), Prof. Vitali Kapovitch (Toronto), and Prof. Francesco Maggi (ICTP-Trieste). The involvement of these established scientists, will increase the prospect of the project yielding high-impact results.

-Research mathematicians working in the field of non-smooth spaces satisfying lower Ricci curvature bounds in a synthetic sense; they will profit directly from the new techniques developed in the proposed research.

-The proposal aims to have an impact on a broader group of mathematicians, in particular, researchers working on geometric analysis, differential geometry, metric geometry, optimal transportation, functional inequalities.

-On a longer time scale advanced techniques in geometric analysis and optimal transportation will have an impact on applied sciences, for example economics, physics and mathematical biology.

-Graduate students from the relevant areas will profit directly from interactions with the PI and his collaborators.

In order to reach these beneficiaries, the PI will do the following:

-the work will be presented at key international conferences, such as the Rolf Nevalinna Colloquium to be held at ETH-Zurich in June 2017, at specialized workshops like the ones in MFO Oberwolfach, and at several departmental seminars in UK universities and overseas.

-The PI will organize an international workshop on geometric analysis and optimal transport with applications, with up to 20 invited speakers, at the University of Warwick.

-To ensure timely availability of the project's results to other academics working in the area, the PI will post these on various preprint servers, like arXiv and cvgmt.

-The work will be published in high quality general journals like Inventiones Mathematicae, Journal of the European Mathematical Society, Proceedings of the London Mathematical Society, and specialized ones like Journal of Differential Geometry, Calculus of Variations and Partial Differential Equations, Journal of Functional Analysis.

-The PI will collaborate on various sections of the project with Dr. Fabio Cavalletti (SISSA-Trieste), Prof. Vitali Kapovitch (Toronto), and Prof. Francesco Maggi (ICTP-Trieste). The involvement of these established scientists, will increase the prospect of the project yielding high-impact results.

## People |
## ORCID iD |

Andrea Mondino (Principal Investigator) |

### Related Projects

Project Reference | Relationship | Related To | Start | End | Award Value |
---|---|---|---|---|---|

EP/R004730/1 | 01/01/2018 | 30/09/2019 | £101,150 | ||

EP/R004730/2 | Transfer | EP/R004730/1 | 01/01/2020 | 31/03/2020 | £16,699 |