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.
Publications
Cavalletti F
(2020)
New formulas for the Laplacian of distance functions and applications
in Analysis & PDE
De Ponti N
(2021)
Sharp Cheeger-Buser Type Inequalities in RCD ( K , 8 ) Spaces.
in Journal of geometric analysis
De Ponti N
(2022)
Entropy-Transport distances between unbalanced metric measure spaces
in Probability Theory and Related Fields
Ikoma N
(2020)
Foliation by Area-constrained Willmore Spheres Near a Nondegenerate Critical Point of the Scalar Curvature
in International Mathematics Research Notices
Kapovitch V
(2021)
On the topology and the boundary of N-dimensional RCD(K,N) spaces
in Geometry & Topology
Mondino A
(2020)
Polya-Szego inequality and Dirichlet p-spectral gap for non-smooth spaces with Ricci curvature bounded below
in Journal de Mathématiques Pures et Appliquées
Mondino A
(2020)
Existence and Regularity of Spheres Minimising the Canham-Helfrich Energy
in Archive for Rational Mechanics and Analysis
Description | The grant permitted the PI to dedicate 50% of his time to research, to invite and visit collaborators and leading scientists. The key findings of this award regard the properties and structure of possibly non-smooth metric measure spaces with Ricci Curvature bounded below in a synthetic sense via optimal transport. We proved interior topological regularity and several new geometric and functional inequalities. In particular, one paper (joint with De Ponti) found some unexpected applications to string theory in theoretical physics. The corresponding papers can be found in the "publications" folder. |
Exploitation Route | The joint paper with Kapovitch (proving interior topological regularity of possibly non-smooth metric measure spaces with Ricci Curvature bounded below in a synthetic sense via optimal transport) contained several conjectures and it served as starting point for further investigation also by other authors (e.g. Brue-Naber-Semola). The joint paper with De Ponti about Cheeger and Buser inequalities in RCD spaces has found some (quite un-expected) applications to string theory. |
Sectors | Aerospace Defence and Marine Education Transport |
URL | http://arXiv.org |
Description | The research performed with the framework of the grant was in pure mathematics, more precisely in Optimal Transport and Geometric Analysis. Nevertheless, the findings obtained during the grant had applications to other fields of science and technology. Indeed: - The paper "Mondino A, Scharrer C. (2020). Existence and Regularity of Spheres Minimising the Canham-Helfrich Energy. Archive for Rational Mechanics and Analysis" found applications in mathematical biology, indeed the paper studies the shape of lipidic bilayer membranes which form cells of most living organisms. - The paper "De Ponti N.- Mondino A. (2021) Sharp Cheeger-Buser Type Inequalities in RCD Spaces. in Journal of geometric analysis" had applications in theoretical physics, more precisely to study the Kaluza-Klein spectrum for gravity compactifications in string theory. - The paper "De Ponti N.- Mondino A. (2022) Entropy-Transport distances between unbalanced metric measure spaces in Probability Theory and Related Fields" had applications in Artificial Intelligence and Machine Learning |
First Year Of Impact | 2023 |
Sector | Digital/Communication/Information Technologies (including Software),Education,Pharmaceuticals and Medical Biotechnology |
Impact Types | Cultural |
Description | Collaboration with Dr. Nicoló De Ponti |
Organisation | International School for Advanced Studies |
Country | Italy |
Sector | Academic/University |
PI Contribution | We co-authored one paper during the timeframe of the grant and several others out of the timeframe of the grant |
Collaborator Contribution | We co-authored one paper during the timeframe of the grant and several others out of the timeframe of the grant |
Impact | De Ponti, Nicolò; Mondino, Andrea Sharp Cheeger-Buser type inequalities in (K,8) spaces. J. Geom. Anal. 31 (2021), no. 3, 2416-2438. |
Start Year | 2019 |
Description | Collaboration with Prof. Vitali Kapovitch |
Organisation | University of Toronto |
Country | Canada |
Sector | Academic/University |
PI Contribution | We co-authored one publication |
Collaborator Contribution | We co-authored one publication |
Impact | Kapovitch, Vitali; Mondino, Andrea On the topology and the boundary of N-dimensional (K,N) spaces. Geom. Topol. 25 (2021), no. 1, 445-495. |
Start Year | 2019 |