Optimal transport and geometric analysis
Lead Research Organisation:
University of Warwick
Department Name: Mathematics
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.
Organisations
- University of Warwick (Lead Research Organisation)
- University of Texas at Austin (Collaboration)
- International School for Advanced Studies (Collaboration)
- Ruhr University Bochum (Collaboration)
- International Centre for Theoretical Physics (Project Partner)
- SISSA - ISAS (Project Partner)
- University of Toronto (Project Partner)
Publications
Cavalletti F
(2018)
Isoperimetric inequalities for finite perimeter sets under lower Ricci curvature bounds
in Rendiconti Lincei - Matematica e Applicazioni
Cavalletti F
(2020)
New formulas for the Laplacian of distance functions and applications
in Analysis & PDE
Cavalletti F
(2018)
Quantitative Isoperimetry à la Levy-Gromov
in Communications on Pure and Applied Mathematics
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
Galaz-García F
(2018)
On quotients of spaces with Ricci curvature bounded below
in Journal of Functional Analysis
Ikoma N
(2020)
Foliation by Area-constrained Willmore Spheres Near a Nondegenerate Critical Point of the Scalar Curvature
in International Mathematics Research Notices
Kell Martin
(2018)
On the volume measure of non-smooth spaces with Ricci curvature bounded below
in ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA-CLASSE DI SCIENZE
Ketterer C
(2018)
Sectional and intermediate Ricci curvature lower bounds via optimal transport
in Advances in Mathematics
Lerario A
(2019)
Homotopy properties of horizontal loop spaces and applications to closed sub-Riemannian geodesics
in Transactions of the American Mathematical Society, Series B
| Description | The grant permitted the PI to dedicate 50% of his time to research, to invite and visit collaborators and leading scientists. The main outputs are the following: 1) "An optimal transport formulation of the Einstein equations of general relativity", joint with Stefan Suhr, submitted for publication in a leading mathematical journal. The goal of the paper is to give an optimal transport formulation of the full Einstein equations of general relativity, linking the (Ricci) curvature of a space-time with the cosmological constant and the energy-momentum tensor. Such an optimal transport formulation is in terms of convexity/concavity properties of the Shannon-Bolzmann entropy along curves of probability measures extremizing suitable optimal transport costs. The result gives a new connection between general relativity and optimal transport; moreover it gives a mathematical reinforcement of the strong link between general relativity and thermodynamics/information theory that emerged in the physics literature of the last years. 2) "Polya-Szego inequality and Dirichlet $p$-spectral gap for non-smooth spaces with Ricci curvature bounded below", joint with Daniele Semola, submitted for publication in a leading mathematical journal. In the paper we study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by $K>0$ and dimension bounded above by $N\in (1,\infty)$ in a synthetic sense, the so called CD(K,N) spaces. We first establish a Polya-Szego type inequality stating that the $W^{1,p}$-Sobolev norm decreases under such a rearrangement and apply the result to show sharp spectral gap for the $p$-Laplace operator with Dirichlet boundary conditions (on open subsets), for every $p\in (1,\infty)$. This extends to the non-smooth setting a classical result of B\'erard-Meyer and Matei; remarkable examples of spaces fitting our framework and for which the results seem new include: measured-Gromov Hausdorff limits of Riemannian manifolds with Ricci$\geq K>0$, finite dimensional Alexandrov spaces with curvature$\geq K>0$, Finsler manifolds with Ricci$\geq K>0$. \\In the second part of the paper we prove new rigidity and almost rigidity results attached to the aforementioned inequalities, in the framework of RCD(K,N) spaces, which seem original even for smooth Riemannian manifolds with Ricci$\geq K>0$. 3) "Foliation by area-constrained Willmore spheres near a non-degenerate critical point of the scalar curvature" joint with Norihisa Ikoma and Andrea Malchiodi, to appear in Int. Math. Res. Not. Let $(M,g)$ be a 3-dimensional Riemannian manifold. The goal of the paper it to show that if $P_{0}\in M$ is a non-degenerate critical point of the scalar curvature, then a neighborhood of $P_{0}$ is foliated by area-constrained Willmore spheres. Such a foliation is unique among foliations by area-constrained Willmore spheres having Willmore energy less than $32\pi$, moreover it is regular in the sense that a suitable rescaling smoothly converges to a round sphere in the Euclidean three-dimensional space. We also establish generic multiplicity of foliations and the first multiplicity result for area-constrained Willmore spheres with prescribed (small) area in a closed Riemannian manifold. The topic has strict links with the Hawking mass. 4) "Sectional and intermediate Ricci curvature lower bounds via optimal transport" joint with Christian Ketterer, published in Advances of Mathematics. The goal of the paper is to give an optimal transport characterization of sectional curvature lower (and upper) bounds for smooth n-dimensional Riemannian manifolds. More generally we characterize, via optimal transport, lower bounds on the so called p-Ricci curvature which corresponds to taking the trace of the Riemann curvature tensor on p-dimensional planes, 1 =p =n. Such characterization roughly consists on a convexity condition of the p-Renyi entropy along L2-Wasserstein geodesics, where the role of reference measure is played by the p-dimensional Hausdorff measure. As application we establish a new Brunn-Minkowski type inequality involving p-dimensional submanifolds and the p-dimensional Hausdorff measure. |
| Exploitation Route | I consider with particular high potential for the future the paper about optimal transport and Ricci curvature; indeed this is opening the way to use optimal transport tools, so useful in the euclidean and Riemannian setting for geometric and analytic purposes, in the Lorentzian setting with the potential of high impact into general relativity. |
| Sectors | Aerospace, Defence and Marine,Education,Transport |
| URL | http://arxiv.org |
| Description | ERC Starting grant CURVATURE |
| Amount | € 1,250,000 (EUR) |
| Funding ID | 802689 |
| Organisation | European Research Council (ERC) |
| Sector | Public |
| Country | Belgium |
| Start | 02/2019 |
| End | 07/2024 |
| Description | Collaboration with Prof. Fabio Cavalletti (SISSA) |
| Organisation | International School for Advanced Studies |
| Country | Italy |
| Sector | Academic/University |
| PI Contribution | We co-authored two papers in the timeframe of the grant |
| Collaborator Contribution | We co-authored two papers in the timeframe of the grant |
| Impact | Cavalletti, Fabio; Mondino, Andrea New formulas for the Laplacian of distance functions and applications. Anal. PDE 13 (2020), no. 7, 2091-2147. Cavalletti, F.; Maggi, F.; Mondino, A. Quantitative isoperimetry à la Levy-Gromov. Comm. Pure Appl. Math. 72 (2019), no. 8, 1631-1677. |
| Start Year | 2012 |
| Description | Collaboration with Prof. Francesco Maggi (University of Texas at Austin) |
| Organisation | University of Texas at Austin |
| Country | United States |
| Sector | Academic/University |
| PI Contribution | We coauthored one paper |
| Collaborator Contribution | We coauthored one paper |
| Impact | Cavalletti, F.; Maggi, F.; Mondino, A. Quantitative isoperimetry à la Levy-Gromov. Comm. Pure Appl. Math. 72 (2019), no. 8, 1631-1677. |
| Start Year | 2018 |
| Description | Collaboration with Stefan Suhr |
| Organisation | Ruhr University Bochum |
| Country | Germany |
| Sector | Academic/University |
| PI Contribution | We wrote the joint paper "An optimal transport formulation of the Einstein equations of general relativity" published in JEMS |
| Collaborator Contribution | Contributed on ideas for the joint paper |
| Impact | A. Mondino and S. Suhr, "An optimal transport formulation of the Einstein equations of general relativity" published in J. Eur. Math. Soc. (Online first), DOI 10.4171/JEMS/1188 |
| Start Year | 2018 |
| Description | International Conference |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | International conference of one week with the top international experts presenting the last results in the field to other experts, postdocs and post-graduate students. |
| Year(s) Of Engagement Activity | 2019 |
| URL | https://otgeoan.wixsite.com/venice |