# Research Area B3

# Singular geometries, optimal transport, and geometry processing

## Research Area Leader: Karl-Theodor Sturm

## PIs: Benny Moldovanu, Martin Rumpf, Karl-Theodor Sturm

## Contributions by Werner Ballmann, Ronan Herry, Leif Kobbelt (RWTH Aachen), Eva Kopfer, Matthias Lesch, Joe Neeman (University of Texas), Chiara Rigoni, Koen van den Dungen

## Topics and goals

Concepts of transport have led to fundamental new insights in the geometry of metric measure spaces and to powerful tools in mechanism design, computer vision, and geometry processing. There is a deep connection between these developments which is a leitmotiv of this research area. We will push forward the geometric calculus on metric measure spaces with synthetic Ricci bounds and extend it to discrete and non-commutative geometries, quantum systems, and kinetic evolution equations. We will study the many facets of the heat trace and optimal design problems associated with small eigenvalues. Tools based on transport models will be developed further and applied to problems of mechanism design, model uncertainty, and image processing. We aim to expand our Riemannian calculus on shape spaces and to combine it with shape statistics, machine learning, and shape semantic.

## State of the art, our expertise

**Singular geometries**. Sturm and Lott–Villani introduced the synthetic geometry of metric measure spaces with lower bounds on Ricci curvature (‘RCD spaces’). Subsequently complemented by a powerful analytic calculus by Ambrosio–Gigli–Savaré, this initiated an enormous ongoing research activity. Erbar–Kuwada–Sturm [EKS15] succeeded in deducing the full Bochner inequality in metric measure spaces and thus opened the door for the extension of the powerful Li–Yau calculus to singular spaces and to impressive rigidity and structure results including splitting (Gigli), maximal diameter theorem (former BIGS student Ketterer), rectifiability (Mondino– Naber), globalization (former HCM postdoc Cavalletti and Milman), and isoperimetric inequality (Cavalletti–Mondino). A novel direction is taken up by Kopfer–Sturm [KS] with the study of heat flow on time-dependent metric measure spaces, establishing a crucial link to the celebrated work on Ricci flows by Hamilton, Perelman, and many others, including recent research on singular Ricci flows by Kleiner–Lott, Haslhofer–Naber, and Bamler–Kleiner. For stratified spaces of iterated edge type – an important and rich class of singular spaces – Lesch [Les13] proved subexponential off-diagonal decay of the heat trace and a gluing formula for the analytic torsion. Lesch et al. [LM16] extended the pioneering result of Connes–Moscovici on heat trace asymptotics for the conformal Laplacian on the noncommutative torus to all vector bundles. Ballmann [BMM16] derived estimates on the number of small eigenvalues on Riemannian surfaces of finite type where ‘smallness’ is measured by a new invariant, the ‘analytic systole’, and the number of eigenvalues is estimated by the negative Euler characteristic.

**Optimal transport**. The optimal transport approach to dissipative evolution equations is emerging as a tool for a broad class of problems ranging from random walks on graphs to quantum systems and kinetic evolution equations. Erbar and Maas [EM12] developed a discrete notion of curvature, which can be applied to discrete models in probability, theoretical computer science, and statistical mechanics. Links to non-commutative geometry are emerging in works by Carlen, Maas, Mielke et al., providing gradient flow structures for dissipative quantum systems. Erbar established a gradient flow interpretation for the spatially homogeneous Boltzmann equation in terms of a new transport geometry. Erbar–Rumpf–Schmitzer–Simon studied a numerical approximation of an analogue of theWasserstein distance on graphs – see Figure (a) – and a minimizing movement scheme for associated gradient flows. Fundamental challenges have been addressed: Gigli–Rajala–Sturm and Cavalletti–Huesmann proved the existence of transport maps in the metric setting. Huesmann–Sturm [HS13] were the first to study optimal couplings of random measures like Poisson point processes. Optimal transport tools have been used to obtain various functional inequalities with sharp constants and rigidity results. Mossel–Neeman [MN15] established dimension free quantitative rigidity for various Gaussian functional inequalities. Powerful concepts from optimal transport were turned into novel tools for problems in economics. For instance, optimal transport was one of the key techniques to identify optimal mechanisms and to analyze certain mean field games; multi-marginal and martingale variants of the optimal transport problem allow for a systematic treatment of the important question of model uncertainty in option pricing as demonstrated by HCM’s postdoc Huesmann et al. [BCH17]; a multiplicative version of cyclical monotonicity led to a new theory of portfolio optimization. Dizdar and Moldovanu [DM16] used the classical ‘twist’ condition to identify efficient matchings in a setting with double-sided multi-dimensional information.

**Geometry processing**. Tools from high- and infinite-dimensional shape manifolds and metric measure spaces have a tremendous impact on new methods in geometry processing and vision. There are remarkable breakthroughs in analysis such as the proof by Mumford et al. of geodesic completeness for certain classes of shape manifolds or the convergence theory of Peyré et al. for entropic transport schemes. Rumpf et al. investigated the metamorphosis model for images [BER15] and the Riemannian manifold of viscous shells; see Figure (b). They developed a fully fletched and consistent discrete calculus including discrete geodesics, exponential map, parallel transport, and discrete splines with applications such as shape animation or detail transfer. Moreover, they combined the metamorphosis model with optimal transport. Kobbelt et al. [MCSK+17] used optimal transport plans to identify low distortion maps between surfaces.

## Research program

**Singular geometries**. A core topic of our future research will be the geometry and analysis on metric measure spaces, in particular, on RCD spaces. One of the most prominent challenges is a precise structure analysis of RCD spaces and their relation to Alexandrov spaces. Is every RCD space a Cheeger–Colding limit of Riemannian manifolds? Of particular interest will be rigidity and quantitative almost rigidity results involving synthetic bounds for Ricci curvature and effec- tive dimension; for the latter the whole range including infinite, fractional, and negative effective dimension will be considered. A related major open problem is the generalized Blaschke conjecture on the classification of spaces with maximal injectivity radius. Ballmann and Grove recently found a serious mistake in related classical work on great circle fibrations. Another challenge is the study of heat flows on time-dependent metric measure spaces with singular time-dependence and changing topological type, closely related to the analysis of Ricci flows passing through singularities. We will derive novel functional inequalities for time-dependent metric measure spaces and study corresponding rigidity and isoperimetry. A particularly demanding aim is to establish the full heat trace expansion of a Laplace-type operator on a stratified space. As far as noncommutative spaces are concerned, we seek a coherent theory unifying the various examples such as noncommutative tori, quantum groups, Podles spheres etc. Concerning the analytic systole, we will study the first Dirichlet eigenvalue in isotopy classes of subsurfaces of Riemannian surfaces as an optimal design problem. Minimizers will be singular with collapsed boundary. We will study whether this boundary is a smooth graph.

**Optimal transport**. We will push forward the concept of Otto’s calculus on discrete spaces with synthetic Ricci curvature bounds, aiming for sharp functional inequalities and rigidity results. We will study the long-term behavior of solutions to kinetic equations, exploiting their gradient flow structure in suitable transport geometries. In the long run, an exciting challenge will be to extend this to the spatially inhomogeneous Boltzmann equation. Moreover, we will promote the application of optimal transport techniques to problems in non-commutative geometry. Promising applications of modified transport are also emerging in numerical analysis and imaging. Starting from our numerical transport scheme on discrete spaces, we will investigate the numerical approximation of the collisional transport associated with the Boltzmann equation. Insight gained via numerics is expected to support the analytical studies of discrete processes and kinetic equations. We will further develop the interplay of optimal transport and Gaussian analysis, in particular regarding measure quantization problems. Starting from our results on Gaussian isoperimetric and noise stability problems, we will use new optimal transport techniques to attack other long standing optimality questions in Gauss space such as the peace sign conjecture and the propeller conjecture. We will exploit UBonn’s strength as a leading center both for mechanism design and for optimal transport. We will advance the systematic treatment of model uncertainty (see also [BCH17]) and, based on the results of Moldovanu et al. [GGK+13], open a new chapter in mechanism design by fully exploiting the structural power of optimal transport to solve complex, multidimensional screening and allocation problems. We will also apply geometric concepts, probabilistic methods, and tools from optimal transport to the analysis of data spaces, in particular to the study of barycentric constructions on metric spaces with synthetic curvature bounds.

**Geometry processing**. We aim to extend the Riemannian calculus to spaces of natural images. To this end, we will consider these images as compositions of texture patches and local cartoons and combine the Riemannian approach with structure classifiers and kernel learning methods. Envisioned applications are dynamic texture processing and editing as well as texture sensitive image blending. We will investigate shape correspondence for triangular meshes, taking into account novel shell deformation energies [IRS17]. We will explore the link to Sturm’s notion of a geodesic distance and gradient flows on the space of metric measure spaces. Furthermore, in the functional map framework, eigenfunctions of PDEs on surfaces will play a particular role in shape correspondences, shape modeling, and animation. The Riemannian perspective on the space of discrete shells will lead to novel tools for surface animation, e.g., for skeleton based surface animation or training based statistics. A major challenge in the processing of highly detailed, complicated surfaces is to use shape space paradigms [ZHRS15] and learning tools to combine geometric and semantic analysis. Here, with the involvement of Kobbelt as a member of HCM, new links will emerge between shape space analysis and computer graphics applications.

## Summary

This research area combines geometry, analysis, and numerical approximation to solve challenging problems in the theory of singular spaces, in mechanism design and model uncertainty, and in image processing and computer graphics. A central and powerful tool is optimal transport, used to analyze synthetic geometries, to derive novel functional inequalities, and to reveal the proper geometry of evolution equations. Techniques developed in this research area are expected to prove their power also in other areas. There are close links to RA B2 with respect to identification of relevant low-dimensional geometric structures in high-dimensional data, to RA C3 with respect to noise stability and planted bisection, to RA A3 in the context of mappings between singular spaces, to RA C1 concerning kinetic models such as the Boltzmann equation, and to the IRUs D3–5 with respect to image analysis.

## Bibliography

[BCH17] M. Beiglböck, A. M. G. Cox, and M. Huesmann. Optimal transport and Skorokhod embedding. *Invent. Math., 208(2):327–400, 2017*.

[BER15] B. Berkels, A. Effland, and M. Rumpf. Time discrete geodesic paths in the space of images. *SIAM J. Imaging Sci., 8(3):1457–1488, 2015*.

[BMM16] W. Ballmann, H. Matthiesen, and S. Mondal. Small eigenvalues of closed surfaces. J. *Differential Geom., 103(1):1–13, 05 2016*.

[DM16] D. Dizdar and B. Moldovanu. On the importance of uniform sharing rules for efficient matching. *J. Econom. Theory, 165:106–123, 2016*.

[EKS15] M. Erbar, K. Kuwada, and K.-T. Sturm. On the equivalence of the entropic curvaturedimension condition and Bochner’s inequality.* Invent. Math., 201(3):993–1071, 2015*.

[EM12] M. Erbar and J. Maas. Ricci curvature of finite Markov chains via convexity of the entropy.* Arch. Ration. Mech. Anal., 206(3):997–1038, 2012*.

[GGK+13] A. Gershkov, J. K. Goeree, A. Kushnir, B. Moldovanu, and X. Shi. On the equivalence of Bayesian and dominant strategy implementation. *Econometrica, 81(1):197–220, 2013*.

[HS13] M. Huesmann and K.-T. Sturm. Optimal transport from Lebesgue to Poisson.* Ann. Probab., 41(4):2426–2478, 2013*.

[IRS17] J. A. Iglesias, M. Rumpf, and O. Scherzer. Shape-aware matching of implicit surfaces based on thin shell energies. *Found. Comput. Math., 2017. DOI:10.1007/s10208-017-9357-9*.

[KS] E. Kopfer and K.-T. Sturm. Heat flows on time-dependent metric measure spaces and super- Ricci flows. *Comm. Pure Appl. Math. to appear*.

[Les13] M. Lesch. A gluing formula for the analytic torsion on singular spaces. *Anal. PDE, 6(1):221– 256, 2013*.

[LM16] M. Lesch and H. Moscovici. Modular curvature and Morita equivalence. Geom. *Funct. Anal., 26(3):818–873, 2016*.

[MCSK+17] M. Mandad, D. Cohen-Steiner, L. Kobbelt, P. Alliez, and M. Desbrun. Variance-minimizing transport plans for inter-surface mapping.* ACM Trans. Graph., 36(4):art. 39, 2017*.

[MN15] E. Mossel and J. Neeman. Robust dimension free isoperimetry in Gaussian space.* Ann. Probab., 43(3):971–991, 2015*.

[ZHRS15] C. Zhang, B. Heeren, M. Rumpf, and W. A. P. Smith. Shell PCA: Statistical shape modelling in shell space. *In 2015 IEEE International Conference on Computer Vision (ICCV), pages 1671–1679, 2015*.