Optimal Transport: Mini-Course and Mini-Workshop

Abstract

A key inequality which underpins the regularity theory of optimal transport for costs satisfying the Ma-Trudinger-Wang condition is the Pogorelov second derivative bound. This translates to an a priori interior modulus of the differential estimate for smooth optimal maps. We describe a new derivation of this estimate with Brendle, Leger and Rankin which relies in part on Kim, McCann, and Warren’s observation that the graph of an optimal map becomes a volume maximizing non-timelike submanifold when the product of the source and target domains is endowed with a suitable pseudo-Riemannian geometry that combines both the marginal densities and the cost. This unexpected links optimal transport to the plateau problem in geometry with split signature, and shows the key difficulty is showing the maximizing non-timelike submanifold is in fact (uniformly) spacelike.

J. Reine Angew. Math. 817 (2024) 251-266 doi:10.1515/crelle-2024-0071, arXiv 2311.10208.

Back to Program

Abstract

While Einstein’s theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually breakdown. It is thus of great interest to study the theory in low regularity settings. In the lecture, we establish a low regularity splitting theorem by sacrificing linearity of the d’Alembertian to recover ellipticity. We exploit a negative homogeneity \(p\)-d’Alembert operator for this purpose. The same technique yields a simplified proof of Eschenberg (1988) Galloway (1989) and Newman’s (1990) confirmation of Yau’s (1982) conjecture, bringing all three Lorentzian splitting results into a framework closer to the Cheeger-Gromoll splitting theorem from Riemannian geometry.

Based on joint work with Mathias Braun, Nicola Gigli, Argam Ohanyan, and Clemens Saemann: [1] arXiv 2501.00702 [2] arXiv 2408.15968 [3] arXiv 2410.12632 [4] arXiv 2507.06836.

Back to Program

Abstract

While Einstein’s theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually breakdown. In positive-definite signature, there is a highly successful theory of metric and metric-measure geometry which includes Riemannian manifolds as a special case, but permits the extraction of nonsmooth limits under dimension and curvature bounds analogous to the energy conditions in relativity: here sectional curvature is reformulated through triangle comparison, while and Ricci curvature is reformulated using entropic convexity along geodesics of probability measures.

This lecture explores recent progress in the development of an analogous theory in Lorentzian signature, whose ultimate goal is to provide a nonsmooth theory of gravity. In a setting which relaxes local compactness, we describe a differential calculus for monotone curves and functions, and applications including a notion of infinitesimal Minkowskianity (that distinguishes Lorentz from Lorentz-Finsler norms), and a \(p\)-d’Alembert comparison theorem which allowed us to prove a Lorentzian splitting theorem on manifolds with limited regularity \(g_{ij} \in C^1\).

Based on joint work with Tobias Beran, Mathias Braun, Matteo Calisti, Nicola Gigli, Argam Ohanyan, Felix Rott and Clemens Saemann: arXiv 2408.15968.

Back to Program

Abstract

In this talk, we introduce a new proof for the interior and boundary regularity of Monge-Ampère equation. The proof is based on the Green functions of the linearised Monge-Ampère equation and applies to both the Dirichlet and the Neumann problems.

Back to Program

Abstract

Trajectory inference arises in important scientific problems. In particular, biological development can be interpreted as a curve in the space of gene-expression distributions, and the goal is to infer this trajectory from observed data. There has been progress by using optimal transport (OT) as a way to interpolate between distributions. More recently, Schrödinger bridges, a stochastic generalization of OT, have been considered. In this talk, we discuss stability of such OT-based methods.

This is joint work with Geoffrey Schiebinger and Rentian Yao.

Back to Program

Abstract

We consider the problem of recovering the Riemannian metric on a compact closed manifold from the optimal transport maps when the underlying cost function is the squared Riemann distance. We show that the metric can be uniquely determined up to a multiplicative constant.

This is joint work with Jian Zhai.

Back to Program

Abstract

There are classical conditions under which optimal transport problems are known to have solutions which are supported on the graph of a mapping. However, if the cost is ambient Euclidean distance squared restricted to the boundary of a convex body, a result of Gangbo and McCann demonstrates there may be nice measures for which there is no singled-valued optimal map. In this talk, I will present a recent result showing that on the boundary of a \(C^1\), convex body, optimal transport between suitably nice measures that are close enough to each other still result in existence of an optimal map. This result is sharp in the sense that we show a counterexample for a uniformly convex, but not \(C^1\) domain.

This talk is based on joint work with Seonghyeon Jeong.

Back to Program

Abstract

We will talk about the following boundary value problem of Monge-Ampère equation

Here

This is joint work with Weiming Shen.

Back to Program

Abstract

We give simple new proofs of two well-known results for the Schrödinger operator: first, the Brunn-Minkowski inequality for Dirichlet eigenvalues and, second, the log-concavity of the first Dirichlet eigenfunction. Our proof of the first applies to a class of domains including \(C^{1,1}\) connected domains and convex potentials. In the special case of convex domains, the second result is a simple corollary. I’ll discuss extension of these results to new operators and a work-in-progress extension to manifolds.

This is joint work with Paul Bryan and Julie Clutterbuck.

Back to Program