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.

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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.

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The principal-agent problem is an important paradigm in economic theory for studying the value of private information: the nonlinear pricing problem faced by a monopolist is one example; others include optimal taxation and auction design. For multidimensional spaces of consumers (i.e. agents) and products, Rochet and Chone (1998) reformulated this problem as a concave maximization over the set of convex functions, by assuming agent preferences are bilinear in the product and agent parameters. This optimization corresponds mathematically to a convexity-constrained obstacle problem. The solution is divided into multiple regions, according to the rank of the Hessian of the optimizer.

If the monopolists costs grow quadratically with the product type we show that a partially smooth free boundary delineates the region where it becomes efficient to customize products for individual buyers. We give the first complete solution of the problem on square domains, and discover new transitions from unbunched to targeted and from targeted to blunt bunching as market conditions become more and more favorable to the seller.

Based on joint work with Kelvin Shuangjian Zhang, Cale Rankin, and Lucas O'€™Brien in various combinations: [1] Math. Models Methods Appl. Sci. 34 (2024) 2351-2394; [2] J. Convex Anal. (Rockafellar 90 Issue), 32 (2) (2025) 579-584; [3] arXiv 2303.04937; [4] arXiv 2412.15505; [5] arXiv 2603.14100.

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TBA

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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.

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TBA

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

Here

This is joint work with Weiming Shen.

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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.

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