Prof. Felix Otto
Max Planck Institute for Mathematics in the Sciences
A variational regularity theory for Optimal Transportation, and its application to matching
Tuesday March 5th, 2025 - 16:30
Room TBA Torre Archimede
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The optimal matching of two large point clouds is a special case of optimal transportation between measures, a ubiquitous variational problem. In statistics, it is natural to consider random points clouds that arise from sampling from a given distribution, like the uniform distribution.
The setting in two space dimension is known to be critical, and its finer behavior has been predicted by Parisi et. al. The predictions rely on the connection between the Monge-Ampère equation, which is the Euler-Lagrange equation for optimal transport, and its linearization, the Poisson equation from electrostatics. Ambrosio et. al. established these predictions rigorously on a macroscopic level.
A variational regularity theory, used as a large-scale regularity theory, allows to establish this connection down to the microscopic level. It mimics De Giorgi's approach to the regularity theory of minimal surfaces in the sense that a harmonic approximation result is at its center: Under a non-dimensional smallness condition, the optimal point displacement is close to the gradient of a harmonic function. Probably the main advantage of this variational regularity theory over the one based on maximum principle for the Monge-Ampère equation -- and attached to the names of Caffarelli and Figalli -- is that it does not require any regularity of the involved measures.
Short Bio
Felix Otto earned a Ph.D. from the University of Bonn in 1993 under S. Luckhaus, followed by an experience in industry and by postdoctoral positions: in Bonn, at the Courant Institute in New York, at CMU in Pittsburgh, and at Carnegie Mellon. After a tenure-track position at the University of California, Santa Barbara, he became a Full Professor there and then at the University of Bonn. From 2006 to 2010 he was managing director of the cluster of excellence "Hausdorff Center for Mathematics". From 2010, Felix Otto has served as a Director at the Max Planck Institute for Mathematics in the Sciences, Leipzig, and hold an honorary professorship at Leipzig University in "Analysis and Mathematical Modelling." His leadership includes chairing SFB 611 on scaling in mathematical models.
Otto is a specialist in materials science and mathematical analysis, known for the Otto-Villani theorem and for his development of the Otto calculus. His work has earned him significant recognition, including the prestigious Gottfried Wilhelm Leibniz Prize from the Deutsche Forschungsgemeinschaft in 2006, the highest award in German research. In 2002, he was invited speaker at ICM, in Beijing. In 2009, he delivered the Gauss Lecture for the German Mathematical Society, and in 2008, he was elected to the German Academy of Sciences Leopoldina. In 2024, he was honored with the Cantor Medal by the German Mathematical Society.
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