报告题目： A geometric approach to apriori estimates for optimal transport maps

报告人：Robert McCann (University of Toronto)

报告时间：2025年12月15日（周一）10:00-11:30

报告地点：成人漫画207报告厅

报告摘要： 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 apriori 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 (2010) 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 submanifold is (uniformly) spacelike.

个人简介：Robert McCann 是多伦多大学数学、经济学与物理学领域的加拿大研究讲座教授(Canada Research chair)。自20世纪90年代以来，他一直是最优运输理论及其应用领域的国际领军人物之一。他曾在2014年国际数学家大会(ICM 2014)上作邀请报告，并于同年当选为加拿大皇家学会院士。2025年，他因在最优运输理论及其应用方面的杰出贡献，荣获AMS·SIAM 诺伯特·维纳奖(Norbet Wiener Prize)。他的研究成果发表在 Annals of Mathematics、Acta Mathematica、Inventiones Mathematicae 和 Journal of the American Mathematical Society等国际顶级数学期刊上。