Abstract: Caffarelli’s contraction theorem states that the Brenier optimal transport map from the standard Gaussian measure to a more log-concave probability measure is 1-Lipschitz. In view of its applications to geometric and functional inequalities, Villani asked whether this theorem extends to Riemannian manifolds. Motivated by results in comparison geometry, E. Milman formulated several conjectures on spheres and on weighted manifolds satisfying the $\mathrm{CD}(K,\infty)$ condition with $K>0$. Such a contracting transport map implies a corresponding spectral comparison. In the spherical setting, this comparison was also conjectured by Colding and Minicozzi in 1998 for closed connected manifolds with a positive Ricci curvature lower bound. In this talk, I will present counterexamples to the spherical conjectures in dimensions $d\geq4$, followed by joint work with Bang-Xian Han and Zhuo-Nan Zhu proving the spherical transport conjecture when $d=2$. Together with a recent counterexample by Shengjie Lin, Haibin Wang, and Guoyi Xu in dimension $d=3$, these results settle the spherical transport and spectral comparison conjectures in every dimension $d\geq2$. If time permits, I will discuss some open problems.

Bio: Shrey Aryan is a graduate student in mathematics at the Massachusetts Institute of Technology, advised by Tobias H. Colding. His research interests include geometric analysis, nonlinear partial differential equations, and optimal transport.

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