Samuël Borza
Summary: Theory of distributions, sub-Riemannian structures, and admissible trajectory, controllability and Chow-Rachevsky theorem, Cauchy-Carathéodory theorem and the endpoint map, necessary conditions for minimality (Pontryagin’s Maximum Principle, normal and abnormal extremals), the Heisenberg group, the Grushin plane, the Martinet flat structure, contact structures, and Carnot groups, metric tangent for sub-Riemannian structures.
Wilhelm Klingenberg
Summary: Necessary background in analysis (density functions and its push forward by a map of domains, convexity and the second derivative condition, Legendre transform, convex dual of a functional, and Jensen’s inequality), Monge minimization problem of transport for a continuous cost function c(x, y) with an example in one space dimension, the dual maximization problem due to Leonid Kantorovich, Brenier Theorem.
Mark Rudelson
Summary: Wigner Semicircle Law for normalized eigenvalues of large random symmetric matrices was proved, which can be viewed as a non-commutative version of the Central Limit Theorem. For this purpose the following technical tools were introduced and developed: Stieltjes transform, Hanson-Wright inequality, self-consistent equation for Stieltjes transform.