Summer School 2026

Abstract

Given a smooth manifold, and a distribution (a lower dimensional subspace of the tangent space on the manifold at each point), we are interested in the question whether there is an embedded submanifold of the manifold whose tangent space is precisely this distribution. In the 1-dimensional case, the answer to this is positive. More precisely, if we are given a non-vanishing vector field on the manifold, then the integral curves of this vector field are immersed submanifolds and locally, we can straighten the vector field to obtain that these integral curves are straight lines. We will quickly review this and standard results on flows.  Next, we consider the question of finding higher dimensional analogues of integral curves (locally embedded submanifolds), given a k-dimensional distribution. There is a necessary condition called involutivity that should be satisfied by the distribution. Frobenius theorem states that the involutivity condition is also sufficient. We will prove local and global versions of Frobenius theorem and give some applications to the study of partial differential equations.

Abstract

Lecture 1&2 — Foundations of Optimal Transport
Monge and Kantorovich formulations; Kantorovich duality; cyclical monotonicity; convex potentials and gradients of convex functions; Brenier–McCann theorem; displacement interpolation; Wasserstein geodesics.

Lecture 3— Regularity of Transport Map
Monge–Ampère equation as a fully nonlinear elliptic PDE; a brief overview of the regularity theory for optimal transport maps; McCann’s pointwise formulation of the Monge–Ampère equation; Caffarelli’s contraction theorem.

Lecture 4&5 — Functional and Geometric Inequalities via Transport
I plan to present transport-based proofs of the following inequalities: Sharp Sobolev inequality in the Euclidean isoperimetric inequality; Gaussian functional inequalities; Talagrand, logarithmic Sobolev, and Poincaré inequalities; transfer of functional inequalities to log-concave perturbations of the Gaussian distribution; The HWI inequality; Brascamp-Lieb inequality, Concentration inequalities.

Abstract

This series of lectures provides an introduction to the Liouville equation , a central model in geometric analysis. We discuss its conformal invariance, explicit radial (bubble) solutions and the classification of finite-mass solutions. The variational structure and the role of the Moser–Trudinger inequality are outlined along with the lack of compactness leading to blow-up and energy quantization in multiples of .