# Video Lectures

In this talk we consider the classical Monge-Amp´ere equation in two dimensions in a low-regularity regime:

(0.1) det D 2u = f on D ? R2 .

We will assume that f is a given strictly positive, smooth function, but we want to assume as...

Let f be an embedding of a non compact manifold into an Euclidean space and p_n be a divergent sequence of points of M. If the image points f(p_n) converge, the limit is called a limit point of f. In this talk, we will build an embedding f of a...

The "c-principle" is a cousin of Gromov's h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that for the MT-theorem, when the base dimensions is not equal four, only the mildest cobordisms...

We describe a geometric framework to study Newton's equations on infinite-dimensional configuration spaces of diffeomorphisms and smooth probability densities. It turns out that several important PDEs of hydrodynamical origin can be described in...

Singularities of smooth maps are flexible: there holds an h-principle for their simplification. I will discuss an analogous h-principle for caustics, i.e. the singularities of Lagrangian and Legendrian wavefronts. I will also discuss applications...

### Chaos in the incompressible Euler equation on manifolds of high dimension

In this talk we will show how to construct finite dimensional families of non-steady solutions to the Euler equations, existing for all time, and exhibiting all kinds of qualitative dynamics in the phase space of divergence-free vector fields, for...

### Lefschetz fibrations on the Milnor fibers of cusp singularities and applications

We introduce Lefschetz fibration structures on the Milnor fibers of simple-elliptic and cusp singularities in complex three variables, whose regular fibers are diffeomorphic to the 2-torus. We know two ways to construct them and explain h-principle...

The field of continuous combinatorics studies large (dense) combinatorial structures by encoding them in a "continuous" limit object, which is amenable to tools from analysis, topology, measure theory, etc. The syntactic/algebraic approach of "flag...

Convex integration and the holonomic approximation theorem are two well-known pillars of flexibility in differential topology and geometry. They each seem to have their own flavor and scope. The goal of this talk is to bring new perspective on this...