I use the same title for different talks, changing the selection of topics and the level of detailedness for each topic:) This is a survey talk based on joint projects with D. Chen, S. Ivanov, late Ya. Kurylev, M. Lassas, J. Lu, and, if time permits...

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Workshop on the h-principle and beyond

Cieliebak and Eliashberg showed that there is a special class of flexible symplectic structures that satisfy an h-principle and hence have `trivial' symplectic topology. In this talk, I will explain that it is fruitful to think of flexibilization...

We describe a construction of a locally conformal symplectic structure homotopic to any given non-degenerate 2-form and whose Lee form can be any non-exact 1-form. Moreover, each connected component of the boundary, if any, may be chosen to be...

Traditionally, objects of study in symplectic geometry are smooth - such as symplectic and Hamiltonian diffeomorphisms, Lagrangian (or more generally, isotropic and co-isotropic) submanifolds etc. However, in the course of development of the field...

In the first half of the talk I will review Gromov's work on convex integration for open differential relations. I will put particular emphasis on comparing various flavours of ampleness and, in particular, I will note that the different flavours...

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 little...

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...