# Video Lectures

Let f:Y→X be a finite covering map of complex algebraic varieties. The essential dimension of f is the smallest integer e such that, birationally, f arises as the pullback of a covering Y′→X′

of dimension e, via a map X→X′. This invariant goes back...

A theorem of Borel says that any holomorphic map from a complex algebraic variety to a smooth arithmetic variety is automatically an algebraic map. The key ingredient is to show that any holomorphic map from the (poly) punctured disc to the Baily...

Let E be a finite degree extension of Qp. Given a mod p representation of the absolute Galois group of E we construct a sheaf on a punctured absolute Banach-Colmez space that should give the first step in the construction of the mod p local Langlands...

Explicit constructions of "random-like" objects play an important role in complexity theory and pseudorandomness. For many important objects such as prime numbers and rigid matrices, it remains elusive to find fast deterministic algorithms for...

All large-scale structure cosmologists are faced with the question: how do we robustly extract cosmological information, such as on dark energy, gravity, and inflation, from observed tracers such as galaxies whose astrophysics is extremely complex...

Lusztig's theory of character sheaves for connected reductive groups is one of the most important developments in representation theory in the last few decades. In this talk, we will describe a construction which extends this "depth zero" picture to...

We develop on a new strategy based on point-set topology, which allows us to produce a purely p-adic statement for the crystallinity properties of rigid flat connections.

Joint with Michael Groechenig.

Let X be a proper, smooth rigid space and G a commutative rigid group. We study the relationship between G-representations of the fundamental group of X and G-Higgs bundles on X. This is joint work with Ben Heuer and Mingjia Zhang.

### Symplectic Capacities of Domains Close to a Ball and Geodesics in the Space of Contact Forms

An old open question in symplectic geometry asks whether all normalised symplectic capacities coincide for convex domains in the standard symplectic vector space. I will show that this question has a positive answer for smooth convex domains which...