This is joint work with S. Dinew (Krakow). We investigate the class of compact complex Hermitian-symplectic manifolds $X$. For each Hermitian-symplectic metric $\omega$ on $X$, we introduce a functional acting on the metrics in a certain cohomology class of $\omega$ and prove that its critical points (if any) must be K\"ahler when X is 3-dimensional.

The development of free probability theory has drawn much inspiration from its deep and far reaching analogy with classical probability theory. The same holds for its operator-valued extension, where the fundamental notion of free independence is generalized to free independence with amalgamation as a kind of conditional version of the former. Its development naturally led to operator-valued free analogues of key and fundamental limiting theorems such as the operator-valued free Central Limit Theorem due to Voiculescu and results about the asymptotic behaviour of distributions of matrices with operator-valued entries.

In this talk, we show Berry-Esseen bounds for such limit theorems. The estimates are on the level of operator-valued Cauchy transforms and the L{\'e}vy distance. We address also the multivariate setting for which we consider linear matrix pencils and noncommutative polynomials as test functions. The estimates are in terms of operator-valued moments and yield the first quantitative bounds on the L{\'e}vy distance for the operator-valued free CLT. This also yields quantitative estimates on joint noncommutative distributions of operator-valued matrices having a general covariance profile.

This is a joint work with Tobias Mai.

I will discuss some simplified models for the shape of liquid droplets on rough solid surfaces, especially Bernoulli-type free boundary problems. In these models small scale roughness leads to large scale non-uniqueness, hysteresis, and anisotropies. In technical terms we need to understand laminating/foliating families of plane-like solutions, this is related to ideas of Aubry-Mather theory, but, unlike most results in that area, we need to consider local (but not global) energy minimizers.

In geometry, one often starts with a base space (e.g. a manifold, or a variety, etc) and is interested in constructing global objects over the base. One of the ways to construct these global objects is to glue them over the base from simpler local data.

For example, one builds global vector bundles by first describing them locally as products and then gluing them via isomorphisms satisfying certain cocycle conditions. Said more abstractly, one tries to recover the `category' of vector bundles on the base by looking at the `category' of vector bundles on `small open' sets on the base. The fact that one can do this is succinctly summarised by saying that the `category' of vector bundles satisfies $\emph{descent}$ on open sets over topological spaces. More provocatively, one says that the `category' of vector bundles is a $\emph{stack}$ over the base. This abstraction is not just decorative—the analogous statement fails for isomorphism classes of vector bundles!

In this talk I will discuss this and other ideas which go under the collective name of descent. The first half will be an informal introduction to descent with minimal prerequisites. The second half will discuss counterparts of these ideas in the context of algebraic geometry and commutative algebra, although again with minimal prerequisites.

Rigidity phenomena in homogeneous dynamics have been extensively studied over the past few decades with several striking results and applications.
In this talk, we will give an overview of recent activities related to quantitative aspect of the analysis in this context; we will also highlight some applications.

Recently, Taelman defined a ``class module" associated to any Drinfeld module defined over a function field. In the spirit of Iwasawa theory, we will study the structure of these class modules in certain p-adic towers of fields. Using the Equivariant Tamagawa Number Formula for Drinfeld modules, we will propose an Iwasawa main conjecture for these class modules.

In this talk, I will discuss ongoing work to develop a method for proving a shrinking target property on primitive square-tiled surfaces that comes from the action of a subgroup $G$ of its Veech group. Our main tool is the construction of a Fourier-like transform which we can use to relate the $L^2$-norm of the Koopman operator induced by $G$ to the $L^2$-norm of a Markov operator related to a random walk on $G$.

Let $X$ be a smooth, affine, geometrically connected curve
over a finite field of characteristic $p > 2$. Let $\rho:\pi_1(X) \to
\mathbb{C}^\times$ be a character of finite order $p^n$. If $\rho\neq
1$, then the Artin $L$-function $L(\rho,s)$ is a polynomial, and a
theorem of Kramer-Miller states that its $p$-adic Newton polygon
$\mathrm{NP}(\rho)$ is bounded below by a certain Hodge polygon
$\mathrm{HP}(\rho)$ which is defined in terms of local monodromy
invariants. In this talk we discuss the interaction between the polygons
$\mathrm{NP}(\rho)$ and $\mathrm{HP}(\rho)$. Our main result states that
if $X$ is ordinary, then $\mathrm{NP}(\rho)$ and $\mathrm{HP}(\rho)$
share a vertex if and only if there is a corresponding vertex shared by
certain ``local" Newton and Hodge polygons associated to each ramified
point of $\rho$. As an application, we give a local criterion that is
necessary and sufficient for $\mathrm{NP}(\rho)$ and $\mathrm{HP}(\rho)$
to coincide. This is joint work with Joe Kramer-Miller.