A Kleinian group is a discrete subgroup $\Gamma\subset PSL_2\mathbb{C}=$ Aut $\mathbb{P}^1$. The regular set $\Omega_\Gamma$ is the maximal set on which all elements of $\Gamma$ form a normal fanimly of maps $\Omega_\Gamma\to \mathbb{P}^1$.  The quotient $\Omega_\Gamma$ is a Riemann surface, and the projection $\pi:\Omega_\Gamma \to \Omega_\Gamma/\Gamma$ is a ramified covering map, unramified is $\Gamma$ is torsion free.

Let $f:\mathbb{P}^1 \to \mathbb{P}^1$ be a rational function of degree $d\ge 2$. The Fatou set $\Omega_f$ is the maximal set on which the iterates $f^{\circ n}$ form a normal family. The Fatou set is invariant under $f$ and $f^{-1}$, so components are mapped to components by proper maps.

The two theorems I want to describe are:

Theorem: [Ahlfors] if $\Gamma$ is a finitely generated Kleinian group, then  $\Omega_\Gamma$ is a finite union of Riemann surfaces of finite type.

Theorem: [Sullivan] If $f$ is a rational function of degree $d\ge 2$, then every component of $\Omega_f$ is eventually periodic.
 

Although the two theorems appear to be unrelated, the proofs are extremely similar.  I will give these proofs and emphasize their similarity.

The question of whether quantization and reduction commute dates back to the early development of formal deformation quantization. For symplectic manifolds with sufficiently well-behaved group actions, this question has been answered positively. In my talk, I aim to broaden the range of examples of star products by considering those that quantize linear Poisson structures on vector bundles, also known as Lie algebroids. I will show that so-called homogeneous star products admit a relatively simple classification in terms of an associated characteristic class. Using this framework, I will explain when a star product can be reduced and how to compute the characteristic class of its reduction in the realm of coisotropic reduction.

The study of connected algebraic subgroups of the Cremona group is a classical way of deepening the understanding of the Cremona group.

Via the Weil regularisation theorem and the Minimal Model Program, to such a group we associate a rational Mori fibre space on which it acts regularly.

In this talk, we will discuss the notion of maximal connected algebraic subgroups of the Cremona group, and its relation with the geometry of the associated Mori fibre spaces.

This is a work in collaboration with A. Fanelli and S. Zimmermann.
 

Scientific results require reproducibility and validation to become trustworthy, reliable, and support decision-making. Provenance can be used to evaluate data quality, reliability, and trustworthiness. In the context of scientific applications, it can provide an audit trail.
But how do we move beyond the concept? How can capturing provenance actually benefit scientific applications?
This talk addresses these essential questions. We will cover the core concepts of provenance, explore its applications in scientific computing, from monitoring to data analysis, and discuss approaches and tools designed to seamlessly capture and manage provenance information within complex simulation environments. Discover how to transition from opaque results to auditable, trustworthy scientific results!

The problem of guided wave propagation is considered. It is shown that in a three-dimensional waveguide with perfectly reflecting boundaries the field can be represented as a superposition of normal modes whose amplitudes satisfy Klein-Gordon equations. This approach can be further extended to the case of more general layered media by introducing certain effective parameters (e.g., effective propagation velocity). The capability of this technique is illustrated by the solution of a realistic sound propagation problem in a shallow-water waveguide. A simple numerical methods for solving Klein-Gordon equations in the context of wave packet propagation is also introduced, and the absorbing boundary conditions required for artificial domain truncation are derived. 

We prove uniqueness results for capillary disks that are modeled by an elliptic PDE, in three-dimensional domains, assuming that the domain contains a family of surfaces that have further properties. Our main result is a generalization of Nitsche's uniqueness theorem for capillary constant mean curvature disks on the Euclidean ball, and is inspired by Gálvez and Mira's generalization of Hopf's uniqueness theorem for constant mean curvature spheres in the Euclidean space.