L3

The model-theoretic construction of the binding group in the presence of internality has proven to be one of the most important group existence theorems. It saw numerous applications, including those in differential Galois theory, envisaged by Poizat, and pursued by Pillay in his generalisation of the strongly normal theory of Kolchin.

We show how to view the binding group construction as an instance of Janelidze's categorical Galois theory. This allows us to apply it in areas beyond model theory, for example to the scheme-theoretic treatment of Pillay's differential Galois theory.

We will discuss connections with Hrushovski’s work on groupoids.

I will discuss an application to probability theory (more precisely, to the theory of exchangeable random variables) of recent developments in affine logic, the fragment of continuous logic in which only affine connectives are allowed.

An affine theory is simplicial if its type spaces, which are always compact convex sets, form Choquet simplices. A remarkable dichotomy holds: for a complete simplicial theory, either all type spaces are Bauer simplices (i.e., the set of extreme points is closed) or they are all Poulsen simplices (i.e., the extreme points are dense).

On the other hand, simplices of invariant measures arising in ergodic theory and probability theory can be represented as *quantifier-free* type spaces of appropriate universal affine theories. I will discuss a quantifier-free adaptation of the dichotomy result mentioned above and its application to the geometry of certain simplices of invariant measures arising in exchangeability theory. Relative property (T) for Polish groups will play an important role.

In this talk, we study the numerical approximation of the stochastic heat equation on the sphere driven by an additive Lévy process. For this, we first prove new regularity results for the solution of the stochastic heat equation under different regularity assumptions on the initial value and driving Lévy process. In these settings, we perform a spectral approximation based on the truncation of the series expansion with respect to the real-valued spherical harmonic functions. Further, we apply a forward resp. backward Euler-Maruyama scheme for the temporal approximation. We prove strong and weak convergence rates for the introduced approximation scheme and present numerical simulations that confirm our theoretical results.

This is joint work with Annika Lang and Andrea Papini.

I will discuss probabilistic and analytic aspects of two-dimensional Coulomb gases confined to Jordan domains, curves, and arcs. In particular, I will describe the asymptotic behavior of particle distributions as the number of particles tends to infinity. I will also explain how the Loewner energy and related quantities arise in the asymptotic expansion of the free energy. A key analytical tool throughout is the Grunsky operator. 

The talk is based on joint works with Kurt Johansson.