Gibbs measures on spaces of functions or distributions play an important role in various contexts in mathematical physics.  They can, for example, be viewed as continuous counterparts of classical spin models such as the Ising model, they are an important stepping stone in the rigorous construction of Quantum Field Theories, and they are invariant under the 
flow of certain dispersive PDEs, permitting to develop a solution theory with random initial data, well below the deterministic regularity threshold. 

These measures have been constructed and studied, at least since the 60s, but over the last few years there has been renewed interest, partially due to new methods in stochastic analysis, including Hairer’s theory of regularity structures and Gubinelli-Imkeller-Perkowski’s theory of paracontrolled distributions. 

In this talk I will present two independent but complementary results that can be obtained with these new techniques. I will first show how to obtain estimates on samples from of the Euclidean $\phi^4_3$ measure, based on SPDE methods. In the second part, I will discuss a method to show the emergence of phase transitions in the $\phi^4_3$ theory.

The homogeneous coordinate ring of a projective variety may be constructed by geometrically quantizing the multiples of a symplectic form, using the complex structure as a polarization. In this talk, I will explain how a holomorphic Poisson structure allows us to deform the complex polarization into a generalized complex structure, leading to a non-commutative deformation of the homogeneous coordinate ring. The main tool is a conjectural construction of a category of generalized complex branes, which makes use of the A-model of an associated symplectic groupoid. I will explain this in the example of toric Poisson varieties. This is joint work with Marco Gualtieri (arXiv:2108.01658).

TBA

The so called Drinfeld conjecture states that the complement to very stable bundles has pure codimension one in the moduli space of vector bundles. In this talk I will explain a constructive proof in rank three, and discuss if/how it generalises to wobbly fixed points of the nilpotent cone as defined by Hausel and Hitchin. This is joint work with Pauly (Nice).

The curve graph of a surface of finite type is a fundamental object in the study of its mapping class group both from the metric and the combinatorial point of view. I will discuss joint work with Valentina Disarlo and Thomas Koberda where we conduct a thorough study of curve graphs from the model theoretic point of view, with particular emphasis in the problem of interpretability between different curve graphs and other geometric complexes.