This talk focuses on the logic side of the following result: the non-definability of free independence in the theory of tracial von Neumann algebras and C*-probability spaces. I will introduce continuous model theory, which is suitable for the study of metric structures. Definability in the continuous setting differs slightly from that in the discrete case. I will introduce its definition, give examples of definable sets, and prove an equivalent ultrapower condition of it. A. Berenstein and C. W. Hanson exposited model theory for probability spaces in 2023, which was done with continuous model theory. It makes it natural for us to consider the definability of the notion of free independence in probability spaces. I will explain our result, which gives an example of a non-definable set.

This is work with William Boulanger and Emma Harvey, supervised by Jenny Pi and Jakub Curda.

I will talk about some recent work with Tingxiang Zou on higher-dimensional Elekes-Szabó problems in the case of an Ind-constructible action of a group G on a variety X. We expect nilpotent algebraic subgroups N of G to be responsible for any such; this roughly means that if H and A are finite subsets with non-expansion |H*A| <= |A|^{1+\eta}, then H concentrates on a coset of some such N.

A natural example is the action of the Cremona group of birational transformations of the plane. I will talk about a recent result which confirms the above expectation when we restrict to the group of polynomial automorphisms of the plane, using Jung's description of this group as an amalgamated free product, as well as some work in progress which combines weak polynomial Freiman-Ruzsa with effective Mordell-Lang, after Akshat Mudgal, to handle some further special cases.

This talk will focus on various definitions of orientability for non-smooth spaces with Ricci curvature bounded from below. The stability of orientability and non-orientability will be discussed. As an application, we will prove the orientability of 4-manifolds with non-negative Ricci curvature and Euclidean volume growth. This work is based on a collaboration with E. Bruè and A. Pigati.

The Ginzburg–Landau energy is often used to approximate the Dirichlet energy. As the perturbation parameter tends to zero, critical points of the Ginzburg–Landau energy converge, in an appropriate (bubbling) sense, to harmonic maps. In this talk I will first explain key analytical properties of this approximation procedure, then show that not every harmonic map can be approximated in this way. This is based on a rigidity theorem: under the energy threshold of 8pi, we classify all solutions of the associated nonlinear elliptic system from S2 to S2, thereby identifying exactly which harmonic maps can arise as Ginzburg–Landau limits in this regime.