In recent papers by Alon et al. and Fox et al. it is demonstrated that families of graphs with a semialgebraic edge relation of bounded complexity have strong regularity properties and can be decomposed into very homogeneous semialgebraic pieces up to a small error (typical example is the incidence relation between points and lines on a real plane, or higher dimensional analogues). We show that in fact the theory can be developed for families of graphs definable in a structure satisfying a certain model theoretic property called distality, with respect to a large class of measures (this applies in particular to graphs definable in arbitrary o-minimal theories and in p-adics). (Joint work with Sergei Starchenko.)[-]

An atomic class $K$ is the class of atomic first order models of a countable first order theory (assuming there are such models). Under the weak $\mathrm{GCH}$ it had been proved that if such class is categorical in every $\aleph_n$ then it is categorical in every cardinal and is so called excellent. There are results when we assume categoricity for $\aleph_1, \ldots, \aleph_n$. The lecture is on a ZFC result in this direction for $n=1$. More specifically, if $K$ is categorical in $\aleph_1$ and has a model of cardinality $>2^{\aleph_0}$, then it is $\aleph_0$-stable, which implies having stable amalgamation, and is the first case of excellence.
This a work in preparation by J.T. Baldwin, M.C. Laskowski and S. Shelah.[-]

Which Noetherian integral domains are NIP? This is a natural question to ask, given the prominence of Noetherian rings in commutative algebra. We cannot hope to answer this question in full generality any time soon, as it includes other hard problems such as the conjectures on stable fields and NIP fields. Nevertheless, we present some interesting partial results which begin to paint a picture of NIP Noetherian rings. Let $R$ be a Noetherian domain which is NIP. Then either $R$ is a field, or $R$ is a semilocal domain of Krull dimension 1 and characteristic 0. Assuming the henselianity conjecture on NIP valued fields, $R$ is a henselian local ring. In the dp-minimal case, one can give a complete classification. Specifically, every dp-minimal Noetherian domain is a finite index subring of a dp-minimal discrete valuation ring. The situation in dp-rank 2 seems to be significantly worse, but a classification may still be possible in terms of differential valued fields.[-]

We extend the notion of beautiful pairs by Poizat to unstable theories via definable types, with a specific interest in such pairs of valued fields. In particular, we establish an analogue of Ax-Kochen-Ershov principles in for certain pairs of valued fields. In the specific case of ACVF, we classify all such pairs and deduce the strict pro-definability of various spaces of definable types, such as the stable completion introduced by Hrushovski-Loeser and a model theoretic analogue of the Huber analytification of an algebraic variety. This is joint with Pablo Cubides Kovacsics and Martin Hils.[-]

If CCM denotes the theory of compact complex spaces in the langauge of complex-analytic sets, then the theory of models of CCM equipped with an automorphism has a model companion, denoted by CCMA. The relationship to meromorphic dynamical systems is the same as that of ACFA to rational dynamical systems. I will discuss recent joint work with Martin Bays and Martin Hils that begins a systematic study of CCMA as an expansion of ACFA. Particular topics we consider include: stable embeddedness, imaginaries, and the Zilber dichotomy.[-]