I'll introduce the basics of the model theory of metric structures with the goal of using operator systems, essentially vector spaces with "higher level" real, ordered structure, as a main example.

Let $C_1,C_2\subseteq\mathbb{G}_m^N(\mathbb{C})=(\mathbb{C}\setminus\{0\})^N$ be geometrically irreducible closed algebraic curves, where $N\geq 3$. Suppose $C_1$ is not contained in a proper algebraic subgroup of $\mathbb{G}_m^N(\mathbb{C})$. Let $\mathcal{N}=\{n\in\mathbb{N}:[n]C_1\subseteq C_2\}$, where $[n]C_1=\{x^n:x\in C_1\}$. It is conjectured that $\bigcup\limits_{n\in\mathbb{N}\setminus\mathcal{N}}\{x\in C_1:x^n\in C_2\}$ is finite. In particular, responding to a question of Aaron Levin, Umberto Zannier showed that this conclusion follows from Boris Zilber's conjecture on intersections with tori (CIT).

Martin Bays, Jonathan Kirby and Alex Wilkie proved an analogue of Schanuel's conjecture for the operation of raising to an exponentially transcendental power. One might hope to apply that to this problem by assuming $\bigcup\limits_{n\in\mathbb{N}\setminus\mathcal{N}}\{x\in C_1:x^n\in C_2\}$ to be infinite and then obtaining a non-standard point on $C_1$ with an infinite power on $C_2$ whose existence would hopefully contradict the Bays-Kirby-Wilkie result and so be impossible. Unfortunately, in the complex exponential field, infinite integers are not exponentially transcendental. However, some version of the idea can be made to work and we obtain the conjecture in the case where at least one of the two curves is not defined over a number field.

Stratifications are a classical tool to study singularities of real and complex algebraic varieties. Working in a larger field extension equipped with a valuation, I. Halupczok introduced a valuative variant which he called t-stratifications. He proved their existence and showed how they can be used to recover Whitney stratifications over $\mathbb{R}$ and $\mathbb{C}$. In a joint work with I. Halupczok, we introduced a strengthening of the notion of t-stratification, which we call t$^2$-stratifications. We proved their existence over algebraically closed valued fields and show how they induce the existence of Lipschitz stratifications of complex algebraic varieties. The aim of this talk is to introduce both t and t$^2$-stratifications, motivated by their relation with the above mentioned classical notions. If time permits, I will say a bit more about stronger variants arising in a similar framework.

Recovering algebraic structures from geometric objects has been a long standing problem in mathematics, with well known theorems such as Hilbert's construction of the real field from his axioms for Euclidean geometry. These ideas were formalized by Zilber in what has been a driving problem in model theory called the "trichotomy principle" which states that in a strongly minimal structure, if one has a large enough definable family of definable curves (in a strongly minimal structure) then there is an underlying definable field.

This is not true in full generality (Hrushovski constructed counterexamples), but it is true in many important settings where one has additional hypotesis about the context in which the strongly minimal set can be defined. In this talk we will introduce this conjecture, give some motivation for it, and present some recent positive results in the context of algebraically closed valued fields.

This talk is meant to be an introduction to the subject. Notions such as algebraically closed valued fields, reducts and strongly minimal sets will be defined.

The model theory of expansions of the ordered real field has been of interest for several decades. In particular Wilkie showed that the expansion of the ordered real field by the real exponential function is o-minimal. By combining Wilkie's methods with a functional transcendence result Bianconi showed that no non-trivial restriction of sine to an interval is definable in this structure.

In this talk I will discuss similar nondefinability results for a function related to exp, namely the Weierstrass P-function. I will conclude with a more general nondefinability result for the exponential maps of abelian varieties, building on previous work of Gareth Jones, Jonathan Kirby and Tamara Servi. This latter result is joint work with Jones and Kirby.

(based on joint works with L.S. Krapp and S. Kuhlmann)

Let k be a field, G a totally ordered abelian group. The maximal field of generalised power series k((G)), endowed with the canonical valuation v, is important in classifying valued fields.

I will study the group v-Aut K of valuation preserving automorphisms of a subfield k(G) ⊆ K ⊆ k((G)), where k(G) is the fraction field of the group ring k[G].

Two important "lifting properties" will allow to decompose v-Aut K into a semi-direct product of notable subgroups, and I will show how to construct fields satisfying said lifting properties.

I will then introduce the strongly additive automorphisms of K, i.e., automorphisms commuting with infinite sums, for which even better descriptions are possible.

To illustrate the methods I will apply them to more familiar examples such as the fields of Laurent Puiseux series.

References:

• [1] L. S. Krapp, S. Kuhlmann and M. Serra, 'Generalised power series determined by linear recurrence relations', arXiv preprint arXiv:2206.04126 (2022).
• [2] L. S. Krapp, S. Kuhlmann and M. Serra, 'On Rayner structures', Comm. Algebra 50 (2022) 940–948, DOI:10.1080/00927872.2021.1976789 .
• [3] S. Kuhlmann and M. Serra, 'The automorphism group of a valued field of generalised formal power series', J. Algebra 605 (2022) 339–376, DOI:10.1016/j.jalgebra.2022.04.023 .
• [4] M. Serra, 'Automorphism groups of hahn groups and hahn fields', Ph.D. thesis, Universitaet Konstanz (2021).