It is a long-standing open problem whether the ring of integers Z has an existential first-order definition in Q, the field of rational numbers. A few years ago, Jochen Koenigsmann proved that Z has a universal first-order definition in Q, building on earlier work by Bjorn Poonen. This result was later generalised to number fields and to global function fields, using classical machinery from number theory and class field theory related to the behaviour of quaternion algebras over global and local fields.

In this talk, I will sketch a variation on the techniques used to obtain the aforementioned results. It allows for a relatively short and uniform treatment of global fields of all characteristics that is less dependent on class field theory. Instead, a central role is played by Hilbert's Reciprocity Law for quadratic forms. Finally, I will touch on quantitative aspects of the method, and, if time allows, discuss other instances in which similar techniques can be applied.

A long standing question in both model theory and group theory has been whether all non-abelian free groups are elementarily equivalent. The question was famously asked by Tarski around 1945 and answered positively by Z. Sela and, independently, by O. Kharlampovich and A. Myasnikov in 2006. In pursuit of the question, Sela studied what he called (hyperbolic) towers. This concept was further investigated by C. Perin and R. Sklinos, among others, who worked out multiple details and further applications of the concept. The central justification for studying hyperbolic towers is the following result due to Sela: If G is a non-abelian torsion-free hyperbolic group and a hyperbolic tower over some non-abelian subgroup H, then H is elementarily embedded in G. And its converse, due to Perin: If a torsion-free hyperbolic group H is elementarily embedded into some torsion-free hyperbolic group G, then G is a hyperbolic tower over H.

The definition of hyperbolic towers relies heavily on concepts in geometric group theory. Therefore we will quickly discuss fundamental groups of complexes and graphs of groups. The latter provide a tool to decompose groups into amalgamated products and HNN-extensions, known as Bass-Serre theory. Towers then consist of multiple layers of such decompositions with certain additional properties. We will work our way through the definitions along multiple examples and state the main results. If time permits, we will also discuss a generalization of classical Fraïssé limits, which was used by Kharlampovich-Myasnikov and later by Guirardel-Levitt-Sklinos to provide a homogeneous group in which all non-abelian free groups (Kharlampovich-Myasnikov) or more generally, all elementarily equivalent torsion-free groups (Guirardel-Levitt-Sklinos) embed elementarily.

Zilber’s quasiminimality conjecture predicts that all subsets of the complex numbers that are definable using the language of rings and the exponential function are either countable or cocountable. Building on Zilber’s work, Bays and Kirby have proved that the quasiminimality conjecture would follow from the exponential-algebraic closedness conjecture, also due to Zilber, which predicts sufficient conditions for systems of equations in polynomials and exponentials to have complex solutions.

In this talk, I will give an introduction to this topic before presenting some recent work which solves the conjecture for a class of algebraic varieties which corresponds to systems of exponential sums. This turns out to be closely related to tropical geometry, a "combinatorial shadow" of algebraic geometry which reduces some questions about algebraic varieties to questions about polyhedral objects.

We will discuss new decidability results for mixed characteristic henselian fields, whose proof goes via reduction to positive characteristic. In order to achieve the reduction, we will use extensively the theory of perfectoid fields/p-adic Hodge theory and also the earlier Krasner-Kazhdan-Deligne principle. Our main results are the following:

(1) A relative decidability result for perfectoid fields. This says that, under a certain natural assumption, a perfectoid field K is decidable relative to its tilt K^♭. As an application, we obtain several decidability results for tame fields of mixed characteristic, transposing a recent result of Lisinski (building on earlier work of Kuhlmann). We also obtain a different application by transposing work of Anscombe-Fehm to mixed characteristic.

(2) An undecidability result for the asymptotic theory of p-adic fields (fixed p). This says that the set of sentences in the language of valued fields with cross-section which are true in all but finitely many finite extensions of Qp is undecidable. This should be contrasted with the Ax-Kochen/Ershov Theorem, saying that each individual p-adic field is decidable in said language.

The model theory of tame fields in the language of valued fields has been extensively studied by Kuhlman. In particular, the theory of an equal characteristic tame field in this language is given by the theory of the residue field and the theory of the value group. Building on Kuhlman's results, we give an AKE-principle for tame valued fields of equal characteristic in L_t, the language of valued fields with a distinguished constant symbol t. Furthermore, we use this principle together with Kedlaya's work on the connection between generalised power series and finite automata to show that a tame Hahn field of equal characteristic is decidable in L_t if it has decidable residue field and decidable value group. In particular, we obtain decidability of F_p((t^{1/p^\infty})) and F_p((t^Q )) in L_t.

Time permitting, we will also see how approximation methods used in this work reveal a condition on algebraicity for generalised power series in terms of the order type of the support.

We study the class of sets definable by first-order formulae in henselian valued fields. The guiding principle, building on classical work by Ax-Kochen and Ershov, is that definable sets in well-behaved henselian valued fields are governed by those of the residue field and the value group. We present a new theorem generalizing these classical ideas.

We then discuss when a valuation is so intrinsic to the field that its valuation ring is definable using just the arithmetic of the field. This is closely linked to a conjecture by Shelah that considers fields for which the class of definable sets has restricted combinatorial complexity, i.e., no formula has the independence property. The conjecture predicts that any infinite such field is separably closed, real closed, or admits a nontrivial henselian valuation.

We present the state of the art regarding the conjecture, including a theorem that any NIP henselian valued field satisfies an Ax-Kochen/Ershov principle. Arithmetic definability of valuations turns out to be a key tool in this context.

In [1] it was shown that the automorphism groups of (1-)order expansions of several homogeneous structures with a stationary independence relation are simple. We follow the basic approach from [1] in order to show that the same holds true for n-order expansions, 1<n. However, the passage from 1-orders to n-orders causes some extra challenge that requires a different proof idea. My proof involves a reduction of the problem to 1-orders that comes, in return, with significant constraints we have to deal with.

[1] F. Calderoni, A. Kwiatkowska, K. Tent. Simplicity of the Automorphism Groups of Order and Tournament Expansions of Homogeneous Structures.

One of the leading questions in many mathematical research programs is whether a certain classification problem admits a “satisfactory” solution. What constitutes a satisfactory solution depends, of course, on the context and it is often subject to change when the original goals are deemed hopeless. Indeed, in recent years several negative anti-classification results have been attained. For example: by the work of Hjorth and Foreman-Weiss we know that one cannot classify all ergodic measure-preserving transformations using isomorphism types of countable structures as invariants; and by the work of Thomas we know that higher rank torsion-free abelian groups do not admit a simple classification using Baer-invariants such as in the rank-1 case.

In this talk I will provide a gentle introduction to *Invariant Descriptive Set Theory*: a formal framework for measuring the complexity of such classification problems and for showing which types of invariants are inadequate for a complete classification. In the process, I will present several anti-classification criteria which come from topological dynamics. In particular, I will discuss my recent joint work with Shaun Allison, in which we provide a new obstruction to classification by (co)homological invariants and use it to attain anti-classification results for Morita equivalence of continuous-trace C∗-algebras and for the isomorphism problem of Hermitian line bundles.

One of the leading questions in many mathematical research programs is whether a certain classification problem admits a “satisfactory” solution. What constitutes a satisfactory solution depends, of course, on the context and it is often subject to change when the original goals are deemed hopeless. Indeed, in recent years several negative anti-classification results have been attained. For example: by the work of Hjorth and Foreman-Weiss we know that one cannot classify all ergodic measure-preserving transformations using isomorphism types of countable structures as invariants; and by the work of Thomas we know that higher rank torsion-free abelian groups do not admit a simple classification using Baer-invariants such as in the rank-1 case.

In this talk I will provide a gentle introduction to *Invariant Descriptive Set Theory*: a formal framework for measuring the complexity of such classification problems and for showing which types of invariants are inadequate for a complete classification. In the process, I will present several anti-classification criteria which come from topological dynamics. In particular, I will discuss my recent joint work with Shaun Allison, in which we provide a new obstruction to classification by (co)homological invariants and use it to attain anti-classification results for Morita equivalence of continuous-trace C∗-algebras and for the isomorphism problem of Hermitian line bundles.

Abstract: The imaginaries in the theory ACVF of non-trivially valued algebraically closed valued fields are classified by the so-called 'geometric' sorts. This is a fundamental result due to Haskell-Hrushovski-Macpherson. We show that the imaginaries in henselian equicharacteristic 0 valued fields may be reduced, under rather general assumptions, to the geometric sorts and to imaginaries in RV together with sorts for certain finite-dimensional vector spaces over the residue field.

In the talk, I will mainly speak about an application of this reduction, which has been the initial motivation of our work: The imaginaries in the theory VFA of algebraically closed non-trivially valued fields of equicharacteristic 0, endowed with a non-standard Frobenius automorphism, are classified by the geometric sorts. This requires an understanding of imaginaries in pure short exact sequences, and a key result from Hrushovski's groupoids paper.

Abstract: A valued field is called maximal if it admits no proper immediate extentions (having the same residue field and value group). Krull observed that every valued field must have some maximal immediate extension; Kaplansky established sufficient conditions for uniqueness. In doing so, Kaplansky proved that a field is maximal if and only if it is spherically complete: that the intersection of any chain of closed (valuative) balls is non-empty. As can be expected, spherical completeness can be convenient for analytic/geometric arguments. Model theoretically, it can be helpful to transfer to a spherically complete model, if at least one exists.

But, while every valued field has a spherically complete extension, this need not be an elementary extension (even as a valued field). Furthermore, it might be important to preserve extra (algebraic/analytic) structure on the field.

Cluckers, Halupczok and Rideau have recently introduced hensel minimality for expansions of valued fields. In this talk, we discuss the existence of spherically complete models of hensel minimal expansions of valued fields; joint work with Immanuel Halupczok.

In diesem Vortrag geben wir zuerst, anhand von Beispielen fundamentaler Begriffe wie Sprache und Struktur, eine kurze Einführung in die Grundlagen der mathematischen Logik.

Anschließend, stellen wir verschieden elementare Klassen von Strukturen vor und betrachten innerhalb dieser algebraischen Objekte (Gruppen, Ringe, Körper). Kurz gefasst sind diese Klassen durch das Verbot von gewissen Kombinatorischen Mustern in den definierbaren Mengen charakterisiert. Dies hat erstaunliche Konsequenzen für Gruppen als auch Körpern. Außerdem gibt es Verbindungen zur Theorie von Gruppen mit Kettenbedingungen auf Zentralisatioren die wir aufzeigen werden.

Abschließend, stellen wir die Hierarchie der n-abhängigen Theorien vor (für jede natürliche Zahl n). Der n-Zufallsgraph ist das kanonische Objekt welches n-abhängig aber nicht (n-1)-abhängig ist. Folglich ist dies eine strikte Hierarchie. Jedoch liegt die Frage ob algebraische Objekte in jeder dieser Klassen existiert nahe. Motiviert durch diese Frage stellen wir alle bisher bekannten Resultate von n-abhängigen Gruppen und Körpern vor.

Let F be a non-principal ultraproduct of finite fields, T its theory, and A a subfield of F which is relatively algebraically closed in F. It is known that any two models of T which contain A as a relatively algebraically closed subfield are elementarily equivalent over A.

Theorem: If A is not pseudo-finite, then T has no prime model over A.

The proof of this result is fairly easy when A is countable: one just uses that prime models over A are atomic. The proof when A is uncountable is more involved, as it involves constructing 2^{|A|} non-isomorphic models of T which are of transcendence degree 1 over A. We also discuss the existence or non-existence of \kappa-prime models of T (i.e., \kappa-saturated models of T containing A and which A-embed into any \kappa-saturated model of T containing A), for regular uncountable \kappa such that \kappa =\kappa^{<\kappa}.

Extremal combinatorics studies how large or how small a collection of finite objects can be, if it has to satisfy certain restrictions. For instance, in an n-element set, what is the largest number of k-element subsets that can pairwise intersect one another? This question, which has been answered by Erdos, Ko and Rado, can be considered as a typical question in this area.

The Erdos-Ko-Rado theorem can be considered in the context of the group actions. In this talk, I will present some results showing how the classification of minimal finite simple groups, established by Thompson, can be applied to answer some problems in the context of extremal group theory.

The family of high rank arithmetic groups is class of groups which is playing an important role in various areas of mathematics. It includes SL(n,Z), for n>2, SL(n, Z[1/p] ) for n>1, their finite index subgroups and many more. A number of remarkable results about them have been proven including; Mostow rigidity, Margulis Super rigidity and the Quasi-isometric rigidity.

We will talk about a new type of rigidity: "first order rigidity". Namely if D is such a non-uniform characteristic zero arithmetic group and E a finitely generated group which is elementary equivalent to it (i.e., the same first order theory in the sense of model theory) then E is isomorphic to D.

This stands in contrast with Zlil Sela's remarkable work which implies that the free groups, surface groups and hyperbolic groups (many of whose are low-rank arithmetic groups) have many non isomorphic finitely generated groups which are elementary equivalent to them.

Joint work with Nir Avni and Chen Meiri.

We study the complexity of the isomorphism relation for various classes of closed subgroups of the group of permutations on N. We use the setting of Borel reducibility between equivalence relations on Polish spaces.

For profinite, locally compact, and Roelcke precompact groups, we show that the complexity is the same as the one of countable graph isomorphism. For oligomorphic groups, we merely establish this as an upper bound.

Joint work with Alexander Kechris and Katrin Tent

Let $S \subset \mathbb{R}^n$ be a smooth algebraic subset. We consider the system $(E)$ of differential equations which describes the motion of a free particle with $n$ degrees of freedom and fixed energy $E_0$ constrained to move on the submanifold $S$. As $S \subset \mathbb{R}^n$ is a algebraic subset, this system is given by polynomial differential equations and gives rise to a definable set of the theory $\mathbf{DCF}_0$ of differentially closed fields.

In my talk, I will be interested in the model-theoretic behaviour of such definable sets. More precisely, I will describe a general criterion of orthogonality to the constants for types over the field of real numbers (with trivial derivation). Then, using this criteria, I will explain how to deduce from structural results on the dynamics of the system $(E)$, that if $S$ is compact and the restriction of the euclidian metric in $\mathbb{R}^n$ to $S$ has strictly negative curvature, then the generic type of the system $(E)$ is orthogonal to the constants.