Vorträge im Sommersemester 2025
10 Apr - Leo Gitin. The Galois characterisation of p-adically closed fields and the function field analogy
A field K has absolute Galois group isomorphic to that of Q_p if and only if K is p-adically closed—a deep result due to Efrat, Koenigsmann, and Pop. While the original proof is difficult, developments over the past three decades allow for simplifications and a more systematic approach. In upcoming joint work with Koenigsmann and Stock, we present a new and more transparent treatment.
I will focus on a key component of the argument: determining the discreteness of fields whose absolute Galois group is isomorphic to that of Q_p. Here, a version of the function field analogy plays a central role. Recent advances by Jahnke and Kartas shed new light on this part of the theory and help us complete the picture.17 Apr - Stefan Ludwig. Model theory of difference fields with an additive character on the fixed field
Motivated by work of Hrushovski on pseudofinite fields with an additive character we investigate the theory ACFA+ which is the model companion of the theory of difference fields with an additive character on the fixed field working in (a mild version of) continuous logic. Building on results by Hrushovski we can recover it as the characteristic 0-asymptotic theory of the algebraic closure of finite fields with the Frobenius-automorphism and the standard character on the fixed field. We characterise 3-amalgamation in ACFA+ and obtain that ACFA+ is simple as well as a description of the connected component of the Kim-Pillay group. We will mention some results on higher amalgamation.
If time permits, we discuss a generalisation of Hrushovski's result in a different direction, namely pseudofinite fields equipped with both an additive and a multiplicative character. Estimates of character sums on curves over finite fields play a crucial role and ultimately allow for a generalisation of the definability of the Chatzidakis-Macintyre-van den Dries counting measure to this context.24 April - Akash Hossain. Forking in pure short exact sequences
Literature on model theory of Henselian valued fields usually establishes,
or relies on transfer principles between the theory of a valued field, and
that of its value group Gamma and residue field k. Recent contributions
often use an alternative approach, which is to study transfer principles
with an intermediate reduct of the valued field: the *leading-term
structure* RV, the expansion of the Abelian group sitting in the pure short
exact sequence (PSES, for short):
1->k*->RV->Gamma->0
The cleanest, most natural and most general framework to study this
structure is that developed in Section 4 of the very influential (and
recent) article by Aschenbrenner-Chernikov-Gehret-Ziegler: the setting of
PSES of *Abelian structures*, with an arbitrary expansion on their term on
the left and the right (such as the order on Gamma and addition on k). The
aforementioned article establishes transfer principles for *quantifier
elimination* and *distality* between the middle term (RV), and the two
other terms (Gamma, k). Thanks to this very general setting, those results
carry over for free to natural expansions of valued field (by a derivation,
an automorphism...).
In this talk, we present our contribution to this work, where we establish
similar transfer principles for *forking and dividing *in the same setting
of expansions of PSES of Abelian structures.8 May - Sebastian Eterović. Likely intersections
The Zilber-Pink conjecture is a central open problem in
arithmetic geometry, which describes the interplay between local
arithmetic behaviour and global geometric behaviour in terms of unlikely
intersections. In joint work with Thomas Scanlon we proved a strong
counterpart of Zilber-Pink describing the presence of likely
intersections. We give an axiomatic set up where the proof works, which
encompasses the many arithmetic varieties in which Zilber-Pink has been
studied. In this talk I will describe the set up and our main theorem on
likely intersections.15 May - Eran Alouf. Towards classifying dp-minimal expansions of (Z,+)
Every known dp-minimal proper expansion of (Z,+) is either:
(1) interdefinable with (Z,+,<),
(2) defines the cyclic order induced by some embedding of Z in R/Z, or
(3) defines the generalized valuation induced by a descending chain of
subgroups.
I will present my work towards proving that this classification holds for
all dp-minimal proper expansions of (Z,+).22 May - Hannah Boß. Hyperbolic groups under HNN extensions
In 1998, Kharlampovich and Myasnikov proved that if the two associated subgroups of an HNN extension of a hyperbolic group satisfy certain conditions, then the extension itself is hyperbolic.
In this talk, I will first familiarize you with this result. Afterwards, I will present an alternative proof of it under more restrictive assumptions.
The basic approach is to show that geodesic triangles in a Cayley graph of the HNN extension are slim. This will be achieved by making use of the observation that the considered Cayley graph is built from a family of copies of a Cayley graph of the base group that are arranged in tree shape.5 June - Zahra Mohammadi Khangheshalghi. On the model theory of the Free Factor Complex of Rank 2
We axiomatize the theory of the Farey graph and show that it is \omega-stable of Morley rank \omega. The Farey graph is isomorphic to the complex of conjugacy classes of free factors of rank 2. We then use the model-theoretic properties of the Farey graph to see what happens in the case of the complex of free factors of rank 2. This is joint work with Katrin Tent.
26 June - Mariana Vicaria. On model theoretic AKE perspectives towards applications in tame geometry
One of the most striking results in the model theory of henselian valued
fields is the AKE principle, which roughly states that the first order
theory of a henselian valued field of equicharacteristic zero is fully
determined by the first order theory of its residue field and the first
order theory of its value group. A model theoretic principle follows from
this theorem: the model theoretic behavior of a henselian valued field is
fully controlled its residue field and its value group. In this talk I will
convey an overview of instances of this principle for applications on
elimination of imaginaries, residue domination (which is in essence a
notion that reflects how a valued field is controlled by its residue
field), on how these tools lead the ground to the study of the topological
space of definable types orthogonal to the value group. The later ones
correspond to geometric spaces arising naturally in geometry: the
berkovich analyfication of a variety, the real analyfication of a
semi-algebraic set, etc.
The talk will introduce all the required background, and questions are more
than welcomed.
This work encompasses joint work (some of it in progress) with : Silvain
Rideau-Kikuchi, Pierre Simon, Pablo Cubides and Jinhe Ye.3 July - Simon André. Homogeneity in Coxeter groups
A group G is said to be homogeneous if any two finite tuples of elements of G that have the same type are in the same orbit under the action of the automorphism group of G. Ould Houcine and Perin - Sklinos proved that finitely generated free groups are homogeneous, but the situation is completely different in the presence of elements of finite order. In this talk, I will present some results on homogeneity of virtually free groups and (hyperbolic) Coxeter groups (a nice class of groups generated by involutions). This is based partly on joint work with Gianluca Paolini.
10 July - Béranger Séguin. Applications of point-counting on difference schemes to the statistics of function fields
Number theorists have long been interested in the quantitative aspects of the distribution of Galois groups of field extensions. Recently, progress has been realized in counting extensions of function fields over (large enough) finite fields by reducing to characteristic zero, and more specifically to the topology of certain varieties which parametrize extensions. However, these methods apply only to "tame" extensions, where the characteristic does not divide the order of the Galois group.
The "wild" case, when the Galois group is a p-group and p is the characteristic of the base field, is very mysterious. In recent work with Fabian Gundlach, we have related extensions of the local function field 𝔽_q((T)) to the solutions to certain equations over the ring W(𝔽_q) of Witt vectors. These equations involve the absolute Frobenius automorphism σ : x ↦xᵖ, making them difference equations. Counting extensions (including questions like reduction to characteristic 0, uniformity in the prime p, etc.) is then related to counting points on difference schemes and to the "asymptotic behavior" of the absolute Frobenius automorphism as p grows, thus connecting the initial problem to Hrushovski-Lang-Weil-type estimates.
In this talk, I will present these connections and our current results. Efforts will be made towards rephrasing some of our questions in model-theoretic language, including open questions which may benefit from a model-theoretic perspective.