19th (Tuesday) | 20th (Wednesday) | 21st (Thursday) | 22nd (Friday) | 23rd (Saturday) | |
---|---|---|---|---|---|

9:30-10:30 | Hils | Ye | Onshuus | Eleftheriou | Ziegler |

10:30-11:00 | Hasson | Castle | Nagloo | Jagiella | Coffee break |

11:00-11:30 | Kuhlmann | ||||

11:30-12:00 | Coffee break | Coffee break | Coffee break | Coffee break | |

12:00-13:00 | Moosa | D'Aquino | Meir | Chatzidakis | End of conference |

13:00-15:30 | Lunch break | Lunch break | Free afternoon | Lunch break | |

15:30-16:30 | Hoffmann | Jaoui | Zou | ||

16:30-17:00 | Coffee break | Coffee break | Boat party | Coffee break | |

17:00-18:00 | Maříková | Tanović | Aschenbrenner | ||

18:00-19:30 | Reception + poster session | ||||

19:40-??? | Conference dinner | ||||

Filling countable gaps in Hardy fields

If *H* is a Hardy field and *A < B* are countable subsets of *H*, then there is an element *f* in a Hardy field extension of *H* with *A < f < B*. Moreover, this statement also holds, not only in the smaller class of smooth Hardy fields, but (somewhat surprisingly) also for analytic Hardy fields. I will outline the proofs of these statements, give some applications, and pose a few related open questions. (Joint work with L. van den Dries and J. van der Hoeven.)

On Zilber's Restricted Trichotomy in Hausdorff Geometric Structures

Continuing from Assaf Hasson's talk, I will discuss results on the Zilber trichotomy for structures interpreted in ACVF. I will introduce a class of structures equipped with a topology, called Hausdorff geometric structures (and including ACVF), and then axiomatize in this setting various stages of the trichotomy proof from the complex field. Restricted to ACVF, the main result says that a non-locally modular strongly minimal structure whose universe is definable (not interpretable) must interpret a one-dimensional group. Time permitting, I will also discuss results for the imaginary sorts, including a full reduction to the real sorts in characteristic zero.

Some questions about difference-differential fields.

In an earlier paper, I showed that while the theory ACFA of existentially closed difference fields does not have prime models, kappa-prime models over algebraically closed subfields with fixed field pseudo-finite and kappa-saturated, exist and are unique in characteristic 0. (Probably never in positive characteristic).

The object of this talk is to discuss a little the proof, and how it could be adapted to the context of difference-differential fields of characteristic 0 (one automorphism, several derivations, everyone commuting). Basically I will mention the differences which makes the proof more difficult than in the case of ACFA, and a strategy of proof. This is joint work with R. Bustamante and S. Montenegro.

Residue rings of models of PA

I will present a join work with A. Macintyre on the model theory of residue rings M/nM where M is a model of Peano Arithmetic. The analysis considers first the case of n a prime power, and then the general case for n composite. In this last case we use Feferman-Vaught classical results on definability in product of structures, which we develop further obtaining an inverse of Feferman-Vaught theorem, i.e. we get a characterization of those unital commutative rings which are elementarily equivalent to a product of unital commutative rings.

Unbounded versions of Zarankiewicz's problem

Zarankiewicz's problem for hypergraphs asks for upper bounds on the number of edges of a hypergraph that has no complete sub-hypergraphs of a given size. Let M be an o-minimal structure. Basit-Chernikov-Starchenko-Tao-Tran (2021) proved that the following are equivalent:

- (1) “linear Zarankiewicz's bounds” exist for hypergraphs whose edge relation is induced by a fixed relation definable in M
- (2) M does not define an infinite field.

We prove that the following are equivalent:

- (1') linear Zarankiewicz bounds exist for sufficiently “distant” hypergraphs whose edge relation is induced by a fixed relation definable in M
- (2') M does not define a full field (that is, one whose domain is the whole universe of M).

We will also report on current work on Zarankiewicz's problem in Presburger Arithmetic.

This is joint work (in progress) with Aris Papadopoulos.

On recent developments around Zilber's Restricted Trichotomy

Recently, Ben Castle proved that any strongly minimal set interpretable in an algebraically closed field of characteristic 0 is either locally modular or interprets a copy of the original field. In the talk, I will discuss a recent proof of the same result in positive characteristic. The proof goes through by extending to ACVF Castle's results as well as some ideas from the proof (with Sustretov) to reducts of algebraic curves (in all characteristics), on which my talk will focus.

The talk will be based on a joint work with Castle and Ye and on the PhD thesis of S. Pinzon.

The amalgamation property for definable types

In order to generalize Poizat's theory of belle paires to unstable theories, one needs to assume the amalgamation property for the class of global definable types. In the talk, we will highlight this connection, exhibit some important classes of theories in which definable types may be amalgamated, and exhibit an example of a theory in which they don't. This talk is based on joint work with Pablo Cubides and Jinhe Ye, and with Rosario Mennuni.

Oscillations in space of types

We generalize a notion of rank from topological dynamics, which was introduced by Glasner and Megrelishvili, to apply it in non-metrizable dynamical systems, like the space of types. This rank counts oscillations of elements of the Ellis semigroup at a given point and, in some sense, refines the well-known Cantor-Bendixson rank. Then we characterize a couple of the main dividing lines in the stability hierarchy by the values our rank can take, values counted in the dynamical system where Aut(M) acts on S(M). A joint work with Alessandro Codenotti.

Bohr compactifications of first-order structures

(Work in progress, joint with K. Krupiński) Among the classical objects associated with a topological group *G* are its \emph{compactifications}, namely homomorphisms into a compact topological group with dense image. These include the universal (or “Bohr”) compactification of *G*. Such compactifications can be obtained via model theory through a more general construction of “definable” compactifications. I will talk about a natural generalization of such construction to the category of (appropriately defined) “topological” ℒ-structures (for a fixed, but arbitrary language ℒ). I will then cover the specific case of (unital) rings, giving the description of their (definable and classical) compactifications, along with some applications.

On the density of strongly minimal algebraic vector fields

I will present a recent result which states that the solution sets of “almost all” autonomous algebraic differential equations are strongly minimal and geometrically trivial. The proof of this result relies on a general construction which produces an algebraic group from a (non linear) algebraic differential equation obtained by applying (linear) differential Galois theory to a linear version of the equation constructed using one-forms.

During my talk, I will describe this construction and explain how certain properties of this group relate to minimality properties of the solution set of the algebraic differential equation under study.

Perfectoid and deeply ramified fields: algebra and model theory

A part of my talk will be devoted to the great recent work of Jahnke and Kartas. By “taming” perfectoid fields they develop Ax-Kochen/Ershov principles which among other applications can be applied to Scholze's tilting construction. To this end they generalize my own model theoretic results on tame fields to the case of roughly tame fields, which in mixed characteristic form a larger class. While the value groups of tame fields with residue characteristic p > 0 are always p-divisible, the value groups of roughly tame fields are only required to be roughly p-divisible in the sense as introduced by Johnson. Then they show that under suitable coarsenings of the valuation, ultraproducts of perfectoid fields become roughly tame fields. In my talk I will sketch some details and main results.

It should be noted that perfectoid fields do not admit AKE principles in the same way as roughly tame fields do (this is why one has to pass to coarsenings of the valuation). This is due to the fact that in contrast to the latter, the former admit finite extensions with nontrivial defect. In joint work with Anna Rzepka we have studied the valuation theory of the larger class of deeply ramified fields and shown that these valued fields admit only a certain type of defects (called “independent”). Joining forces with Steven Dale Cutkosky, we have revealed their connection with Kähler differentials which play a role in almost ring theory. This has led to further work in which we show that modules of Kähler differentials in Galois extensions of prime degree that were only known to be almost zero are actually zero. This (ongoing) work as well as the work of Jahnke and Kartas sheds new light on important results by mathematicians like Gabber-Ramero, Faltings, Scholze and others.

Matchings and a possible Følner condition in the o-minimal setting

Higher arity non-Ramsey

The Ramsey property of a structure, or a class of structures, has been useful in studying and defining dividing lines in model theory. For example, the Ramsey property of certain tree structures was used to complete the proof that simplicity (NTP) is equivalent to the conjunction of NTP_{1} and NTP_{2}. The Ramsey property of ordered graphs can be used to give equivalent definitions of NIP. In it's most basic form, the use of the Ramsey property of linear orders is prevalent throughout model theory, and can be traced back to the work of Frank Ramsey himself.
We will review some basic definitions from structural Ramsey theory, and their connections to model theory.
We will then show some surprising results about structures, and classes of structures, *without* the Ramsey property, and how the lack of the Ramsey property follows from their behaviour as indexing sets for indiscernibles. These surprises are joint work with Aris Papadopoulos and Pierre Touchard.
The motivation leading to the results depicted above comes from an attempt to find new dividing lines, outside the NIP_{n} hierarchy, defined using structures of arity greater than 3 with the Ramsey property. The question of existence of such dividing lines was also asked, in a different phrasing, in combinatorics circles, and is open, as of the date of writing this abstract.

Bounding nonorthogonality

In the context of finite rank types, nonorthogonality can be replaced by non-weak-orthogonality if one replaces the type in question by a sufficiently high (finite) Morley power. Several results over the last couple of years have had to do with determining how high a power one needs to consider, mostly in the context of differentially closed fields and compact complex manifolds. I will survey these results, and then discuss the latest installment appearing in recent work with James Freitag and Leo Jimenez.

A differential approach to the Ax-Schanuel

In this talk I will discuss recent work, joint with D. Blázquez-Sanz, G. Casale and J. Freitag, on proving Ax-Schanuel type results using a combination of tools from differential geometry, differential algebra and the model theory of differentially closed fields. I will focus on the “curves” case and highly the role of model theory in obtaining the full Ax-Schanuel Theorem with derivatives for uniformizers of simple projective structure on curves including the unifomizers of any Fuchsian group of the first kind and any genus.

Lie maps between groups definable in o-minimal expansions of the real field

Pillay proved in the late eighties that every group definable in an o-minimal expansions of the real field can be endowed with a topology that makes it a (C1)-Lie group. More recently we (with Conversano and Post) classified precisely which Lie groups appear as (i.e. are Lie isomorphic to) groups definable a o-minimal structures.

In this talk we will continue to explore the relation between the category of Lie groups and the category of groups definable in o-minimal expansions of the real field, showing for example that in ℝ_{an,paff} any Lie homomorphism between definable groups is definable.

Vaught’s conjecture: classifying countable models of weakly quasi-o-minimal theories

Let T be a complete, countable theory with infinite models and with a definable linear order <. T is (weakly) quasi-o-minimal if any parametrically definable subset is a Boolean combination of unary 0-definable sets and (convex sets) intervals. The talk will focus on the classification of countable models of wqom T with fewer than continuum countable models. In our earlier work the case of binary T has been completed, and now we extend it to a wider class which includes all quasi-o-minimal theories and all rosy wqom theories; in particular, we confirm Vaught’s conjecture for these theories. This research is supported by the Science Fund of the Republic of Serbia, Grant No. 7750027: Set-theoretic, model-theoretic and Ramsey-theoretic phenomena in mathematical structures: similarity and diversity–SMART.

A finiteness result on tropical functions on skeleta

Skeleta are piecewise-linear subsets of Berkovich spaces that occur naturally in a number of contexts. We will present a general finiteness result, obtained in collaboration with A. Ducros and E. Hrushovski and F. Loeser, about the ordered abelian group of tropical functions on skeleta of Berkovich analytifications of algebraic varieties. The proof uses the stable completion of an algebraic variety, a model theoretic version of analytification previously developed by Hrushovski and Loeser.

Pairs of algebraically closed fields are noetherian (together with Amador Martin-Pizarro)

A theory T is “noetherian” if it has quantifier elimination down to a “noetherian” family F of formulas, which means that F is closed under conjunctions and F-definable sets have the DCC in every model of T.

For example the theory of differentially closed field of characteristic 0 is noetherian by quantifier elimination and Ritt-Raudenbush.

We will show that the theory of pairs of algebraically closed fields is noetherian with respect to “tame” formulas. Also we relate Morley rank to the foundation rank of tame formulas.

The Elekes-Szabó’s theorem and approximate subgroups

The Elekes-Szabó’s theorem says very roughly that if a complex irreducible subvariety V of X*Y*Z has “too many” intersection with cartesian products of finite sets, then V is in correspondence with the graph of multiplication of an algebraic group G. In this talk, I will talk about some results around the Elekes-Szabó’s theorem and its relation with approximate subgroups, this is joint work with Martin Bays and Jan Dobrowolski.