Summer school: Exponential fields and valued fields in model theory

This is the second event of the INdAM intensive period Model theory of tame expansions of topological fields.

The lectures will be in Aule T Edificio 8. The campus entrance is on Via Cintia, Monte S. Angelo (see also the campus map).

Lecturers

The lecturers will give four minicourses on the following topics: quasiminimality and exponential algebraic closedness, Hensel minimality and valued fields, o-minimality of analytic functions, model theory of differential fields. Titles and abstracts will appear soon.

• Jonathan Kirby (University of East Anglia) – Quasiminimality and exponential algebraic closedness

  This course will begin by looking at the axioms of pseudoexponentiation, and then we will focus specifically on the question of exponential algebraic closedness. We will review some of the known results surrounding this problem, and then we will describe the theorem of Bays-Kirby which states that if one can solve exponential algebraic closedness, then the complex exponential field is quasiminimal.

• Silvain Rideau-Kikuchi (École normale supérieure – PSL) – Hensel minimality

  As exemplified by o-minimality, imposing strong restrictions on the complexity of definable subsets of the affine line can lead to a rich tame geometry in all dimensions. There has been multiple attempts to replicate that phenomenon in non-archimedean geometry (C, P, V, b minimality) but they tend to either only apply to specific valued fields or require geometric input. In this talk I will present another such notion, h-minimality, which covers all known well behaved characteristic zero valued fields and has strong analytic and geometric consequences. By analogy with o-minimality, this notion requires that definable sets of the affine line are controlled by a finite number of points. Contrary to o-minimality though, one has to take special care of how this finite set is defined, leading to a whole family of notions of h-minimality.

  This notion has been developed in the past years by a number of authors and I will try to paint a general picture of their work and, in particular, how it compares to the archimedean picture. During the course we will go over the original example of henselian fields in characteristic zero and derive the definition of h-minimality from the properties of their definable sets. We will then derive the main consequence of h-minimality in terms of differentiability on the one hand and cell decomposition on the other. Finally, we see how this relates to point-counting in the non-archimedean setting.

• Tamara Servi (Paris Cité University) – A geometric proof of the o-minimality of ${ℝ}_{\mathrm{an},\exp }$

  The o-minimality of the reals equipped with all restricted analytic functions and the full exponential function is one of the foundational results behind various applications of o-minimality, such as the proof of the André-Oort conjecture. We will present a geometric proof of the o-minimality of this structure, which provides a strong form of quantifier elimination relevant to applications in asymptotic analysis. We will also discuss generalisations of the method to wider classes of functions, including for instance the Riemann zeta function.

• Marcus Tressl (University of Manchester) – Model theory of differential fields

  This course will be given in two parts. The first part will be an introduction to the subject, requiring only basic knowledge of model theory and algebra, targeting a good basic understanding of differentially closed fields. The second part will discuss and outline some of the highlights and recent developments of the subject (like Zilber Trichotomy in DCF, Functional Transcendence, Differential Galois Theory).

Further details will appear soon.

Registration & financial support

Please register by 1 April via the registration form.

Thanks to the sponsorship of the Association for Symbolic Logic, ASL student members can apply for an ASL Student Travel Award. Note that you must apply 3 months before the start of the event. If you are participating in more than one event, you should submit a single application with a combined budget rather than several applications.

Some funding is available to cover accommodation for students attending the Summer School. Students wishing to attend the Summer School must also:

• submit a short research statement (300 words) and a one page CV;
• state if they have access to other sources of funding to cover accommodation;
• ask their supervisor to send a short reference letter;

to naplesmodeltheory2025@gmail.com by the registration deadline.

Shortly after the deadline, we will inform all participants whether we are able to cover their accommodation.

Contact

You can reach us at naplesmodeltheory2025@gmail.com. All communications will be exclusively from that address or from the organisers' institutional emails. Please be aware that scammers are already pretending to be part of the organisation. If you receive a suspicious message and are unsure about it, please get in touch with us at the above address.