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 Aula T10, 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.

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

  In this series of talks I will survey some of the work done towards understanding the model theory of exponential fields, including the real exponential and Zilber’s approach to the complex exponential field. We know from Wilkie that the real exponential field is not too complicated (it is o-minimal) and this has good consequences in geometry, in number theory, and even in machine learning. For the complex exponential, we do not know if it is tame (quasiminimal) or whether it is maximally complicated (interpreting both reals and integers). I will explain progress towards proving that it is tame.

  1. The o-minimal approach and the pregeometry of exponential algebraic closure.
  2. The algebra of exponential fields: kernels and strong extensions.
  3. Zilber's exponential fields and the conjectures.
  4. More about quasiminimality.
• 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.

  1. Slides
  2. Slides
  3. Slides
  4. Slides
• 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).

Schedule

Monday 23^rd June

Tuesday 24^th June

Wednesday 25^th June

Thursday 26^th June

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.

How to reach the Monte Sant'Angelo campus

Note that the University of Naples is scattered across several locations in the city. The Department of Mathematics is in the Monte Sant'Angelo campus.

How public transport works. Public transport is run my several operators, and even the Metro lines are run by different companies. Here are a few recommendations to help you navigate the system.

• The weekly integrated ticket (€16) is the easiest way to move around Naples. It is valid on all means of transport (bus, metro, suburban trains, etc) within the city limits for 7 days. The Alibus airport shuttle is not included.
• There are several apps, such as such as MooneyGO, to help you plan trips, check what ticket you need, and even buy digital QR tickets for all companies serving Naples.
• Moovit (most lines) and Gira Napoli (ANM buses only) can show you the realtime location of trains, metro, and buses.
• The 'Tap & Go' system is only available on certain lines and it is inactive on most buses. We recommend you buy tickets instead.
• Please remember that integrated tickets are valid on all lines; normal tickets are only valid with the company you bought them from. You can incur substantial penalties if you travel with the wrong ticket.
• Integrated tickets can be bought digitally with the above apps (the tickets seem to require an internet connection on every use), or physically at authorized resellers of the Unico Campania circuit (typically kiosks and tabaccherie), ANM Points, and ticket machines inside metro line 1, 6, 11 and funicular stations. Non-integrated tickets can be bought at a few more places, depending on the company, and sometimes directly on the bus.
• The Alibus shuttle connects the airport to the central railway station (15 minutes) and to the port (35 minutes). It runs every 15–30 minutes, and the €5 ticket can be bought at vending machines, online, on the mobile apps, or directly from the driver.

From the city center. From most locations, the best way to reach campus is a Metro line up to Piazzale Tecchio (drop off at Campi Flegrei for Line 2 or Mostra for Lines 6, 9), followed by one of the many buses (S1, S2, R6, 180) towards Monte Sant'Angelo.

From the airport. You can take the Alibus shuttle towards Naples Garibaldi - Stazione Centrale, then use the Metro Line 2 as above. At certain times, you can also take the direct bus 180 towards Piazzale Tecchio with a normal bus ticket.

Taxi from the airport. A much quicker option. The journey takes about 20–30 minutes, depending on traffic, and the fare is approximately €25–30. Ask the driver for: Università Federico II, Dipartimento di Matematica, Monte Sant'Angelo, via Cintia.

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.