The 3rd Regional Days in Model Theory and Applications will take place in Münster, on 8th of June, 2018.

Please register if you are interested in participating.

The RDMTA takes place roughly twice a year, in varying places in the region: Düsseldorf, Leuven, Mons, Münster and maybe others.
• Schedule: The talks are 80 min long, followed by 10 min for questions/discussion.
• 10.00 – 10.30: Welcome Coffee
• 10.30 – 12.00: Krzysztof Krupinski – Galois groups as quotients of Polish groups
• 12.00 – 13.30: Lunch
• 13.30 – 15.00: Kien Huu Nguyen –  Conjectures on exponential sums and conjectures on numerical data
• 15.00 – 15.30: Coffee break
• 15.30 – 17.00: Florian Severin – Towards motivic integration in
elementary extensions of  $\mathbf{Q}_p$
• Abstracts:

Krzysztof Krupinski – Galois groups as quotients of Polish groups:

This is a joint work with Tomasz Rzepecki. We work in a monster model of an arbitrary countable theory $T$. Our main result yields a presentation of the space of classes of a given bounded invariant equivalence relation $E$ defined on the set of realizations of a single complete type over $\emptyset$ as a quotient of a compact, Polish group by some subgroup (which inherits the good descriptive set theoretic properties of $E$). This presentation is in two senses: as a topological space and also in terms of Borel cardinality (with a stronger conclusion under NIP). In order to get this, we use topological dynamics to present the Lascar Galois group of $T$ as a quotient of a compact Polish group by an $F_\sigma$ normal subgroup.
During the talk, I will discuss in details a much easier particular case, when $E$ is coarser than the relation of having the same Kim-Pillay type. We will see basic descriptive set theoretic tools which are involved. In this special case, the compact Polish group from the main theorem is just the Kim-Pillay Galois group of $T$, and so no topological dynamics is needed. In the general case, we use and develop a non-trivial machinery from topological dynamics in order to produce the desired compact Polish group; I will discuss some ideas related to this, if time permits.
Our presentation of the space of classes of $E$ yields, as a quick corollary, the main result from our earlier paper (joint with Anand Pillay) whose main part says that $E$ is smooth (in the sense of descriptive set theory) if and only if it is type-definable

Kien Huu Nguyen – Conjectures on exponential sums and on numerical data

This talk is a joint work with Raf Cluckers. Exponential sums modulo $p^m \$ of a polynomial was investigated by Igusa in his famous lecture. Igusa conjectured a uniform upper bound on $p \$ and $m \$ of the global exponential sums modulo $p^m \$ with respect to a non-constant homogeneous polynomial. Denef and Sperber considered local exponential sums modulo $p^m \$ of a non-constant polynomial at the original point of the affine space $\mathbb{A}^n \$ and gave a conjecture on these sums by the spirit of Igusa. Cluckers and Veys stated a version of Igusa’s conjecture for all non-constant polynomials and a uniform version at all integer points of $\mathbb{A}^n \$ of Denef-Sperber’s conjecture. In this talk, we will establish conjectures on numerical data associated to an embedded resolution of singularities and relate them to Cluckers-Veys’s conjecture on exponential sums. We also connect the global exponential sums and the local exponential sums. In the remains of this talk, we will discuss about some special cases of these conjectures.

Florian Severin – Towards motivic integration in elementary extensions of  $\mathbf{Q}_p$

In the 1980s, Denef and Pas fruitfully combined $p$-adic integration (i.e., integration based on the Haar measure on $\mathbb{Q}_p$) with model-theoretic methods and obtained uniform rationality results about a general type of Zeta functions. More recently, an integration theory in valued fields of equicharacteristic 0 has been developed by Cluckers-Loeser in the henselian case and by Hrushovski-Kazhdan in the algebraically closed case.

Since all of what is mentioned above relies on a model-theoretic framework, a natural question to ask is: What happens if we consider elementary extensions $\mathbb{Q}_p^{\ast}$ of $\mathbb{Q}_p$ ? Following a similar approach as Hrushovski-Kazdhan, the first step in this direction is to define a “universal measure” on $\mathbb{Q}_p^{\ast}$ — and the second step is to describe and understand this measure (and the ring it takes values in) more concretely.

In this talk, I will conjecturally describe this ring of measures and give some indication as to why it should be the right one.

• Venue:
• Talks will take place at Orleans-Ring 10, 48149 Münster, Room N2 at ground floor (next to the main math building, the tallest building).
• Coffee breaks : Room SR0 (Sitzungszimmer), main building, ground floor. Einsteinstr. 62,48149 Münster.
• Travel information:
• Getting from Münster main train station to the Institute: We recommend you to take bus number 1,5,12 or 13 from the stop Hauptbahnhof B1 (see picture) and get off at the stop P+R Coesfelder Kreuz B. Then you can walk to the Institute  (walk in the opposite direction to the bus driving direction, see picture, follow the dots). You can buy tickets inside the bus (3,10 Euros/ticket).  Taxis are also available in front of the Train Station (~ 10-12 Euros to the Institute). By foot it takes about half an hour.
• To buy train tickets to “Münster (Westf Hbf)” you can consult the following sites:

https://www.deutschebahn.com/

https://en.oui.sncf/en/

https://www.b-europe.com/EN

Note that the prices -for the same travel- can vary according to the sites. Sometimes it is also convenient to buy different connections from different sites. For instance, you can buy personal tickets from b-europe to Cologne/Dusseldorf and take a “SchönerTagTicket NRW”  from deutschebahn (45 Euros up to 5 persons) if you travel together with other participants.  Some tickets may allow you to take a bus connection, please be sure about the conditions before buying your ticket (cf. https://www.bahn.de/westfalenbus/view/tickets/nrw-tarif.shtml).

• Accommodation:  The closest hotels to the Institute are the followings:
• Contact: $on\mathfrak{a}y @ ww\mathfrak{u}.de$