# Lancashire-Yorkshire Model Theory Seminar

### Abstracts

• Interpreting modules in modules, Mike Prest

• Families of definable sets in the ordered group $\mathbb{Z}$, Immanuel Halupczok
There is a simple classification of definable sets in the ordered group $\mathbb{Z}$ up to definable bijection. I will present generalizations of this to definable families of sets and to definable sets in elementary extensions of $\mathbb{Z}$. A nice result is the following. If $X_y$ and $X'_y$ are two definable families of finite sets (where $y$ runs over some definable set $Y$), then there exists a bijection $X_y \longrightarrow X'_y$ which is definable uniformly in $y$. This is joint work with Raf Cluckers.

• On Transformation in the Painlevé family, Joel Nagloo
The Painlevé equations are nonlinear 2nd order ODE and come in six families P1-P6, where P1 consists of the single equation $y''=6y^2 + t$, and P2-P6 come with some complex parameters. They were discovered strictly for mathematical considerations at the beginning of the 20th century but have arisen in a variety of important physical applications including for example random matrix theory and general relativity. In this talk I will explain how one can use model theory to answer the question of whether there exist algebraic relations between solutions of different Painlevé equations from the families P1-P6.

• Applications of the twisted theorem of Chebotarev, Ivan Tomašić
The twisted theorem of Chebotarev is a result about difference polynomial equations. More precisely, it is a difference function field analogue of the celebrated theorem of Chebotarev. We shall discuss some of its classical number-theoretic applications, and focus on some exceptionally nice examples.