14:30 - 15:15: Ulla Karhumäki
University of Manchester
Small infinite simple groups of finite Morley rank admitting a tight automorphism whose fixed point subgroup is pseudofinite
[Slide]
We explain in detail a recent approach, developed by Pınar Uǧurlu, towards the well-known Cherlin-Zilber Conjecture. The main aim of this approach is to prove that the Cherlin-Zilber Conjecture is equivalent to another conjecture called the Principal Conjecture, which is due to Ehud Hrushovski; Let $G$ be an infinite simple group of finite Morley rank with a generic automorphism $α$. Then the fixed point subgroup $C_G(\alpha)$ is pseudofinite. We prove a partial result supporting the expected equivalence between these two conjectures. Namely, we prove that, under suitable conditions, a “small” infinite simple group of finite Morley rank G admitting a tight automorphism α whose fixed point subgroup $C_(\alpha)$ is pseudofinite is isomorphic to $PSL_2(K)$ over an algebraically closed field $K$ of characteristic different from 2. This is joint work with Pınar Uǧurlu.15:30 - 16:15: Blaise Boissonneau
Universität Münster
NIPity in algebraic extensions of Qp
Understanding structures in terms of what combinatorial properties they can express is an important part of Model Theory. One such combinatorial property is called independence, and structures not expressing it are called NIP. A prototypical example of a NIP field is Qp, which is NIP as a field, but also as a henselian valued field. Any non-trivial example of a NIP field is henselian, but not all henselian fields are NIP: they have recently been classified by Sylvy Anscombe and Franziska Jahnke, and based on this result, one can derive a classification of NIP algebraic extensions of Qp. 16:30 - 16:45: COFFEE BREAK!
16:45 - 17:30: Brian Tyrrell
University of Oxford
A Geometric Hilbert's Tenth Problem for Global Fields
If $K$ is a function field with constant subfield $k$, we may consider it as a structure in the "geometric" language of rings $L_F = \{0, 1, +, x, F\}$, where $F$ is a predicate for nonconstancy (i.e. $F(x) \iff x \not\in k$). Until recently, the decidability of the existential theory of $K$ in $L_F$ (without parameters) - a geometric analogue of Hilbert's Tenth Problem over $K$ - was very open. In this talk we will partially resolve this problem by proving the existential $L_F$-theory of any high genus global field (of sufficiently large characteristic) is undecidable. Equivalently, as the predicate $F$ is existentially definable in this context, this is an undecidability result in the language of rings without parameters. We will also use this machinery to conclude the AE-theory of a rational function field is undecidable in the language of rings without parameters, thus partially answering a question of Fehm. Finally, by uniformising our arguments we will give an example of a non-global function field whose existential $L_F$-theory is undecidable.
17:45 - 18:30: Alexi Block Gorman
University of Illinois at Urbana-Champaign
Characterizing companionability for expansions of o-minimal theories by a dense, proper subgroup
[Slide]
Recent works in model theory have established natural and broad criteria concerning the existence of model companions and the preservation of certain neostability properties when passing to the model companion. In this talk, we restrict our attention to the o-minimal setting. By doing so we are able to isolate the sort of necessary and sufficient condition that can be elusive in more general settings. The central result is a full characterization for when the expansion of a complete o-minimal theory (with quantifier elimination) by a unary predicate that picks out a dense, divisible subgroup has a model companion.We provide examples both in which the predicate is an additive subgroup, and in which it is a mutliplicative subgroup.The o-minimal setting allows us to provide a full and geometric characterization for companionability, with an especially nice dividing line when the group operation is multiplication. We conclude with a brief discussion of neostability properties, and give examples that illustrate the lack of preservation for properties such as strong, NIP, and NTP2, though there are also examples for which some or all three of those properties hold.18:45 - 20:00: DINNER BREAK!
20:15 - 21:00: Aaron Anderson
UCLA
Combinatorial Bounds in Distal Structures
Many proofs in geometric combinatorics make use of cuttings: Given some family of semialgebraic curves, such as lines, circles, or hyperplanes, we try to partition space into as few pieces as possible, under the condition that each piece intersects only a small number of curves. If the curves are instead definable in some distal structure, then we can still produce cuttings, using distal cell decompositions. In this talk, we will introduce the framework of distal cell decompositions, present new bounds on their sizes in some o-minimal and p-adic contexts, and deduce some combinatorial implications. 21:15 - 22:00: Shi Qiu
University of Manchester
High dimensional integer-valued definable functions
This talk concerns the growth at infinity of functions which are analytic and definable in real exponential field, and which take integer values on $\mathbb{N}$. Recently, there are several results focus on the Polya-type problem with respect to these functions. For example, in 2016, Wilkie showed that if such functions are bounded by $2^{rx}$ ($0 < r < 1$) for all sufficiently large x, then there exists a polynomial $P$ such that $f(x) = P(x)$ for sufficiently large $x$. In this talk, I will introduce proof from Jones, Thomas and Wilkie (2012) which gives us Polya-type theorem for high dimensional functions ($f : [0,\infty)^n → \mathbb{R}$ such that $f(\mathbb{N}^n) \subseteq \mathbb{Z}$). If time permits, I will mention more results and conjectures about high dimensional cases.