Juan Pablo Acosta: I will give a list of all one dimensional groups definable in the p-adic numbers up to a finite index subgroup and a quotient by a finite subgroup. The starting point of this are results that obtain an algebraic group from a definable group, which start from Hrushovski, with the specific one used obtained by Samaria Montenegro, Alf Onshuus and Pierre Simon.
Sylvy Anscombe: TBA
Tim Clausen: We study profinite groups G together with a fundamental system K_i , i \in I of open subgroups as a two-sorted structure (G,I), where the second sort is used to make the fundamental system uniformly definable. The profinite groups for which we can find a fundamental system such that the associated two-sorted structure is dp-minimal turn out to be essentially vector spaces over finite fields or strongly complete abelian.
Katharina Dupont: pdf
Pantelis Eleftheriou: Let N be an expansion of an o-minimal structure M that admits a good dimension theory. Several NIP pairs N=(M, P) fall under this category, such as when P is a dense elementary substructure of M, or it is a multiplicative Mann group, or a dense independent set. We prove: (1) a Weil's group chunk theorem that guarantees a definable group with an o-minimal group chunk is o-minimal, (2) a full characterization of those definable groups that are o-minimal as those groups that have maximal dimension; namely, their dimension equals the dimension of their topological closure, (3) as an application, if N expands M by a dense independent set, then every definable group is o-minimal.
Jakub Gismatullin: I will discuss groups equipped with invariant norms and their metric ultraproducts. I will discuss simplicity and amenability of metric ultraproducts, and some open problems from algebra, where metric ultraproducts are useful.
Yatir Halevi: Shelah's conjecture on NIP fields states that every infinite NIP field is either algebraically closed, real closed or admits a henselian valuation. This conjecture is deeply related to the topology that valuations can define on NIP fields and to the family of valuations on such fields. We will survey the brief history of Shelah's Conjecture, and sketch a proof of the following generalization of past results: If (K,v,u) is a field with two incomparable non-trivial valuations and v is henselian then (K,v,u) has IP.
Franziska Jahnke: TBA
Grzegorz Jagiella: In definable topological dynamics, one considers a group $G$ definable over a model $M$ and the natural action of $G(M)$ on its space of (external) types over $M$. It is a $G(M)$-flow in the sense of classical topological dynamics and it is universal in the appropriate category of \emph{definable} $G$-flows. Associated with this flow is its \emph{Ellis group}. Similarly, given $N$ an elementary extension of $M$ one can consider the universal $G(N)$-flow and consider its Ellis group. A number of conjectures relate the groups obtained this way and speculate that (under some assumptions) they are isomorphic, making the Ellis group a property of the ambient theory rather than an accidental (model-dependent) object. I will talk about some properties that help in describing the dynamics of a given group (such as definable amenability, fsg, dfg) and give some results about topological dynamics of linear matrix groups definable over NIP fields.
Itay Kaplan: Joint work with Pierre Simon and Martin Bays. A type p(x) in S(A) is compressible if for every phi(x,y) there is some psi(x,z) such that for every finite A0 contained in A, there is an instance of psi in p which isolates the phi-part of p restricted to A0. I will present a proof that compressible types are dense in countable NIP theories, , based on work of Chen, Tang and Cheng.
Ulla Karhumaki: We prove that infinite definably simple locally finite groups of finite centraliser dimension are simple groups of Lie type over locally finite fields. Then, we axiomatise some of the key properties of Frobenius maps under the name of finitary automorphism groups. This allows us to classify definably simple stable groups in the specific case when they admit such automorphism group.
Krzysztof Krupiński: I will introduce the notion of amenable theory as a natural counterpart of the notion of definably amenable group. Roughly speaking, amenability means that there are invariant (under the action of the group of automporphism of a sufficiently saturated model), Borel, probability measures on various types spaces. I will discuss several equivalent definitions and give some examples. Then I will discuss the theorem which says that each amenable theory is G-compact. This is a part of my recent paper (still in preparation) with Udi Hrushovski and Anand Pillay.
Gabriel Lehéricy: pdf
Vincenzo Mantova: The theory of real exponentiation can be essentially axiomatised by describing separately the local behaviour (via its Taylor series) and the global behaviour (a group homomorphism that grows faster that any polynomial). In restricted Hahn fields which are also "omega-fields" (i.e., their additive group is isomorphic to the monomial group), the entirety of the global behaviour of exp can be encoded in a single order isomorphism from the field to its positive part; vice versa, any such isomorphism defines a global exp. This captures very well how exponentiation is defined on surreal numbers; but it also leads to the bizarre situation where we build an exponential field which is o-minimal (and a nonstandard model of real exponentiation) if and only if the starting isomorphism stays below a certain test function. This is joint work with A. Berarducci, S. Kuhlmann and M. Matusinski.
Rosario Mennuni: The notion of "equidominance" between global types in an arbitrary theory was introduced in [1], where it was studied in the case of ACVF and it was shown that every invariant type is equidominant with a product of types concentrating in the residue field or in the the value group. This is a motivation to study the interaction of equidominance and product of invariant types but, unfortunately, nor this equivalence relation, nor the coarser relation of "domination-equivalence" are in general congruences with respect to this product. The aim of this talk is to present an instance of this incompatibility, along with a first development of the general theory of this interaction, and discuss some open questions in the NIP unstable case.
Margarita Otero: Let G be a group. A subgroup H of G is a Cartan subgroup of G if H is a maximal nilpotent subgroup of G, and for every normal finite index subgroup X of H, X has finite index in its normalizer in G. We consider Cartan subgroups of definably connect groups definable in an o-minimal structure. In [BB0] we proved that in this context Cartan subgroups of G exist, they are definable and they fall in finitely many conjugacy classes. In this talk I will present some recent results on Cartan subgroups. In particular that the union of the Cartan subgroups is dense in the group, which was the main question left open in [BBO]. (Joint work with Elías Baro and Alessandro Berarducci.)
Giuseppina Terzo: Assuming Schanuel's Conjecture, we prove that for any variety V of dimension n contained in C^n x (C*)^n over the algebraic closure of the rational numbers, and under some natural hypothesis, there exists a generic point in V of the form (a, exp(a)), i.e. strong exponential closure axiom for (C, e^x) holds in many cases.
Frank Wagner; Let A be a connected abelian group in a dimensional theory, and let R be its ring of definable endomorphisms. Assume S is an infinite invariant subring of R such that - the centralizer C of S in R is infinite, - the normaliser N of S in R is unbounded, - A is N-minimal. Then there is a definable skew field F such that A is a finite-dimensional vector space over F, and S, C and N embed into twisted matrix rings over F. This is work in progress joint with Adrien Deloro.