# Junior seminar

Model Theory, Combinatorics and Valued fields

A trimester at the Institut Henri Poincaré, 8 January - 6 April 2018

The aim of this seminar is for graduate students and young researchers to give a research talk accessible to a broad audience in the spirit of the trimester. Basics of model theory will be assumed.

Practical information:

The seminar will take place (most) Tuesdays and Thursdays at (IHP), from **16:45 to 17:45 (please check the agenda agenda to know the room in which the seminar will take place).
**

For any questions you can write to the organizers, Gabriel Conant and Pablo Cubides.

Programme:

Ove Ahlman: Cofinitely homogeneous multigraphs and hypergraphs |
January 23 |

Daoud Siniora: Automorphism groups of homogeneous structures |
February 6 |

Tingxiang Zou Two constructions based on pseudofinite fields |
February 13 |

Vincenzo Mantova: Factorising generalised power series (and omnific integers) |
February 20 |

Nadav Meir: Products of Homogeneous Structures |
February 22 |

Lorenzo Fantini: Motivic and non-archimedean invariants of singularities at infinity |
February 27 |

Jan Dobrowolski: Localized Lascar-Galois groups |
March 1 |

Philip Dittmann: The Model Theory of Galois Groups, with Applications to the p-adic Spectrum |
March 13 |

Alfonso Ruiz : Non-reduced algebraic varieties and model theory. |
March 15 |

Ningyuan Yao On 1-dim definable p-adic groups |
March 19 |

Nick Ramsey Keisler measures in simple theories |
March 22 |

Mariana Vicaria : Definable groups in Presburger Arithmetic |
April 3 |

Abstracts:

*Cofinitely homogeneous multigraphs and hypergraphs*

Ove Ahlman

Homogeneous structures are an important part of model theory and have applications in many related areas. Cofinitely homogenous structures are homogeneous but with a finite number of exceptions

. In this talk I will present the recent progress in classifying the cofinitely homogeneous graphs, hypergraphs and multigraphs. Although there is no known classification of the homogeneous hypergraphs and multigraphs, there is surprisingly much we can say about the cofinitely homogeneous structures and in some cases we can even provide (semi-)classifications.

*Automorphism groups of homogeneous structures*

Daoud Siniora

A special class among the countably infinite relational structures is the class of homogeneous structures. These are the structures where every finite partial isomorphism extends to a total automorphism. A countable set, the ordered rationals, and the random graph are all homogeneous. We will see some connections between the automorphism group of a homogeneous structure M and certain combinatorial properties of its age (the class of finite structures embedded in M). In particular, in a joint work with Solecki, we build on the work of Herwig-Lascar and Hodkinson-Otto by strengthening the notion of the extension property for partial automorphisms (EPPA) to the new notion of ‘coherent EPPA’. We will show that coherent EPPA implies the existence of a dense locally finite subgroup of Aut(M). We will also discuss other topological properties of Aut(M) such as ample generics and the small index property.

*Two constructions based on pseudofinite fields*

Tingxiang Zou

The class of finite fields falls into the category of “One-dimensional asymptotic classes of finite structures”, where the naïve counting measure and dimension behave in an extremely nice way for all definable sets. Any non-trivial ultraproduct of finite fields will give rise to a “pseudofinite field”, which are model theoretically tame. In this talk, I will present two constructions based on pseudofinite fields, both of which essentially only use the nice behavior of counting measure and dimension. The first one is an algorithm that produces “H-structures” of pseudofinite fields in the sense of Bernstein and Vassiliev, such that the new structures stay pseudofinite. The second construction I will talk about is expending pseudofinite fields with a generic automorphism. We will show that a certain class of pseudofinite fields with the non-standard Frobenius automorphism will be model theoretically wild. However, the pseudofinte coarse dimension of definable sets behave well. We will also make a partial connection between the coarse dimension and an algebraic notion in difference algebra, namely the “transformally transcendental degree”.

*Factorising generalised power series (and omnific integers)
*

Vincenzo Mantova

Inside surreal numbers, Conway introduced the subring of "omnific integers", and conjectured that some of them are irreducible, as well as a kind of uniqueness of the factorisations. Berarducci eventually answered affirmatively to the first conjecture by creating an ordinal-valued (semi)valuation on rings of generalised power series with real exponents. In a joint work with S. L'Innocente, we tack on an extra step to produce a finer ordinal-valued valuation called "degree", and we prove that every power series with real exponents factors into some irreducibles and a unique factor with finite support. The factorisation is obtain by playing with the RV monoid associated to the degree, which exhibits a rather complicated structure, especially when compared with the RV group of valued fields.

*Products of Homogeneous Structures
*

Nadav Meir

We will define the “lexicographic product” of two structures and show that if both structures admit quantifier elimination, then so does their product. As a corollary we get that nice (model theoretic) properties such as (ultra)homogeneity, homoge- nizability, stability, NIP and more are preserved under taking such products. It is clear how to iterate the product finitely many times, but we will introduce a new infinite product construction which, while not preserving quantifier elimination, does preserve (ultra)homogeneity. As time allows, we will use this to give a negative answer to the last open question from a paper by A. Hasson, M. Kojman and A. Onshuus who asked “Is there a rigid elementarily indivisible* structure?” * A structure M is said to be elementarily indivisible structure if for every coloring of its universe in two colors, there is a monochromatic elementary substructure N of M such that N is isomorphic to M.

*Motivic and non-archimedean invariants of singularities at infinity*

Lorenzo Fantini

In this talk we will explain how motivic integration can be used to define a motivic Milnor fiber at infinity for a complex polynomial *f*, and how this invariant measures the lack of equisingularity at infinity in the fibers of *f*. If time allows, we will also introduce a non-archimedean analytic avatar of the motivic Milnor fiber of *f*, a Berkovich space over the field of complex Laurent series.
This is a joint work with Michel Raibaut.

*Localized Lascar-Galois groups*

Jan Dobrowolski

The notion of the localized Lascar-Galois group *Gal _{L}(p)* of a type

*p*appeared recently in the context of model-theoretic homology groups, and was also used by Krupinski, Newelski, and Simon in the context of topological dynamics. After a brief introduction of the context, we will discuss some basic properties of localized Lascar-Galois groups. Then, we will focus on the question about how far

*Gal*can be from

_{L}(tp(acl(a)))*Gal*. This is a joint work with B. Kim, A. Kolesnikov and J. Lee.

_{L}(tp(a))*The Model Theory of Galois Groups, with Applications to the p-adic Spectrum*

Philip Dittmann

We give a new perspective on the model theory of absolute Galois groups, using a categorical approach. We use this to gain new insights about the space of p-valuations of a field under some assumptions on the p-Pythagoras number.

*Non-reduced algebraic varieties and model theory*

Alfonso Ruiz

I will report on a ongoing project that tries to capture the non-reduced geometry of algebraic varieties using Zariski Goemetries ( a structure of finite Morley rank). Some remarks about intersection theory will be made.

*On 1-dim definable p-adic groups*

Ningyuan Yao

The 1-dim groups definable in o-minimal structures has been well understood. For example, Adam W. Strzebonski proved that every definably connected 1-dim group definable in the field of reals is either isomorphic to (R,+) or isomorphic to S^1. However, up to now, there have been very few analogous results for p-adic context. In this talk, I will present our recent results on this subject.

*Keisler measures in simple theories*

Nick Ramsey

With the development of NIP theories, Keisler measures became a central object of study and, in the NIP setting, there were many tools available to analyze them. It's natural to ask if there is similarly a theory of measures in the context of simple theories, where counting measures on pseudo-finite structures, e.g. fields, give natural examples. In joint work with Itaï Ben Yaacov and Artem Chernikov, we prove the independence theorem for measures in a simple theory, which provides a beginning to a theory of measures in this context. The tools involved are Keisler randomization and the theory of Kim-independence for NSOP_1 theories in the continuous setting.

*Definable groups in Pressburger Arithmetic (Joint with A. Onshuus).*

Mariana Vicaria

Let T be the theory of Presburger Arithmetic, i.e. the theory of the intergers considered as
an abelian ordered group with predicates for each congruence n ∈ **N**. Plenty is known about this theory, for example it is folklore that it has quantifier elimination in L = {+, −, <, 0, 1, {≡_{n}}_{n∈N}}, it is a quasi o-minimal theory (therefore NIP) and R. Cluckers proved a cell decomposition for the definable sets, which is very similar to the o-minimal setting.
Our main interest lied in understudying the definable groups in models of Presburger Arith-
metic, we will sketch in particular a proof of the following theorems:
Theorem 1: Any definable group in Presburger Arithmetic is abelian-by-finite.
Theorem 2: Let (G, ·) be any bounded group definable in a model of Presburger arithmetic
(**Z**, +). Then there is some G_{0} an abelian subgroup of G of finite index, some finite integer k and a local lattice Λ in **Z**^{k} such that G_{0} is definably isomorphic to (**Z**, +)^{k}/Λ.