$\omega$ $\aleph$ $\infty$
Mathematical Logic at Fudan

## Model Theory Mini Course in Shanghai

Speaker: Anand Pillay

Title: On pseudofinite structures Slides

Time and Location:

May 13: H6204, 14:30 - 17:30

May 14: H6209, 14:30 - 17:30

May 15: H6205, 15:25 - 18:00

May 16, 17: H6209, 14:30 - 17:30

Pre-requisites: Fisrt-order Logic (Compactness Theorem)

PS: All attendants will cover their local expense by themself.

## Model Theory Workshop in Shanghai

Time: May 18 - May 19, 2019

Location: HGW2301, Fudan University, Handan Campus, Shanghai

###### PROGRAM

TIME

SPEAKER

INSTITUTION

TITLE

2019-5-18

9:00-9:45

Byunghan Kim

YONSEI UNIVERSITY

Transitivity and lowness in NSOP1 theories

TEA BREAK

10:00-10:45

Jinhoo Ahn

YONSEI UNIVERSITY

Mekler's construction and tree properties

TEA BREAK

11:00-11:45

William Johnson

FUDAN UNIVERSITY

Parity quantifiers in modular arithmetic

LUNCH

15:00-15:45

Junguk Lee

University of Wrocław

Interpretability of Galois groups of first order structures

TEA BREAK

16:00-16:45

Daniel Hoffmann

University of Warsaw

Weak independence theorem for PAC structures

DINNER

2019-5-19

9:00-9:45

KOBE UNIVERSITY

On the automorphism groups of Hrushovski's pseudoplanes in rational cases

TEA BREAK

10:00-10:45

Anand Pillay

UNIVERSITY OF NOTRE DAME

Amenable first order theories

TEA BREAK

11:00-11:45

JINHE YE

UNIVERSITY OF NOTRE DAME

Sheaf cohomology of \hat{X}

LUNCH

SPEKER: Byunghan Kim

TITLE : Transitivity and lowness in NSOP1 theories

ABSTRACT We study consequences of transitivity of Kim-independence over sets in NSOP1 theories. In particular we show that Lascar types are strong types in any low NSOP1 theories with non-forking existence, extending S. Buechler's result in simple theories. This is a joint work with A. Chernikov and N. Ramsey.

SPEAKER: Jinhoo Ahn

TITLE Mekler's construction and tree properties

ABSTRACT Mekler developed a way to produce a pure group from any given structure where the construction preserves κ-stability for any cardinal κ. Not only the stability, it is known that his construction preserves various model-theoretic properties such as simplicity, NIP, and NTP2. Inspired by the last result, we show that the construction also preserves NTP1(NSOP2) and NSOP1. As a corollary, we obtain that if there is a theory of finite language which is non-simple NSOP1, or which is NSOP2 but has SOP1, then there is a pure group theory with the same properties, respectively.

SPEAKER: William Johnson

TITLE Parity quantifiers in modular arithmetic

ABSTRACT It is a classic result of Ax that the first-order theory of the fields Z/pZ is decidable. In other words, there is an algorithm which inputs a first-order sentence and determines whether the sentence holds in all the fields Z/pZ. Weispfenning, Derakhshan, and Macintyre showed that the same fact holds for the rings Z/nZ. We generalize these results to first-order logic expanded with parity quantifiers'' (there are an even/odd number of x such that P(x)). Many interesting topics show up in the proof, such as difference fields, ell-adic cohomology, finite group schemes, categorical logic, motivic integration, and Feferman-Vaught theory.

SPEAKER: Junguk Lee

TITLE Interpretability of Galois groups of first order structures

ABSTRACT Let T be a complete theory with uniform EI and QE in a language L. Let C be a monster model of T and let K be a small substructure of C. We show that the sorted complete system of Galois group of K is uniformly interpretable in the L_P-structure (M,K) for any elementary substructure of C containing K, where the language L_P is an expansion of L by adding a new unary predicate P. Using this interpretability result, we give two applications for PAC structures:

- A description of relationship of elementary substructures of PAC structures in terms of sorted complete systems.

- A description of types in PAC structures in terms of sorted complete systems.

This is a joint work with Daniel M. Hoffmann.

SPEAKER: Daniel Hoffmann

TITLE Weak independence theorem for PAC structures

ABSTRACT I would like to present several highlights from a current project with Junguk Lee. We are studying PAC substructures (e.g. existentially closed substructures) of some monster model of a stable theory and our goal is to describe such substructures with the Galois theory. It turns out that there is a possibility for defining independence relation in a saturated PAC substructure and this perspective independence relation descends from the absolute Galois group.

Hirotaka Kikyo

TITLE On the automorphism groups of Hrushovski's pseudoplanes in rational cases

ABSTRACT Hrushovski constructed pseudoplanes corresponding to irrational numbers which refute a conjecture by Lachlan. Hrushovski's construction is valid for any real numbers $\alpha$ with $0 < \alpha < 1$. We show that the automorphism groups of pseudoplanes corresponding to rational numbers $\alpha$ with $1/2 < \alpha < 2/3$ are simple groups. It seems that the statement holds for any rational numbers $\alpha$ with $0 < \alpha < 1$.

SPEAKER: Anand Pillay

TITLE Amenable first order theories". (joint with Hrushovski and Krupinski.)

ABSTRACT: We introduce the notion of an amenable first order theory, and prove that such theories are G-compact.

SPEAKER: JINHE YE

TITLE Sheaf cohomology of \hat{X}.

ABSTRACT: We develop a cohomology theory of sheaves on $\hat{X}$, where $X$ is a definable subset (with the underlying theory ACVF) of $V\times \Gamma_\infty^n$. The approach is similar to the o-minimal sheaf cohomology. In particular, this cohomology theory satisfies a version of the homotopy axiom, which includes the deformation retraction of $\hat{X}$ developed by Hrushovski and Loeser. From this, we will derive the finiteness and invariance of the cohomology. This is joint work with Pablo Cubides-Kovacsics and Mario Edmundo.