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
Target Audience: Undergraduate/Graduate students
Prerequisites: Fisrtorder 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 
2019518 

9:009:45 
Byunghan Kim 
YONSEI UNIVERSITY 
Transitivity and lowness in NSOP1 theories 
TEA BREAK 

10:0010:45 
Jinhoo Ahn 
YONSEI UNIVERSITY 
Mekler's construction and tree properties 
TEA BREAK 

11:0011:45 
William Johnson 
FUDAN UNIVERSITY 
Parity quantifiers in modular arithmetic 
LUNCH 

15:0015:45 
Junguk Lee 
University of Wrocław 
Interpretability of Galois groups of first order structures 
TEA BREAK 

16:0016:45 
Daniel Hoffmann 
University of Warsaw 
Weak independence theorem for PAC structures 
DINNER 

2019519 

9:009:45 
KOBE UNIVERSITY 
On the automorphism groups of Hrushovski's pseudoplanes in rational cases 

TEA BREAK 

10:0010:45 
Anand Pillay 
UNIVERSITY OF NOTRE DAME 
Amenable first order theories 
TEA BREAK 

11:0011:45 
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 Kimindependence over sets in NSOP1 theories. In particular we show that Lascar types are strong types in any low NSOP1 theories with nonforking 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 κstability for any cardinal κ. Not only the stability, it is known that his construction preserves various modeltheoretic 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 nonsimple 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 firstorder theory of the fields Z/pZ is decidable. In other words, there is an algorithm which inputs a firstorder 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 firstorder 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, elladic cohomology, finite group schemes, categorical logic, motivic integration, and FefermanVaught 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_Pstructure (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 Gcompact.
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 ominimal 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 CubidesKovacsics and Mario Edmundo.