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

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	Hirotaka Kikyo	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.