Lectures on Large Stable Fields

Dave Marker

Spring 2021

Description

I will give a weekly series of lectures leading up to the new result of Johnson, Tran, Walsberg and Ye that large stable fields are separably closed. Lectures will be Mondays at 4:00pm beginning Monday January 25.

A long outstanding conjecture is that every infinite stable field is separably closed. The class of large fields was introduced in field arithmetic and includes all of the known examples of fields where there is a reasonable model theory.

I will begin with an introduction to stable groups, discuss earlier work done on stable fields and review the necessary background on large fields.

I plan to give one lecture/week on Zoom, Each session will be at most 75 minutes, but more likely an hour or less. Please contact me (marker@uic.edu) if you need the Zoom link.

Prerequisites

Ideally students will have seen the basics of stability as presented, say, in Tent-Ziegler, but I will try to review all the results I'm using.

The chapter of my text on omega-stable groups is also useful background.

References

A. Chernikov, Lectures on Stability Theory
F. Delon, Separably closed fields, Model theory and algebraic geometry, E. Bouscaren ed., Lecture Notes in Math., 1696, Springer, 1998.
J.-L. Duret, Les corps faiblement algébriquement clos non séparablement clos ont la propriété d'indépendence, {\em Model theory of algebra and arithmetic}, pp. 136–162, Lecture Notes in Math., 834, Springer, Berlin-New York, 1980.
Y. Halevei, A. Hasson and, F. Jahnke, Definable V-topologies, henselianity and NIP, J. Math. Log. 20 (2020), no. 2, 2050008, 33 pp.
W. Johnson, C-M. Tran, E. Walsberg and J.Ye, Etale open topology and the stable field conjecture
I. Kaplan, T. Scanlon and F. Wagner, Artin-Schreier extensions in NIP and simple fields, Israel J. Math. 185 (2011), 141--153.
D. Marker, Model Theory: An Introduction, Springer, 2002.
D. Marker, Lecture Notes on Model Theory of Valued Fields Fall 2018.
M. Messmer, Some model theory of separably closed fields, Model theory of fields, D. Marker, M. Messmer, A.Pillay ed., Lecture Notes in Logic, 5. Association for Symbolic Logic, A. K. Peters, 2006.
A. Pillay and E. Walsberg, Galois groups of large simple fields
B. Poizat, Groupes stables, avec types génériques réguliers, J. Symbolic Logic 48 (1983), no. 2, 339--355.
B. Poizat, Stable Groups, AMS, 2001.
F. Pop, Embedding problems over large fields. Ann. of Math. (2) 144 (1996), no. 1, 1--34.
F. Pop, Little survey on large fields
A. Prestel and M. Ziegler, Model-theoretic methods in the theory of topological fields. J. Reine Angew. Math. 299(300) (1978), 318–341.
T. Scanlon, Infinite stable fields are Artin--Schrier closed
K. Tent and M. Ziegler, A Course in Model Theory, Cambridge University Press, 2012
E. Walsberg, Topological approach to the theory of large fields?, in progress.

Lecture Notes

Lecture 1 Introduction + chain conditions in stable groups
Lecture 2 generic types in stable groups
Lecture 3 more on generic types in stable groups
Lecture 4 stable fields
Lecture 5 large stable fields
Lecture 6 the etale open topology

Last Revised: 3/15/21