Fall 2009 Math 507: Stability Theory

Course Meeting: MWF 12 in 311 AH Call Number: 29843
Instructor: Alice Medvedev E-mail: alice@math.uic.edu
Office Hours: Tue 1:30-3:40 or by appointment, in 716 SEO
course webpage: http://www.math.uic.edu/~alice/math507F09


Prerequisites and General overviews of stability theory: bits from Baldwin, Shelah, and Pillay
Approximate syllabus so far: We assumed you have made friends with imaginary monsters as a child.
We covered Shelah's Unstable Formula Theorem (II.2.2) (Here are the definitions we need, and very condensed slogans of the 10 equivalent conditions.) and everything necessary for it: the three hard implications are 2->3, 5->6, and 8->7 (9->8 is only a tricky compactness). For 2->3, we needed I.2.11, which is a special case of I.2.10. For the other two, we needed a few basic facts about R-ranks (II.1.1-II.1.5). Then 5->6 is II.2.7 which needs II.1.11; and 8->7 is half of II.2.12.
We covered Pillay's local analysis of stable formulae and definitional extensions (I.2.4 - I.2.14) and proved that the notion of independence arising from definitional extensions of type in a stable theory satisfies Isomorphism Invariance, Symmetry, Transitivity (including Baldwin's monotonicity axioms), Extension (including Baldwin's Existence), Local character of freeness, Finite Character of non-freeness (including Baldwin's stronger version), and Independence Theorem (actually better: stationarity over models). Thus, we may apply Wagner's Theorem 2.6.1 to conclude that definitional extensions are preceisely the nonforking extensions.
We will now work through (most of) Wagner's Chapter 2 to get comfortable with forking. In the meantime, you should read through Baldwin's Chapter II on abstract independence notions.

Problem sets:
First problem set, due friday, september 11th.
Second problem set, due friday, september 25th.
Third problem set, due friday, october 16th.
Fourth problem set, due friday, october 30th.
Fifth problem set, due monday, november 16.
Sixth and last problem set, due wednesday, december 2nd.


Content: This course is a thorough and technical introducion to stability theory, the heart of modern model theory. The aim is to give students a solid working knowledge of the basics so that they can approach more advanced topics (further classification theory, or geometric stability theory, or...) and variants (simple theories, dependent theories, stable domination, abstract elementary classes, ...). I hope to get to a detailed discussion of various ranks and to stable groups, if time permits.

References: Many books have been written on this subject. This is a list of the books I enjoyed learning from, on which I am likely to base my lectures. Conspicuously missing are Pillay's "Introduction to Stability Theory" (which I happen to dislike), Buechler's "Essential Stability Theory" (which I don't know at all), and several model theory books with substabtial stability content (Marker, Poizat, Lascar, ...); and numerous books on stable groups. I am told that there are also numerous wonderful expository papers on the subject.
  • Fundamentals of Stability Theory by Jonh Baldwin is the official reference for this course. I will follow at least Chapter 2 on Abstract Independence relations quite closely.
  • Geometric Stability Theory by Anand Pillay is more advanced than this course, but Chapter 1 is a very dense summary of this course, with some deep insights.
  • Classification Theory by Saharon Shelah is where it all began, more or less. Unlike Pillay's summary, here are all the details you could wish for, and then some. I may follow this exposition of the Stable Formula Theorem, and the Stable Theory Theorem.
  • Simple Theories by Frank Wagner has the cleanest introducion to forking in the absence of stability, and to the connection between forking and abstract independence relations, which I may follow.
The first books is available from Project Euclid; I will make a few bound paper copies of Part A, all that we are likely to use. Real copies of it, and any form of the other three, are barely available and absurdly expensive.

Grading: As the audience is very heterogeneous, the course will be graded on a very sliding scale. You should discuss with me at the beginning of the semester what you intend to put into this course and what you expect to get out of it.


Last updated 01/15/2009