Fall 2008 Math 512: Model Theory of Fields I

Course Meeting: MWF 2pm in SEO 636. Call Number: 28296
Instructor: Alice Medvedev E-mail: alice@math.uic.edu
Office Hours: Tue 1:30-3:40, Wed 1-2 or by appointment, in 716 SEO
course webpage: http://www.math.uic.edu/~alice/math512S09


Content:

Model theory of fields is a vast subject that can hardly be introduced, much less covered, in one semester. This course is about algebraically closed fields with extra structure; we may also touch on separably closed fields. We will not mention real closed fields, valued fields, or exponential fields.

We assume basic familiarity with groups and fields, on the level of Dummit and Foote's Abstract Algebra; and with first-order logic (as in the first article in Bouscaren's book). Although we will develop basic algebraic geometry (as summarized in sections 1.1 - 1.4 of Hartshorne's Algebraic Geometry) and basic model theory (on the level of Hodges' A Shorter Model Theory), the student who has not encountered any of these will struggle with the course.

We aim towards Hrushovski's work on the Mordell-Lang conjecture and the Manin-Mumford conjecture. I expect to cover rather thoroughly the material in the first four articles of Bouscaren's book. We will develop the model theory of differential fields in far more detail than is done in Bouscaren's book. We will simultaneously look at difference fields, as the two settings have much in common. I am afraid that after this, we will barely have time to define all the words in Bouscaren's outline of Hrushovski's proof of the Mordell-Lang conjecture for function fields (the almost-last article in Bouscaren's book), and perhaps say a few words about the Manin-Mumford conjecture.

One would wish for a thorough discussion of abelian varieties, one-based groups, Zariski geometries, and separably closed fields, as well as a proper discussion of stable and simple theories and groups; that is not possible. However, we will prove some shiny theorems along the way, and come very close to a few others: Morley's categoricity theorem, Reineke's theorem about strongly minimal groups, Macintyre's theorem about strongly minimal fields, Hurshovski's work around Zilber's Trichotomy conjectures, and maybe more.

References:
  • Lecture notes, courtesy of Charles Moss.
  • Abstract Algebra, by Dummit and Foote, is an excellent introductory algebra book.
  • A Shorter Model Theory by Hodges is a book I know and like.
  • Model Theory by marker is also rumored to be good, and is available online through the UIC library.
  • Algebraic Geometry by Hartshorne is a good reference and an awful textbook.
  • I don't have a favourite algebraic geometry textbook.
  • Model Theory of Fields by Marker is an excellent, detailed exposition of the model theory of algebraically closed and differentially closed fields. We will cover a lot of the material in it, though we will approach it somewhat differently. Through the end of January, ASL members can buy this one at a steep discount.
  • Model Theory and Algebraic Geometry by Bouscaren is our inspiration; we will not cover all of it, though you should be able to read the rest after this course.
  • The Notre Dame Lectures contains an article on the model theory of difference fields by Chatzidakis, which is more readable but less complete than the original paper by Chatzidakis and Hrushovski. The book is also part of the January ASL sell-off.
  • a Manin-Mumford reference
Grading: As the audience is very heterogeneous, the course will be graded on a very sliding scale. There will be many excercises, investigating examples or filling in details of proofs. I will also suggest several harder problems and several reading projects, suitable for a student seminar presentation.

Excercises

Projects and problems:

Last updated 01/15/2009