MAGIC004: Applications of model theory to algebra and geometry

Course details

Semester

Spring 2008
Monday, January 21st to Friday, March 14th; Monday, April 28th to Friday, May 16th

Hours

Live lecture hours
20
Recorded lecture hours
0
Total advised study hours
0

Timetable

Mondays
11:05 - 11:55
Tuesdays
11:05 - 11:55

Description

The course will discuss and survey some classical and recent applications of model theoretic techniques to various other areas of mathematics. In addition to introducing basic notions of model theory, the course will also introduce in a soft manner notions from algebraic and diophantine geometry as well as valued fields. As such the course is aimed at the general postgraduate audience. But it would also be essential for students aiming to work in model theory and related subjects. The applications will go from elementary things (Ax's theorem) to more sophisticated ones (definable integration, diophantine geometry over function fields).

Prerequisites

No prerequisites information is available yet.

Syllabus

Note that I may end up covering less material than planned below. In particular the basics of model theory may end up being only covered in the first 8 lectures.
  • Lectures 1 to 6: Basics of model theory with examples. (First order structures and theories, compactness, nonstandard models, quantifier elimination.)
  • Lecture 7 and 8: Affine algebraic varieties and polynomial maps. Ax's theorem that an injective morphism from a complex variety to itself is surjective.
  • Lectures 9 to 12: Logic and valued fields. Statement and explanation of the theorem of Ax-Kochen-Ershov on the first order theory of Henselian valued fields. Application to the asymptotic solution of a conjecture of Artin on p-adic solutions of equations.
  • Lectures 13 to 16: Grothendieck groups of first order theories. Algebraically closed valued fields. Formulation and explanation of main theorem (Hrushovski-Kazhdan) of "definable (or motivic) integration" and applications to birational invariants of varieties.
  • Lectures 17 to 20: The model theory of differential fields and the number and structure of points on varieties over function fields.

Lecturer

  • AP

    Anand Pillay

    University
    University of Leeds

Bibliography

No bibliography has been specified for this course.

Assessment

Attention needed

Assessment information will be available nearer the time.

Files

Only consortium members have access to these files.

Please log in to view course materials.

Lectures

Please log in to view lecture recordings.