18.575: Model Theory (Spring 2007)

MeetingsMW 3:00-4:30, 2-142
InstructorEric Rosen
Office2-172
Emailrosen (at) math (dot) mit (dot) edu
Office HoursWed. 2-3, Thur. 11-12, and by appointment
Syllabus Model theory is a branch of mathematical logic that considers properties of mathematical structures expressible in first-order logic. This course will provide a basic introduction to the subject, while emphasizing applications to other areas, including algebra and number theory. Topics to be covered include the compactness theorem, quantifier elimination, realizing and omitting types, saturated and homogeneous models, indiscernibles, and a discussion of algebraic examples. We will also prove Morley's categoricity theorem and develop the foundations of stability theory, including Morley rank, &omega-stable groups, and strongly minimal sets, which will provide a glimpse of more advanced areas.

The course will be designed to provide the necessary model-theoretic background to understand significant recent applications to, e.g., diophantine geometry and motivic integration, in the work of Hrushovski, Kazhdan, Scanlon, Cluckers, Denef, and Loeser.

TextModel Theory: An Introduction, by David Marker (Springer GTM).

The introduction is available here. More information, including the table of contents, can be found on amazon.

Recommended reading Some other good introductions to the subject include Model Theory and A Shorter Model Theory, both by Wilfrid Hodges, and A Course in Model Theory by Bruno Poizat. These will be put on reserve in the library.

An elegant description of the subject, also by Hodges, can be found here.

Lecture schedule: (tentative) My aim for the semester is to prove Morley's famous categoricity theorem, which was really the starting point for contemporary model theory. Much of the material covered in chapters 2, 4, and 5 gets used in the proof of this theorem. The ideas, tools, and techniques developed in these chapters are also fundamental to all further developments in the subject. Along the way, we will also examine connections with other areas of mathematics, especially algebra. In particular, chapter 3 contains an extended discussion of the model theory of algebraically closed fields, with glimpses at basic ideas in algebraic geometry.
  1. W 2/07: Languages and structures. Section 1.1
  2. M 2/12: Theories. Section 1.2
  3. W 2/14: Definable sets. Section 1.3
  4. TUESDAY 2/20: Compactness theorem. Section 2.1
  5. W 2/21: More on compactness.
  6. M 2/26: Complete theories. Section 2.2
  7. W 2/28: Lowenheim-Skolem theorem. Section 2.3
  8. M 3/5: Elementary chains. Section 2.3
  9. W 3/7: Back-and-forth arguments. Section 2.4
  10. M 3/12: Ehrenfeucht-Fraisse games. Section 2.4
  11. W 3/14: Quantifier elimination. Section 3.1
  12. M 3/19: Quantifier elimination continued.
  13. W 3/21: Algebraically closed fields. Section 3.2

    SPRING BREAK

  14. M 4/2: Types. Section 4.1
  15. W 4/4: Stone spaces and examples.
  16. M 4/9: Omitting types. Section 4.2
  17. W 4/11: Prime, atomic, and homogeneous models.
    M 4/16: PATRIOT'S DAY, no class.
  18. W 4/18: More on prime models.
  19. M 4/23: Saturated models. Section 4.3.
  20. W 4/25: Homogeneous and universal models.
  21. M 4/30: Kappa-stability, Vaught's two-cardinal theorem.
  22. W 5/2: Countably categorical theories. Section 4.4
  23. M 5/7: Partition theorems. Section 5.1
  24. W 5/9: Order indiscernibles. Section 5.2
  25. M 5/14: Morley's theorem. Section 6.1
  26. W 5/16: More on Morley.
Homework: Homework will be assigned on a regular basis. You should hand in solutions to most of them, though you are not required to work on every single one. Problems marked with a * are ones that either introduce important concepts not covered elsewhere in the text, or that I consider particularly interesting or significant. I strongly recommend that you look carefully at each such problem, and at least attempt a solution.

Last updated May 2007