Week 1 - 2, Jan. 9 - 18: A basic introduction to model theory including a discussion of language, structure and ultraproducts. We will look at both classical and continuous model theory with an eye to the applications in the course.

• D. Marker, Model theory: an introduction, Springer Graduate Text in Mathematics, 217, 2002.
  
• K. Tent and M. Ziegler, A course in model theory, Cambridge University Press, 2012.
  

Continuous model theory

• I. Ben Yaacov, A. Berenstein, C. W. Henson, and A. Usvyatsov. Model theory for metric structures. In Z. Chatzidakis et al, editor, Model Theory with Applications to Algebra and Analysis, Vol. II, number 350 in London Math. Soc. Lecture Notes Series, pages 315–427. Cambridge University Press, 2008.
• B. Hart, An introduction to continuous model theory, in I. Goldbring, editor, Model theory of operator algebras, DeGruyter series on Logic and its applications, 2023.

Week 3 - 4, Jan. 23 - Feb. 1: The random graph and the Urysohn sphere.  These two structures are examples of Fraisse construction in the classical and continuous settings respectively.  We will look at axiomatizations of both structures and probabilistic results which arise.

• You can find the material on the random graph in Marker.
• A. M. Vershik, The universal Urysohn space, Gromov metric triples and random metrics on the natural numbers, Uspekhi Mat. Nauk, 53:5(323) (1998), 57-64; Russian Math. Surveys, 53:5 (1998), 921-928
• I. Goldbring, B. Hart and A. Kruckmann, The almost sure theory of finite metric spaces, Bulletin of the London Mathematical Society, Volume 53, Issue 6, December 2021, pp. 1740-1748.

Assignment 2 due by email in pdf format (no pictures please) on Feb. 13; here are solutions.

Week 5 - 6, Feb. 6 - Feb. 15: The Szemeredi regularity lemma and applications in additive combinatorics.  Szemeredi proved his famous lemma in the interest of finding long arithmetic progressions in sets of natural numbers with positive density.  The regularity lemma has many proofs and we will look at a particularly simple one arising from model theoretic considerations.

Here is the complete details of the necessary integral argument in the regularity lemma.

• I. Goldbring and H. Towsner, An approximate logic for measures , Israel J. of Math., 199 (2), 2014, 867-913.
• T. Tao, blog post on ultraproducts and model theory 2013

Week 7 - 8, Feb. 27 - Mar. 7: The model theory of finite fields.  James Ax initiated the model theory of the class of finite fields in the late 1960's.  He provided an axiomatization of what are called pseudo-finite fields.  We will look at this theory with an eye toward quantifier simplification and decidability.

• J. Ax, The Elementary Theory of Finite Fields, Annals of Mathematics, 88 (2), 1968, pp. 239-271. This can be found on jstor.
• Z. Chatzidakis, various notes on finite fields; I was mostly looking at her Helsinki notes.
• Z. Chatzidakis, L. van den Dries and A. Macintrye, Definable sets in finite fields, J. reine angew. Math. 427 (1992), 107-135.

Week 9 - 10, Mar. 12 - Mar. 21: The model theory of matrix algebras and its relationship to quantum complexity and decidability in the continuous setting.

Week 11 - 12, Mar. 26 - Apr. 9: In-class presentations and other topics as time permits.