The class will meet Tuesdays and Thursday from 12:00  1:20 beginning
Jan. 9. The course will roughly follow the outline below.
There will be 5 assignments and in addition, each student will make a
presentation on a topic of their choice related to the course material.
Slides, reference material and other relevant links will appear on this
page.
Course Outline
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.
References:
Classical
model theory
 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), 5764; Russian
Math. Surveys, 53:5
(1998), 921928
 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. 17401748.
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.
Assignment
3 due by email in pdf format (no pictures please) on Mar. 12
Here is
the complete details of the necessary integral argument in the
regularity lemma.
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 pseudofinite 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. 239271. 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), 107135.
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: Inclass presentations and other
topics as time permits.
Assignment
4 due by email in pdf format (no pictures please) on Apr. 25