Logic 2

Notes

• PDF Notes

Syllabus and Logistical Information

• PDF Syllabus

• This class meets Tθ 3:30-5 in Williams 216.

• Office hours will be scheduled weekly, and available by appointment, in DRL 3N8E. Currently those hours are Wednesday, 1-3 PM.

• Contact me at awanders@sas.upenn.edu.

Topics

We will cover a variety of topics in model theory, highlighting connections to combinatorics and algebra. We will start with an exploration of quantifier elimination in a number of classic theories. We will then dive deeper into the model theory of real closed fields (such as $\mathbb{R}$) in particular, generalizing to o-minimal structures, covering properties such as o-minimal cell decomposition. There are then several topics which we can pursue, depending on time and interest, such as

• Pregeometries/matroids/dimension in strongly minimal and o-minimal structures
• Semialgebraic incidence combinatorics and distal cell decompositions
• NIP, VC-dimension and connections with statistical learning theory

Texts

I will be using a variety of sources for this class, so the authoritative source for lecture materials will be my lecture notes. For the first part of the class, my main source will be Marker, which should be available at that link through Penn. I may assign some homework problems from Marker - if you have trouble accessing it, email me.

Here is a larger list of books/notes that may be useful:

• Marker. Model Theory: An Introduction.
• Tent, Ziegler. A Course in Model Theory.
• van den Dries. Tame topology and o-minimal structures.
• Pierre Simon. A Guide to NIP Theories.

Grading

Students will be graded either on presentations or homework. Undergraduate students should enroll in LGIC 3200 or PHIL 4722 to be graded on homework, while graduate students should enroll in MATH 5710 or PHIL 6722 to be graded on presentations.

Homework

Students enrolled in LGIC 3200 / PHIL 4722 will be graded on biweekly homework. Homeworks will be posted here, and are to be submitted on Gradescope. The Gradescope course will require an access code, for which you can email me.

• Homework 1
• Homework 2
• Homework 3
• Homework 4
• Homework 5

Presentations

Students enrolled in MATH 5710 / PHIL 6722 will be graded on giving a presentation.

One 45-minute in-class seminar presentation will suffice to meet the presentation requirement. Presentations can be on any relevant research paper or similar topic with instructor approval. Here is a preliminary list of recommended papers, which will grow, sorted by topic. Feel free to ask me about other potential topics with connections to model theory - I can try to find you a paper to present.

Papers that are crossed out have been claimed.

Combinatorics

• Artem Chernikov, David Galvin, Sergei Starchenko. Cutting lemma and Zarankiewicz’s problem in distal structures
• Mervyn Tong. Zarankiewicz bounds from distal regularity lemma.
• Jacob Fox, Matthew Kwan, Hunter Spink. Geometric and o-minimal Littlewood-Offord problems.
• Hunter Spink. Multiplicative structures and random walks in o-minimal groups.
• József Balogh, Anton Bernshteyn, Michelle Delcourt, Asaf Ferber, Huy Tuan Pham. Sunflowers in set systems with small VC-dimension
• Saugata Basu. Combinatorial complexity in o-minimal geometry.

Learning Theory

• Hunter Chase and James Freitag. Model Theory and machine learning.

General O-Minimality

• Ya’acov Peterzil and Sergei Starchenko. A trichotomy theorem for o-minimal structures.
• Patrick Speissegger. The Pfaffian closure on an o-minimal structure.

O-Minimal Complex Analysis

• Ya’acov Peterzil and Sergei Starchenko. Tame complex analysis and o-minimality.
• Ya’acov Peterzil and Sergei Starchenko. Complex analytic geometry and analytic-geometric categories.

Algebraic Geometry and Number Theory

• Neer Bhardwaj and Lou van den Dries. On the Pila–Wilkie theorem.
• Spencer Dembner and Hunter Spink. Algebraic and o-minimal flows beyond the cocompact case.

Field Theory

• Alice Medvedev, Ramin Takloo-Bighash. An invitation to model-theoretic Galois theory.

Categorical Fraïssé Theory

• Wiesław Kubiś. Fraïssé sequences: category-theoretic approach to universal homogeneous structures.

Continuous Logic

• Itaï Ben Yaacov. Fraïssé limits of metric structures.
• Itaï Ben Yaacov. Continuous and random Vapnik-Chervonenkis classes.

Calendar Information

• First class: 1/15
• No class: 2/3

Presentation Schedule