This course will cover recent applications of pseudofinite model theory in the setting of group theory and arithmetic combinatorics. We will begin with a brief introduction to several key ingredients from model theory, including:
- the ultraproduct construction and the pseudofinite counting measure,
- the compact group topology on quotients of saturated groups by type-definable subgroups
- NIP formulas and definable set systems of bounded VC-dimension.
We will then focus on the main result of arXiv 1802.04246
, which gives a structural approximation of NIP sets in finite groups by Bohr neighborhoods of bounded complexity in subgroups of bounded index.
Prerequisites and Resources
- Lectures: I will give live lectures over Zoom, using a note-writing app (shared screen). The notes will be posted here (in pdf form), and recordings of the lectures will be made available in the Cambridge Part III Moodle. If you are attending the course, but not officially registered in the Moodle, send me an email and I can provide access to the recordings.
Show/hide links to lecture notes.
- Exercises: (updated 15 May) A range of (optional) exercises will be given throughout the course during lectures. A master list can be found here (updated periodically as we go through the material). Solutions to individual exercises will be made available upon request.
- Appointments: I am happy to make additional Zoom appointments to discuss the course material and exercises. Email me to schedule a time. (I am currently in Chicago, which is 6 hours earlier than the UK, so these appointments will be in the afternoon or early evening in Cambridge.)
I will assume prior knowledge of first-order logic and basic model theory, including the notions of first-order languages, structures, and the Compactness Theorem. In addition, the following is suggested reading:
- Keisler's notes on Ultraproducts. The ultraproduct construction will be briefly reviewed on the first day, so I recommend looking over Sections 1-3 of these notes before the start of the course.
- A. Notes on Model Theory: This is a brief list of basic definitions and results on type spaces and saturated models. For further details and exposition of these topics, see the sources below.
- B. Miscellaneous Notes: (updated 28 April) This is a list of basic definitions on important concepts for the course. It is meant to be a reference guide to keep everyone on the same page.
- C. List of Tools: (updated 13 May) This is a collection of the results used during lectures as "black boxes", along with discussion and citations.
- D. Addendum to Lecture 6 (G00
A and Exercises 5 & 6): These notes provide details on the proof of Corollary 6.6 and underlying tools.
In general, Model Theory: An Introduction
(by Dave Marker) is a good resource for the model theoretic prerequisites of the course. I believe this book is available online from the Cambridge university library (via Proquest). The following are some other online notes on model theory.
- Dave Marker's MSRI notes, which are written in a similar style as his book.
- Anand Pillay's course notes from 2002.
The only other prerequisite for the course is basic topology, including topological groups. Some parts of the course will use more sophisticated results on the structure of compact groups (e.g., the Peter-Weyl Theorem), which can mostly be treated as black boxes. The following resources may also be helpful:
- The Structure of Compact Groups by Hofmann and Morris (the latest edition can be accessed online through the Cambridge university library).
- Profinite Groups by Ribes and Zalesskii, especially Chapter 1 on inverse limits and projective systems, and Sections 2.1 and 2.2 on the basics of profinite groups (I haven't yet found online access to this or something comparable; other suggestions are welcome).