### 2023 Fudan Logic Summer School

Time: Jul 31 - Aug 11, 2023

Location: Fudan University (Handan Road Campus)

####
The first week (Jul 31 - Aug 4):

Rizos Sklinos: Some model theory of nonabelian free groups.

####
The second week (Aug 7 - Aug 11):

Jin Renling: Nonstandard Analysis and Combinatorial Number Theory.

Room: HGX208 (光华楼西辅楼)

Hours:

- Lecture1: 9:00 - 10:15 (GMT+8)
- Lecture2: 10:45 - 12:00 (GMT+8)
- Section: 15:00 - 17:00 (GMT+8)

On Aug 9, we have two talks given by Chieu-Minh Tran and Yifan Jing.

### Some model theory of nonabelian free groups

In 1946 Tarski asked whether nonabelian free groups share the same first-order theory. Despite the fact that free groups were a well studied class of groups and techniques from different disciplines of mathematics have been developed to understand them, Tarski’s question proved very hard to tackle. It was only after more than fifty years, in 2001, that Sela and Kharlampovich-Myasnikov answered the question in the positive. Both works were voluminous and still to this day have not been fully absorbed by the mathematical community. It is worth mentioning that the tools developed allowed Sela to understand, model theoretically, also the class of torsion-free hyperbolic groups.

Surprisingly this common first-order theory has been proved, by Sela, to be stable. This is considered by many one of the most profound result in the model theory of groups.

In this mini-course we will explore what is model theoretically known about nonabelian free groups and more generally torsion-free hyperbolic groups. We will first give a mild introduction to both hyperbolic groups and stability theory and then develop all the adequate tools to explore this demanding topic that lies in the intersection of model theory with geometric group theory.

More precisely, we plan to show that the first-order theory of non abelian free groups is connected, unsuperstable and non-equational. It is not AE axiomatizable and only has QE down to boolean combinations of AE formulas. In addition, it does not have elimination of imaginaries, but one can add some easily understood families over which it has. Moreover, we will show that free groups are homogeneous, yet most surface groups are not. Using homogeneity we will understand forking independence in these standard models through JSJ decompositions (a tool of geometric group theory) and prove that this theory is ample. Finally we will show that no infinite field is interpretable in nonabelian free groups. The above list of results is not exhaustive, but mostly indicative of the topics we will touch on.

#### Program

- Day 1: Basic concepts and results on the model theory of nonabelian free groups. Video 01 Video 02.
- Day 2: $F_\omega$ is not equational. Video 01 Video 02.
- Day 3: Break
- Day 4: Elementary free groups and elementary embeddings in torsion free hyperbolic groups. Video 01 Video 02.
- Day 5: Forking independence and ampleness in nonabelian free groups. Video 01 Video 02.

#### Lecturer:

Rizos Sklinos is an associate professor at Chinese Academy of Sciences. His research interests lie in the intersection of model theory and geometric group theory. He has published articles in Duke Mathematical Journal, Memoirs of the American Mathematical Society, Journal of the European Mathematical Society, Journal of the London Mathematical Society and others.

### Nonstandard Analysis and Combinatorial Number Theory

The course is for the students with some knowledge on mathematical logic such as having had one-semester undergraduate level course on mathematical logic. In the first two days, we will cover basic ideas, concepts, properties, principles, etc. in nonstandard analysis. In the last two days, we will focus on applications of nonstandard methods to the problems in combinatorial number theory. Lecture Notes, Exercise Solutions (Aug 12).

#### Program:

- Day 1: First-order logic and ultrapower of superstructure. Slides Video 01 Video 02.
- Day 2: Properties, principles, Loeb spaces and an application to finance. Slides Video 01 Video 02.
- Day 3: Chieu-Minh Tran and Yifan Jing will give two talks.
- Day 4: Easy applications to combinatorial number theory. Slides Video 01 Video 02.
- Day 5: Sophisticated applications to combinatorial number theory. Slides Video 01 Video 02.

#### Lecturer:

金人麟 Jin Renling is a professor at the Department of Mathematics, at College of Charleston. His main research interests are among nonstandard analysis, additive-combinatorial number theory, set theory, model theory, measure theory, general topology. Prof. Jin has published articles in Journal of Symbolic Logic, Advances in Mathematics, The Transactions of American Mathematical Society, and many others.

- Zhaokuan Hao 郝兆宽
- Ruizhi Yang 杨睿之
- Ningyuan Yao 姚宁远
- Will Johnson

Contact: logic@fudan.edu.cn

2023复旦大学数理逻辑暑期学校由复旦大学教务处主办，复旦大学哲学学院逻辑学教研室承办