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Mathematical Logic at Fudan

### 2019 Fudan Logic Summer School

Time: August 5 - August 16, 2019

Location: Fudan University (Handan Road Campus)

Schedule

Room: HGX505 (光华楼西辅)

Hours:

• Lecture 1: 8:30 - 9:45
• Lecture 2: 10:00 - 11:15
• Section: 14:00 - 16:00

The first week (Aug 5 - Aug 9): Set theory - introduction to forcing (Liuzhen Wu 吴刘臻)

The second week (Aug 12 - Aug 16): Model theory - introduction to stability theory（Will Johnson）

Set theory

### Introduction to forcing

Forcing is the basic method to resolve independent problems in axiomatic set theory. Independency is a special subject in the research of mathematical logic. It has many applications in mathematics. In this course, we will focus on the basic properties and techniques of forcing, we will introduce Cohen's classic work on the Continuum Hypothesis and the Axiom of Choice. Finally, we will discuss how forcing can be used as a method not only for an independence proof, but for proofs of general mathematical statements.

#### Program:

• Day 1: Independeny, partial order, dense set, filter, forcing names, forcing extension
• Day 2: The forcing theorems
• Day 3: The independency of the Continuum Hypothesis and the Axiom of Choice, the definition of the Cohen's Forcing
• Day 4: The chain conditions, the closure conditions, and preserving cardinalities
• Day 5: Shoenfield's absoluteness, Forcing absoluteness, and their application

#### Lecturer:

Liuzhen Wu 吴刘臻 is an associate professor at Academy of Mathematics and Systems Science, at Chinese Academy of Sciences. His main research area is set theory. Liuzhen Wu was working at Kurt Gödel Research Center for Mathematical Logic (KGRC), Vienna and the department of mathematics at Nationl University of Singapore as visiting scholar. Liuzhen Wu has published articles in journals of general mathematics such as Advances in Mathematics, Proceeding of American Mathematical Society, and the major journals of mathematical logic such as Journal of Symbolic Logic, Annals of Pure and Applied Logic.

Teaching Assistant: Zhixing You 游志兴

Model theory

### Introduction to stability theory

An algebraic structure is stable if it satisfies one of several equivalent model-theoretic conditions: types are definable, indiscernible sequences are indiscernible sets, no formula has the order property. Stability emerged from the abstract study of classification theory--the structures of interest in Morley's theorem and Shelah's classification theory are all (super)stable. Moreover, many of the model-theoretically tractable structures from number theory and algebra are stable, or nearly stable. Consequently, stability theory has played a central role in modern model theory, both applied and abstract. In this course, we will focus on the special class of "strongly minimal" structures. Strong minimality is essential to the modern understanding of Morley's theorem, and is the setting for the Cherlin-Zilber conjecture on definable groups. We will go through the basic theory of strongly minimal structures, discussing Morley rank and degree of definable sets, the associated pregeometry, and the structure theory of definable groups. We will also verify the elementary facts about stability. At the end of the course, we will go through the classification of group actions on strongly minimal sets, which is the simplest known case of the Cherlin-Zilber conjecture.

Reference: Bruno Poizat 2001, Stable Groups, American Mathematical Society.

#### Program:

• Day 1: Strongly minimal theories. Notes.
• Day 2: Algebraic and definable closure, Pregeometries. Notes.
• Day 3: Rank of tuples, Dimension, Morley degree. Notes.
• Day 4: Macintyre's theorem, interpretable sets, minimal groups and Zilber indecomposability. Notes.
• Day 5: Action of groups, structure theory of groups of finite Morley rank. Notes.

The final set of Notes on the bigger picture.

#### Lecturer:

Will Johnson is a research fellow at the School of Mathematical Sciences, at Fudan University. He has received PhD degree in Mathematics at University of California Berkeley in 2016, advised by Tom Scanlon. His research interest includes model theory of fields and valued fields, neostability (strongly dependent theories, $\textrm{NTP}_2$ theories, $dp$-rank and $o$-minimality), application of model theory to arithmetic geometry, pseudofinite fields and ACFA, etc. Will Johnson has been honoured the 2016 Sacks Prize and Herb Alexander Prize for outstanding dissertation in mathematical logic and pure mathematics respectively.

Teaching Assistant: Jiaqi Bao 包佳齐

Visitors

#### Visitors

• Liuzhen Wu 吴刘臻
• Shichang Song 宋诗畅
• Zhiguang Zhao 赵之光
• Huimin Dong 董惠敏
• Chen Peng 彭程

2019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