The first week (Jul 21 - Jul 25):

喻良:

The second week (Jul 28 - Aug 1):

Ben Castle:

Room: TBA

Hours:

• Lecture 1: 9:15 - 10:30 (GMT+8)
• Lecture 2: 11:00 - 12:15 (GMT+8)
• Section: 15:00 - 17:00 (GMT+8)

递归论基础及其应用

本课程内容主要是递归论基础及其应用。我们计划讲授递归论中的基本技巧，包括优先方法，力迫法，能行超穷归纳等技术。然后我们会介绍一些递归论在描述集合论，算法信息论以及其它数学分支的应用。

Program:

• Day 1: 递归论基础
• Day 2: 能行力迫法
• Day 3: 高阶递归论入门
• Day 4: 算法信息论基础
• Day 5: 一些其它的应用

Lecturer:

喻良，南京大学数学学院教授，主要研究领域为数理逻辑，尤其是递归论，集合论以及算法随机性理论。通过与一些合作者合作，解决了例如 r.e. 度是否是 d.r.e 度的 $E$ 的初等子结构，随机性的刻画问题，发展了高阶递归论与随机性理论以及回答了决定性公理与可数选择公理的关系等一系列公开问题。

Strong Minimality and Geometric Stability Theory

A subfield of model theory, geometric stability theory refers to a body of results and methods aimed at recovering familiar algebraic structures from purely logical data (here ‘stability’ refers to model-theoretic stability, which provides many of the tools used; and ‘geometric’ refers to the use of combinatorial ‘geometries’ to sort out the different types of algebraic structures). For example, one of the earliest results along these lines is a theorem of Zilber on totally categorical theories (theories with the ‘fewest’ models possible): in any model of such a theory, there is an infinite interpretable set with either no structure at all, or precisely the structure of a vector space over a finite field. The remarkable feature of this result is that an assumption seemingly far removed from algebra (namely a restriction on the number of models of a theory) is ultimately intrinsically tied to vector spaces. This course will provide an introduction to geometric stability through the single case of strongly minimal theories: a complete theory T is strongly minimal if for every M ⊧ T, we have (i) M is infinite, but (ii) every definable subset of M is finite or cofinite. The main examples of such theories are pure sets, vector spaces, and algebraically closed fields. The main goals of the course will be: 1. Develop the basic machinery of strongly minimal theories, particularly dimension theory of definable sets and types, and the related notions of canonical bases and Zilber’s trichotomy. 2. As a main goal, prove the locally modular case of the group configuration theorem (one of the most influential theorems of stability theory) and the associated structure theorems for locally modular groups. These theorems will demonstrate the main ideas model theorists use to construct groups and vector spaces out of abstract data. On the one hand, strong minimality is a very strong restriction on a theory; on the other hand, fundamental results of Baldwin and Lachlan, and later Buechler, show that much larger classes of theories can be understood in terms of strongly minimal ones. For these reasons, strongly minimal theories continue to play a fundamental role in modern model theory, and many modern definitions are abstracted from the strongly minimal case. Thus, this course can also be treated as a hands-on tour of some of the core ideas of modern model theory (namely stability and its generalizations).

Program (subject to change):

• Day 1: The Basics.
  • Historical overview of uncountably categorical and totally categorical theories, focusing on the finite axiomatizability problem.
  • Basic properties and examples of strongly minimal theories.
  • The dimension of a definable set, and the decomposition into stationary components.
• Day 2: Dimension via Pregeometries
  • Review of saturated models
  • The algebraic closure operator; pregeometries and geometries.
  • Dimension theory for tuples and types, and the duality with dimension theory for definable sets.
  • Proof that strongly minimal theories are uncountably categorical.
• Day 3:
  • Review of imaginary elements and canonical parameters, and weak elimination of imaginaries in the strongly minimal case.
  • Definition and construction of canonical bases; relation to ‘normal’ families of sets.
  • Definition and examples of 1-based strongly minimal theories.
  • Families of plane curves, and the three levels of Zilber’s trichotomy.
• Day 4:
  • The groupoid of germs.
  • Construction of an infinite family of germs in a non-trivial 1-based strongly minimal theory.
  • Interpreting a group from a family of germs.
  • If time, survey the more general group configuration theorem.
• Day 5:
  • Abelianity of the group, and the structure theorem for definable sets in 1-based groups
  • Complete classification of totally categorical strongly minimal theories up to finite covers (assuming Zilber’s theorem that such theories are 1-based)
  • Informal description of Hrushovski’s proof of Zilber’s theorem above on 1-basedness
  • Informal account of how and when one can interpret a field rather than a group.
  • Informal discussion of applications to other areas of mathematics.

Lecturer:

Ben Castle is a mathematical logician at Department of Mathematics, University of Illinois, Urbana-Champaign. He is specializing in model theory and its interactions with algebra, geometry, and combinatorics. Before coming to Illinois, Castle held postdoctoral positions at the Fields Institute, Notre Dame University, Ben-Gurion University of the Negev, and the University of Maryland; during this time, his work focused on studying reconstruction theorems in geometry from a model-theoretic lens. He holds a PhD in mathematics from UC Berkeley.

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

Contact: logic@fudan.edu.cn