Schedule

The first week (Jul 4 - Jul 8):

叶谨赫 Jinhe Ye (Vincent): 赋值域 Valued Fields (in Chinese).

• Lecture1: 15:00 - 16:15 (GMT+8)
• Lecture2: 16:45 - 18:00 (GMT+8)
• Section: (Next day) 10:00 - 12:00 (GMT+8)

The second week (Jul 18 - Jul 22):

Gabriel Goldberg: Large cardinals beyond the Axiom of Choice.

• Lecture 1: 12:00 - 13:15 (GMT+8)
• Lecture 2: 13:45 - 15:00 (GMT+8)
• Section: 9:00 - 10:00 (GMT+8)

The section of the second week will be lead by Jason Chen (陈泽晟) and start from Jul 18, which is before the first lecture.

Model Theory

赋值域 Valued Fields

赋值域（valued fields）是数学的重要研究对象，Zariski 通过引入赋值的概念来研究代数曲面的奇点消解理论，而经典数学中p进数（p-adic numbers）以及各类函数域正是赋值域重要的例子。模型论上对于赋值域的研究最早起源于 Robinson 关于代数闭赋值域（algebrically closed valued fields）的量词消去的结果，并在其后的发展中出现很多对于主流数学的应用：如 Denef, Loeser 等人的 motivic integration 理论和 Hrushovsk, Loeser 对于 Berkovich space 的研究等。本课程将从赋值域的代数基础讲起，并借用这些代数工具去理解赋值域的模型论，我希望介绍其中的一些如 Ax-Kochen/Ershov 定理等经典结果。时间允许的话，我们也会介绍 p-进数的模型论以及一些基础的 p-进积分理论（p-adic integration）。

The lectures will be in Chinese.

Reference:

• Anand Pillay, Lecture notes - Model Theory.
• David Marker, Model Theory : An Introduction.
• Lou van den Dries, Lectures on the Model Theory of Valued Fields.

Program:

• Day 1: Normed fields, valued fields and Henselianity. Video
• Day 2: Extensions of valuations. Video
• Day 3: Algebraically closed valued fields. Video
• Day 4: Henselian fields and Kaplansky theory. Video
• Day 5: Ax-Kochen-Ershov and Artin conjecture. Video

Lecturer:

叶谨赫，巴黎第六大学，博士后研究员，博士毕业于美国圣母大学，师从 Sergei Starchenko。曾于美国 MSRI 从事博士后工作。主要研究领域是模型论与其在代数几何和非阿几何中的应用。在 J. Eur. Math. Soc，J. Inst. Math. Jussieu，Isr. J. Math，J. Sym. Log 等杂志上被接受和发表文章多篇。

Set Theory

Large cardinals beyond the Axiom of Choice

Program:

• Day 1: Introduction to choiceless large cardinals. Video Slides Exercise
  Section: Prerequisite (by Jason Chen). Video Notes
• Day 2: Measurable cardinals and constructibility, ordinal definablility and the HOD conjecture. Video Slides Notes Exercise
  Section: Exercise 01 (by Jason Chen). Video
• Day 3: Periodicity in the cumulative hierarchy. Video Notes Exercise
  Section: Exercise 02 (by Jason Chen). Video
• Day 4: Simulating the Axiom of Choice, the structure of the club filter. Video Notes Exercise
  Section: Exercise 03 (by Jason Chen). Video
• Day 5: Measures on ordinals, the $\theta_\alpha$ sequence. Video
  Section: Exercise 04 (by Jason Chen). Video

Reference:

• R. M. Solovay, W. N. Reinhardt and A. Kanamori, Strong Axioms of Infinity and Elementary Embeddings, Annals of Mathematical Logic 13 (1978), pp. 73-116.
• J. Bagaria, P. Koellner and W. H. Woodin, Large Cardinals Beyond Choice, Bulletin of Symbolic Logic 25 (2019), issue 3, pp. 283–318.
• Background reading for Lecture 2: Woodin-Davis-Rodriguez The HOD dichotomy, and G. Goldberg preprints Strongly compact cardinals and ordinal definability and The uniqueness of elementary embeddings.
• Background reading for Lecture 3: Goldberg-Schlutzenberg Periodicity in the cumulative hierarchy.
• Background reading for Lecture 4: G. Goldberg Measurable cardinals and choiceless axioms.
• Background reading for Lecture 5: G. Goldberg Choiceless cardinals and the continuum problem.

Lecturer:

Gabriel Goldberg is an assistant professor at the department of mathematics, at UC Berkeley. He is working in set theory, interested in large cardinals, inner models, and infinite combinatorics. He has published some important papers in large cardinal and inner models in JML and JSL in recent years. He is also writing the book The Ultrapower Axiom.

Organizers

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

Contact: logic@fudan.edu.cn

Photos