6月21日 - 6月25日, 7月26日 - 7月30日,  

8月2日 - 8月6日

騰訊會議/Zoom會議（會議ID將根據註冊情況以郵件形式發送）

宋詩暢 Shichang Song: Continuous First Order Logic 

課時一: 9:00 - 10:15 (GMT+8)

課時二: 10:45 - 12:00 (GMT+8)

討   論: 14:00 - 16:00 (GMT+8)

授課教師：宋詩暢（Shichang Song），北京交通大學，數學系，副教授。博士畢業於伊利諾伊大學香檳分校。曾在中國科意昂3數學研究所從事博士後研究。學術研究領域是模型論及其在概率論，群論和組合數學的應用。在Fund. Math., MLQ Math. Log. Q., J. Korean Math. Soc.雜誌發表論文多篇。

Rehana Patel: Model Theory 

課時一: 14:30 - 15:45 (GMT+8)

課時二: 16:15 - 17:30 (GMT+8)

討   論: (第二天) 9:30 - 11:30 (GMT+8)

授課教師⚧🌂：Rehana Patel is a mathematician working in the area of mathematical logic. Her research involves applications of model theory, a branch of mathematical logic, to the study of combinatorics and random structures.

Antonio Montalban: Scott complexity of countable structures

課時一: 8:00 - 9:15 (GMT+8)

課時二: 9:45 - 11:00 (GMT+8)

討   論: 14:00 - 16:00 (GMT+8)

授課教師：Antonio Montalban is a mathematician at the University of California, Berkeley. He works in Mathematical Logic and, within Logic, he focuses on Computability Theory.  In general terms, his research studies the interplay between complexity and mathematics. Logicians have developed techniques to analyze and understand the complexity of many sorts of objects, including sets, structures, constructions, proofs, and so forth.  Computability Theory deals with the complexity measures used on countable objects, in contrast with, for instance, computer science or set theory, which mainly deal with the complexity of finite objects or uncountable objects, respectively.

Continuous First Order Logic

宋詩暢 Shichang Song

本課程講述的連續一階邏輯(continuous first order logic)🦹🏿‍♀️，有時候也叫做度量結構的模型論(model theory for metric structures），連續模型論，或者就叫連續邏輯🩰。它是由Ben Yaacov, Berenstein, Henson, Usvyatsov 在2008年前後發展起來的一種多值邏輯𓀝🛝。連續一階邏輯跟經典一階邏輯最大的不同是，連續一階邏輯的真值表是整個[0,1]區間。連續模型論作為模型論的一種推廣，保持了很多模型論的特性，比如🫰🏼，連續模型論滿足🚟，緊致性定理，Lowenheim-Skolem 定理，可以定義型空間，討論量詞消解⚫️，範疇性和穩定性🚶🏻。本課程將從連續一階邏輯的語法和語義出發，詳細地講述連續一階邏輯的基礎知識🚣🏼‍♂️。最後，作為例子，介紹連續一階邏輯在概率論中的應用🤱🏽。本課程只需要基礎的一階邏輯知識。

The lectures will be in Chinese.

參考文獻:

■I. Ben Yaacov, A. Berenstein, C. W. Henson and A. Usvyatsov, Model theory for metric structures, in: Model Theory with Applications to Algebra and Analysis, Volume 2, London Math. Society Lecture Note Series, 350, Cambridge University Press, 2008, 315-427.

■I. Ben Yaacov and A. Usvyatsov, Continuous first order logic and local stability, Trans. Amer. Math. Soc. 362 (2010), 5213-5259.

課程計劃：

Day 1: Metric structures, signatures, formulas, semantics

Day 2: Ultraproducts, Compactness Theorem, connectives

Day 3: Lowenheim-Skolem Theorem, types, definability

Day 4: Omitting types theorem, separably categoricity, quantifier elimination, stability

Day 5: Application to probability theory; probability algebras, random variable structures

Model Theory

Rehana Patel

TBA

Scott complexity of countable structures

Antonio Montalban

The lectures will be based on the following books with concentration on the Chapter II of the second book.

■Computable Structure Theory: Within the Arithmetic.

■Computable Structure Theory: Beyond the Arithmetic.

課程計劃👩‍🌾🫃🏿：

Day 1: Overview of the notions and the main results

Day 2: The Infinitary language

Day 3: (break)

Day 4: The back-and-forth relations

Day 5: Computable categoricity