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Mathematical Logic at Fudan

### 2021 Fudan Logic Summer School

Date: Jun 21 - 25, Jul 26 - 30, and Aug 2 - 6

Location: Tencent Meeting / Zoom Meeting (Meeting ID To be send via email according to the registration)

Schedule

#### The first week (Jun 21 - Jun 25): 宋诗畅 Shichang Song: Continuous First Order Logic (in Chinese)

• Lecture1: 9:00 - 10:15 (GMT+8)
• Lecture2: 10:45 - 12:00 (GMT+8)
• Section: 14:00 - 16:00 (GMT+8)

#### The second week (Jul 26 - Jul 30): Rehana Patel: Probabilistic Constructions in Model Theory

• Lecture 1: 14:30 - 15:45 (GMT+8)
• Lecture 2: 16:15 - 17:30 (GMT+8)
• Section: (Next day) 9:30 - 11:30 (GMT+8)

#### The third week (Aug 2 - Aug 6):Antonio Montalban: Scott complexity of countable structures

• Lecture 1: 8:00 - 9:15 (GMT+8)
• Lecture 2: 9:45 - 11:00 (GMT+8)
• Section: 14:00 - 16:00 (GMT+8)

Week 1

### Continuous First Order Logic

The lectures will be in Chinese.

#### Reference:

• 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.

#### Program:

• Day 1: Metric structures, signatures, formulas, semantics. Notes Video
• Day 2: Ultraproducts, Compactness Theorem, connectives. Notes Video
• Day 3: Lowenheim-Skolem Theorem, types, definability. Notes Video
• Day 4: Omitting types theorem, separably categoricity, quantifier elimination, stability. Notes Video
• Day 5: Application to probability theory; probability algebras, random variable structures. Notes Video

Week 2

### Probabilistic Constructions in Model Theory

Probabilistic constructions of countable objects play an important role in combinatorics and logic. Perhaps the most well known such construction is the Erdos-Rényi process, in which edges on pairs of distinct natural numbers are determined by independent tosses of a fair coin, producing the Rado graph up to isomorphism with probability 1. The Erdos-Rényi process has the symmetry property of “exchangeability", that is, it does not depend on the order in which the pairs of distinct natural numbers are considered. These lectures will aim to give students a taste of probabilistic constructions that arise in model theory, with a focus on ones that are exchangeable. Topics will include first order zero-one laws, invariant measures via sampling from Borel structures, and sparse random graphs; these may be modified given time constraints and student interest.

#### Reference

• N. Ackerman, C. Freer, and R. Patel, Invariant measures concentrated on countable structures, Forum Math. Sigma 4 (2016), no. e17, 59 pp.
• J. T. Baldwin and S. Shelah, Randomness and semigenericity, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1359–1356.
• K.J. Compton, “Laws in logic and combinatorics”, in: I. Rival (ed.), Algorithms and order, NATO ASI Series (Series C: Mathematical and Physical Sciences), vol. 255, Springer, Dordrecht, 1989, 353–383.
• R. Fagin, Probabilities on finite models, J. Symb. Log. 41 (1976), no. 1, 50–58.
• H. Gaifman, Concerning measures in first order calculi, Israel J. Math. 2 (1964), 1–18.
• Yu.V. Glebskii, D.I. Kogan, M.I. Liogon’kii, and V.A. Talonov, Range and degree of realizability of formulas in the restricted predicate calculus, Cybernet. Systems Anal. 5 (1969), no. 2, 142–154.
• W. Hodges, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.
• P.G. Kolaitis, H.J. Prömel, and B.L. Rothschild, Kl+1-free graphs: asymptotic structure and a 0-1 law, Trans. Amer. Math. Soc. 303 (1987), no. 2, 637–671.
• F. Petrov and A. Vershik, Uncountable graphs and invariant measures on the set of universal countable graphs, Random Structures Algorithms 37 (2010), no. 3, 389–406.
• J. Spencer, The strange logic of random graphs, Algorithms and Combinatorics, vol. 22, Springer, Berlin, 2001.

#### Program:

• Day 1: Model-theoretic preliminaries, Fraissé theory, and the Erdos-Rényi process.
• Day 2: First order zero-one laws for combinatorial classes. Notes
• Day 3: Exchangeable constructions of countable structures. Notes
• Day 4: Exchangeable constructions of first-order theories. Notes
• Day 5: Random constructions of sparse Hrushovski structures. Notes

#### Lecturer:

Rehana Patel is a model theorist. Her main research interests are in applications of model theory to the study of combinatorics and random structures. She has held faculty positions at Harvard University, Olin College of Engineering, and other institutions in the United States, and is a visiting lecturer at the African Institute for Mathematical Sciences, Senegal.

Week 3

### Scott complexity of countable structures

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

Slides Slides with notes Video

#### Program:

• 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.

#### Lecturer:

Antonio Montalban is Professor at the department of mathematics, at the University of California, Berkeley.

Organizers

2021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