Room: TBA

The first week (Aug 17 - Aug 21):

Jan Dobrowolski: Some applications of model theory to algebraic combinatorics

The second week (Aug 24 - Aug 28):

Joan Bagaria: The world of large cardinals

Hours:

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

Some applications of model theory to algebraic combinatorics

The pseudo-finite dimension is a tool, introduced by Hrushovski, that facilitates application of various model-theoretic techniques to finite combinatorics. Roughly speaking, it measures the rate of growth of a sequence of (natural) numbers with respect to some fixed non-principal ultrafilter, hence yielding a notion of dimension of an ultraproduct of finite sets. In this mini-course, I will discuss some applications of model-theoretic techniques employing this tool to so-called Elekes-Szabó-type problems.

If $V$ is a complex algebraic variety of dimension $d$, we say that $V$ admits power-saving if there are positive constants $C$ and $\varepsilon$ such that the intersection of $V$ with Cartesian products of subsets of $\mathbb{C}$ of size $N$ can be bounded above by $CN^{(d-\varepsilon)}$. An important theorem by Elekes and Szabó established that, in 3 coordinates, a surface admits no power-saving if and only if it projects to a curve (in 2 coordinates), or is (essentially) the graph of the group operation in a one-dimensional connected complex algebraic group.

In 2018, Bays and Breuillard applied model-theoretic techniques involving pseudo-finite dimension to generalise the theorem of Elekes-Szabó to higher arity, and to prove a certain even more general version of the result. I will discuss in detail the proofs of these theorems, along with some particular instances of Elekes-Szabó-type problems.

Program (subject to change):

• Day 1: Overview of the material. The Elekes-Szabó theorem. The pseudo-finite dimension.
• Day 2: Group configuration theorem.
• Day 3 and 4: Higher dimensional Elekes-Szabó theorem (after Bays-Breuillard).
• Day 5: Some instances of the (generalised) Elekes-Szabó problem."

Lecturer:

Jan Dobrowolski is a mathematical logician and Associate Professor at the Department of Mathematics, Xiamen University Malaysia. He is specializing in model theory and its interactions with algebra, combinatorics, and topology. Before joining Xiamen University Malaysia, Dobrowolski held research positions at the University of Manchester, the University of Muenster, the University of Leeds (as a Marie Curie Individual Fellow), and Yonsei University; during this time, his work focused on studying independence in $NSOP_1$ theories and the construction of definable sets, groups, and fields. He holds a PhD in mathematics from the University of Wroclaw.

The world of large cardinals

The so-called Gödel's program in the foundation of mathematics consists on the identification and study of new set-theoretic axioms that would settle natural mathematical questions that cannot be decided on the basis of the standard ZFC (Zermelo-Fraenkel with Choice) system of set theory. Gödel proposed that any new axioms should take the form of strong reflection principles about the set-theoretic universe. This kind of principles are known as axioms of large cardinals and form a hierarchy of increasing consistency strength. In this course we'll give an overview of the world of large cardinals under the general unifying principle of structural reflection, stopping at key milestones in the large-cardinal hierarchy, such as Woodin cardinals, supercompact cardinals, or Vopěnka's Principle. The course will also include a presentation of a recently discovered new kind of large cardinals which imply the failure of strong forms of the Axiom of Choice, which creates a tension between large cardinal axioms and principles asserting global simplicity of the mathematical universe. The new results suggest that large cardinals should not only be studied on the basis of their consistency strength, but also according to the degree to which they enforce failures of forms of the Axiom of Choice. Moreover, they represent a revolutionary alternative to the conventional viewpoint about what the mathematical universe should look like, as expressed in the HOD Conjecture of Woodin.

Program:

• Day 1: Large cardinals as principles of Structural Reflection: An overview. Patterns in the large-cardinal hierarchy.
• Day 2: Below supercompactness. From strong to Woodin cardinals. Reflecting measures and extenders.
• Day 3: Strongly compact and supercompact cardinals. From supercompactness to Vopěnka's Principle, and beyond.
• Day 4: Large cardinals at the edge: exacting and ultraexacting cardinals. Large Cardinals Beyond HOD.
• Day 5: Large Cardinals Beyond Choice and the HOD Conjecture. Rethinking the large-cardinal hierarchy.

References:

• A basic introduction to set theory. Bagaria, J., Set Theory. Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/set-theory
• A general overview of set theory. Bagaria, J., Set Theory. Princeton Companion to Mathematics. Timothy Gowers, June Barrow-Green, and Imre Leader, Editors. Princeton. 2008.
• The standard reference on large cardinals. Kanamori, A., The Higher Infinite, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
• A survey on Structural Reflection and large cardinals. Bagaria, J., Large Cardinals as Principles of Structural Reflection. Bulletin of Symbolic Logic, 29(1), 19-70. 2023. doi: 10.1017/bsl.2023.2
• Bagaria, J., Koellner, P., and Woodin, W.H., Large Cardinals Beyond Choice. Bulletin of Symbolic Logic. September 2019, Volume 25, Issue 3, 283-318.
• Bagaria, J., Goldberg, G., Reflecting measures. Advances in Mathematics. January 2024. Volume 443, May 2024, 109586. https://doi.org/10.1016/j.aim.2024.109586
• Aguilera, J.P., Bagaria, J., and Lücke, P., Large cardinals, structural reflection, and the HOD Conjecture. https://arxiv.org/abs/2411.11568
• Aguilera, J.P., Bagaria, J., Goldberg, G., and Lücke, P., Large cardinals beyond HOD. https://arxiv.org/abs/2509.10254

Lecturer:

Joan Bagaria is a mathematical logician and ICREA Research Professor at the University of Barcelona. He specializes in set theory, logic, and the foundations and philosophy of mathematics. Before his current appointment, Bagaria held positions at several Catalonian universities and was a Fulbright Fellow at UC Berkeley; during this time, he served as the first President of the European Set Theory Society and was a Simons Foundation Fellow at the Isaac Newton Institute. He holds a PhD in Logic and the Methodology of Science from the University of California, Berkeley.

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

Contact: logic@fudan.edu.cn