Date/Time : 01 Dec 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Wenkui Du, Massachusetts Institute of Technology

In this talk, I will discuss the singularity theory of mean curvature flow. As the most natural evolution equation in extrinsic geometry, mean curvature flow has striking applications in geometry, topology and image processing. A central challenge for these applications is understanding the structure of singularities. This talk surveys recent progress on the classification of singularity models of mean curvature flow and their applications. In particular, ancient noncollapsed solutions naturally arise when we consider blow-up limits near singularities. Several results about classification of ancient noncollapsed solutions will be presented. I will also mention my future plan of singularity analysis in geometric variational problems. The talk is based on my joint works with collaborators B. Choi, Daskalopoulos, Haslhofer, Šešum, Zhao and Zhu.

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Date/Time : 24 Nov 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Leonardo Nagami Coregliano, The University of Chicago

The classic de Finetti’s Theorem from 1931 says that an infinite sequence of random variables is invariant under permutations (exchangeable) if and only if it is a mixture of i.i.d. random variables. For a long time, this was considered a curiosity of classic probability theory. In the 80s, exchangeability was revived with the Aldous–Hoover Theorem which completely characterized exchangeable distributions of random matrices (i.e., distributions invariant under simultaneous permutations of rows and columns), and more generally tensors. Since the resulting Aldous–Hoover representation theorem yielded complicated representations, Hoover asked which distributions admitted simpler representations, called $\ell$-independent representations and noticed that a natural assumption, called $\ell$-locality, was necessary but not sufficient.

In this talk, I will cover surprising connections of exchangeability theory with combinatorics and learning theory. In the former connection, exchangeability can be used to construct semantic/geometric limits of dense combinatorial objects, which capture the asymptotic behavior of densities of subobjects; in turn, these limit objects are used to study quasirandomness and connect back to Hoover’s question on independence vs. locality. In the latter connection, exchangeability forms the backbone of high-arity probability approximately correct (PAC) learning theory, which is a natural learning theory for learning hypergraphs. In turn, high-arity PAC learning theory can be used to obtain better bounds in tame hypergraph regularity lemmas, which connect back to Aldous–Hoover representations of low rank (a notion dual to $\ell$-independence).

No background in exchangeability theory, limit theory or learning theory is required to follow the talk.  This talk is based on several joint works with Maryanthe Malliaris, Alexander Razborov, Caroline Terry and Henry Towsner.

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Date/Time : 19 Nov 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Siqing Zhang, Yale University

The Hodge Theorem reveals a fundamental phenomenon in algebraic geometry: the de Rham cohomology can often be identified with the Dolbeault cohomology. The Non Abelian version of this phenomenon “identifies” the moduli of flat connections with the moduli of Higgs bundles. The arithmetic version of this phenomenon echoes these identifications in prime characteristics. In this talk, I will introduce recent advances on the interplay among the Non Abelian Hodge Theories in different characteristics. By bringing moduli theory into the prime characteristic version, we reveal new structures over the complex numbers.

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Date/Time : 17 Sep 2025 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Luo Tao, Shanghai Jiao Tong University

Artificial Intelligence for Mathematics (AI4Math) is a rapidly advancing field with significant implications for both academic research and industrial applications. In this talk, we will review the evolution of AI-driven mathematical reasoning, tracing the transition from intuitive, natural language-based approaches to the rigorous, verifiable proofs enabled by formal languages. After providing a foundational overview, we conduct a technical survey of three key research frontiers within the formal systems: Automated Theorem Proving, Autoformalization, and Automated Semantic Evaluation. The primary focus is placed on our contributions to the latter two areas, which are crucial for bridging the gap between human mathematical knowledge and machine-verifiable logic.

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Date/Time : 25 Jun 2025 10:00
Venue : IMS Executive Seminar Room (S17 Level 3)
Speaker : François Loeser, Sorbonne University

We will provide a general overview of our work with E. Hrushovski on tame geometry over valued fields, using the stable completion of algebraic varieties over valued fields. A key result is the existence of retractions to piecewise-linear objects called skeleta, which are definably isomorphic to definable subsets of the value group. We will conclude by presenting a recent finiteness result obtained with A. Ducros, E. Hrushovski and J. Ye for tropical functions on such skeleta.

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Date/Time : 25 Apr 2025 14:00
Venue : S17 #04-04 (SR3)
Speaker : Xu Tianyi, Peking University (BICMR)

This talk explores the latest developments and emerging trends in the application of AI to mathematics, focusing on leveraging generative models for writing formalized mathematical proofs. We will introduce the idea of formalizing mathematics using proof assistants—computer programs designed to verify the correctness of mathematical proofs. We will then compare various approaches to formalization, highlighting the unique advantages of using large language models (LLMs) for this task. Additionally, we will address key challenges that naturally arise in this process and present our strategies for overcoming them.

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Date/Time : 14 Mar 2025 11:00
Venue : S17 #04-04 (SR3)
Speaker : Wang Yutong, National University of Singapore

In the previous sections, we have introduced some recent LLM tools in Lean 4, especially the ones for auto-formalization and auto-proving. In this lecture, I will show you the basic concepts and techniques behind these tools, and how you can use them to improve your efficiency in both formal and informal research. I will also try to introduce to you some recent progress in the field of LLM reasoning in mathematics.

Zoom link: https://nus-sg.zoom.us/j/84932999912?pwd=tSka7Th1z2McUW3zhjquuUQ7hcLfab.1
Zoom ID: 849 3299 9912
Pwd: 475364

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Date/Time : 07 Mar 2025 11:00
Venue : S17 #04-04 (SR3)
Speaker : Wang Yutong, National University of Singapore

In this lecture, we will continue our unfinished discussion on structures and typical proof logic. We will resume our journey to introduce more advanced techniques in recursion and induction. In addition to theorem proving, I will also talk about how can you design syntax and notations to make your formalization more readable and maintainable.

Zoom link: https://nus-sg.zoom.us/j/84932999912?pwd=tSka7Th1z2McUW3zhjquuUQ7hcLfab.1
Zoom ID: 849 3299 9912
Pwd: 475364

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Date/Time : 21 Feb 2025 11:00
Venue : S17 #04-04 (SR3)
Speaker : Wang Yutong, National University of Singapore

In the third lecture, we will learn how to construct new mathematics objects in Lean. Although Mathlib4 has provide numerous amount of objects and definitions, we always need to build our own infrastractures when formalizing a new aspect. In this course, we will use examples in undergraduate calculus and Gaokao to demonstrate how to properly design the infrastracture. Specifically, we will together explore the usage of class, structure, as well as their combination with definitions and theorems to compose a reasonable formalization project.

This mini-course will live stream. Here are the zoom ID and the link:
https://nus-sg.zoom.us/j/81322620719?pwd=Wg8FVTZbfX4QpZ7vbiYwaBN3n1MO7A.1

Meeting ID: 813 2262 0719
Passcode: 359631

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