Date/Time : 21 Jan 2025 14:00
Venue : S17 #06-11 (SR6)
Speaker : Johannes Lederer, University of Hamburg

Artificial intelligence is arguably the hottest topic in all of science. Computer science and engineering currently set the agenda in this field, sidelining mathematics to a large extent. This talk, however, will highlight that mathematics has a lot to offer. We will introduce mathematical guarantees that provide deep insights into the inner workings of AI, and we will show how statistical principles can make AI more efficient. More generally, we will discuss the role of mathematics, especially statistics, in AI and data science.

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Date/Time : 21 Jan 2025 14:00
Venue : S17 #05-11 (SR5)
Speaker : Hui Gao, Southern University of Science and Technology

Using the Breuil–Kisin module attached to an integral crystalline representation, one can define an integral Hodge filtration whose behavior is closely related to arithmetic and geometry of the representation. In this talk, we discuss vanishing and torsion bound on graded pieces of this filtration, using a filtered integral Sen theory as key tool. This is joint work with Tong Liu.

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Date/Time : 20 Jan 2025 09:00
Venue : Zoom link TBA
Speaker : Yangyang Li, The University of Chicago

Minimal surfaces, which generalize geodesics to higher dimensions, naturally model various physical phenomena and are fundamental tools in geometric analysis. In this talk, I will survey recent advances on the existence, regularity, and Morse-theoretic study of minimal (hyper)surfaces. Highlights include our recent results on the enumeration of minimal surfaces, the generic regularity of minimal hypersurfaces, and their anisotropic counterparts. I will also discuss how these findings illuminate the interplay between the space of minimal hypersurfaces and the ambient Riemannian manifold.

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Date/Time : 17 Jan 2025 15:30
Venue : S17 #04-05 (SR2)
Speaker : Hao Wu, University of Zurich

Dynamical systems are deterministic systems. However, in chaotic systems where the entropy is positive, the ergodic sums often behave similarly to the sums of independent random variables and satisfy the spatial central limit theorems (CLT).

In zero-entropy systems, the spatial CLT often fails due to the lack of (fast) mixing properties. But in some cases, such as irrational rotations of bounded type, we can retrieve the central limit theorem if we study single orbit statistics. Here we fix the starting point but randomise time, hence the word “temporal” in the title. In this talk, I will present an ongoing joint work with Bromberg and Ulcigrai, where we make effective the method in their previous work, coding and Markov chains, to obtain a temporal central limit theorem for a broader class of observables over bounded irrational rotations.

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Date/Time : 17 Jan 2025 15:30
Venue : S17 #04-04 (SR3)
Speaker : Tang Lan, Central China Normal University

We mainly study regularity theory for the Monge-Ampère equations. Firstly, we establish the optimal global C^(1,α) regularity results of generalized solutions to the degenerate Monge-Ampère equations; secondly, we give the free boundary regularity for a class of obstacle problem.

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Date/Time : 16 Jan 2025 14:00
Venue : S17 #04-05 (SR2)
Speaker : Weisheng Wu, Xiamen University

Geodesic flows in (and beyond) nonpositive curvature are important examples of nonuniformly hyperbolic dynamical systems. We discuss some ergodic properties of geodesic flows including the ergodicity of Liouville measure and the uniqueness of measure of maximal entropy, and the applications to the problem of counting closed geodesics.

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Date/Time : 15 Jan 2025 15:30
Venue : S17 #04-04 (SR3)
Speaker : Ye Zikai, Chinese University of Hong Kong

Yang-Mills-Higgs equation plays an important role in modern physics and geometry. In this talk, I will introduce the existence and stability of certain solutions to the U(1)-Yang-Mills-Higgs equation in R^4. I will begin with the standard vortex solutions in R^2 and how Liu-Ma-Wei-Wu used them to construct a solution in R^4 that concentrates near the Arrezo-Pacard surface by the Lyapunov reduction method. Then I will show this solution is stable due to the stability of the standard vortex solution and the Arrezo-Pacard surface as a holomorphic curve. Finally, I will mention several ongoing projects on this topic.

This is the joint work with Yong Liu and Juncheng Wei.

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Date/Time : 15 Jan 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : Yash Lodha, The University of Hawaii at Manoa

The groups of homeomorphisms of the real line and the circle exhibit a remarkable subgroup structure, and consequently, a plethora of topological, algebraic and combinatorial phenomena. In this talk I shall provide some historical motivation to study such groups, and describe some concrete examples that illustrate the theory. I will also present some recent advances in the area.

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Date/Time : 15 Jan 2025 14:00
Venue : S17 #04-05 (SR2)
Speaker : Anh Duc Vo, Harvard University

In this talk, I will discuss the notion of Du Bois complexes and provide an overview of classical notions such as rational and Du Bois singularities. I will then discuss their recent extensions to k-rational and k-Du Bois singularities, both for local complete intersections (LCIs) and non-LCI varieties. This talk is based on joint work with Wanchun Shen, and Sridhar Venkatesh.

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