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 : 21 Jan 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : Yusuf Baris Kartal, University of Edinburgh

Morse theory provides a method for studying the topology of smooth manifolds through the critical points of a function and its gradient flow. In the 1980s, Andreas Floer applied ideas from Morse theory to the infinite dimensional setting, producing an invariant in symplectic topology that has since become a central tool in this field, and later understood to be closely related to enumerative geometry and mirror symmetry. Inspired by Floer’s insight, Cohen-Jones-Segal proposed a framework to obtain finer information from the data arising in Morse and Floer theory. However, this setup is not a priori compatible with the compact group actions, which limits their usefulness in the study of equivariant topology and geometry. In this talk, I will explain how, in joint work with Laurent Cote, we extend the setup and the construction of Cohen-Jones-Segal to study equivariant invariants. I will then discuss applications to topology and symplectic geometry, and explain how tools from homotopy theory can be used to relate symplectic invariants to the topology of the underlying manifold.

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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 10:00
Venue : S17 #05-11 (SR5)
Speaker : Ma Jiajun, Xiamen University and Xiamen University Malaysia

This mini-course focuses on the fundamental aspects of local theta correspondence. In the first lecture, we will examine the Weil representation in detail. We will explore various realizations of the Weil representation and definitions of the Metaplectic group. Additionally, we will discuss the definition of the Weil index and related concepts. The course material is based on Gan-Kudla-Takada’s book The Local Theta Correspondence. This mini-course is designed for senior undergraduate students and graduate students.

This mini-course will live stream.  Here is the zoom ID:
Meeting ID: 848 8485 0855
Passcode: 226224

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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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