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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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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Date/Time : 15 Jan 2025 16:00
Venue : S17 #04-05 (SR2)
Speaker : Han Zhang, Soochow University

In 1984, Mahler proposed the following question on Diophantine approximation : How close can irrational numbers in the middle-thirds Cantor set be approximated by rational numbers? One way to reformulate Mahler’s question is to ask if Khintchine’s theorem extends to the middle-thirds Cantor set.

It turns out that random walks on homogenous sapces and slicing theorems in fractal geometry play crucial roles in answering Mahler’s question. In this talk, I will survey works regarding Khintchine’s theorem on fractals, discuss the connection between homogeneous dynamics and Diophantine approximation, and sketch the proof of Khintchine’s theorem on the middle-thirds Cantor set using tools from homogeneous dynamics and certain slicing theorem from fractal geometry.

This is based on a joint work with Timothée Bénard and Weikun He.

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Date/Time : 14 Jan 2025 14:00
Venue : S17 #05-11 (SR5)
Speaker : Yu Xuejun, Rafael Cabral, Huawei Singapore Research Center

This talk explores the applications of Large Language Models (LLMs) in the domain of mathematical reasoning. We introduce the technical advancements that enable LLMs to perform tasks ranging from basic arithmetic to advanced theorem proving. Specifically, we present two pioneering research projects being developed at Huawei’s Singapore Research Center: the Formal Proof Agent and the Geometry Problem Solving Agent. The Formal Proof Agent is designed to solve mathematical proof questions without logical errors. It leverages LLMs to automate the generation of formal proofs by interacting with interactive theorem provers like Lean 4, ensuring reliable and rigorous logical reasoning. Meanwhile, in Geometry Problem Solving, there is the additional challenge of interpreting geometry diagrams, which current multimodal LLMs struggle with. Our work leverages computer vision tools to interpret geometry diagrams and, as before, solves geometry problems step-by-step, by combining formal symbolic languages with the creative capabilities of LLMs. These projects highlight the versatility of LLMs in addressing diverse and multimodal mathematical challenges, paving the way for future innovations in AI-driven reasoning.

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Date/Time : 13 Jan 2025 11:00
Venue : S17 #05-11 (SR5)
Speaker : Stephen Wright, University of Wisconsin-Madison

Complexity analysis in optimization seeks upper bounds on the amount of work required to find approximate solutions of optimization problems in a given class with a given algorithm. There is also study of lower bounds, usually in the form of a worst-case example from a given problem class for a given algorithm class. The relationship between theoretical complexity bounds and practical performance of algorithms on “typical” problems varies widely across problem and algorithm classes, and relative interest among researchers between the theoretical and practical aspects of algorithm design and analysis has waxed and waned over the years. This talk surveys complexity analysis and its relationship to practice in optimization, with an emphasis on linear programming and convex and nonconvex nonlinear optimization, providing historical (and cultural) perspectives on research in these areas.

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Date/Time : 13 Jan 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : Yap Jitwu, Harvard University

Arithmetic dynamics is an emerging field that translates problems from arithmetic geometry into a dynamical framework. In this talk, I will provide an overview of arithmetic dynamics and demonstrate how the equidistribution of Galois orbits, due to Szpiro-Ullmo-Zhang, and Yuan, can be applied within our context. I will then discuss a quantitative version of Yuan’s theorem and along with some implications.

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Date/Time : 10 Jan 2025 14:00
Venue : S17 #04-06 (SR1)
Speaker : Govind Menon, Brown University

We study the training dynamics of the Deep Linear Network (DLN) from the perspective of the geometric theory of dynamical systems. Rigorous results by several authors are unified into a thermodynamic framework for deep learning. The analysis begins with a characterization of the invariant manifolds and Riemannian geometry in the DLN. This is followed by exact formulas for a Boltzmann entropy, as well as stochastic gradient descent of free energy using a Riemannian Langevin Equation. Several links between the DLN and other areas of mathematics are discussed, along with some open questions.

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