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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Date/Time : 10 Jan 2025 16:00
Venue : S17 #04-04 (SR3)
Speaker : Yufeng Wu, University of Connecticut

Suppose a person collected his/her own genome and wants to learn something about his/her ancestry. Further suppose this person’s genome originates from multiple ancestral populations, a phenomenon called population admixture in genetics. The genome of an individual from an admixed population consists of segments originated from different ancestral populations. Most existing ancestry inference approaches can provide an estimate of the percentage (called admixture proportion) of the genome originated from a specific ancestral population. Now what if someone wants to estimate admixture proportions of recent ancestors from his/her own genome? For example, let us say an individual has 50% European and 50% Asian ancestry. This individual’s parents can be also 50-50 European/Asian ancestry, but it is also possible one is 100% European and the other is 100% Asian. More generally, can one learn more about the ancestry of recent ancestors from his/her own genome?

In this talk, I will present research results on ancestry inference for estimating admixture proportions of recent ancestors from an extant genome. I will first briefly describe PedMix, one of the first methods for such inference. A major downside of PedMix is that it can only estimate admixture proportions for parents and grandparents, and cannot scale for more distantly related ancestors. I will then focus on PedMix2, a general approach that can estimate admixture proportions of all recent ancestors of this individual. This approach was published recently in PNAS (2024). To the best of our knowledge, there is no other method that can practically infer ancestors beyond grandparents from an extant individual’s genome. I will present results on both simulated and real data show that PedMix2 performs well in ancestry inference.

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Date/Time : 09 Jan 2025 15:00
Venue : S17 #04-05 (SR2)
Speaker : Gianluca Ceruti, University of Innsbruck

The curse of dimensionality presents a formidable challenge in many fields, from molecular quantum dynamics to computer vision and kinetic equations. This phenomenon, characterized by an exponential growth in parameters with increasing system complexity, demands innovative approaches to enable efficient computation and data handling. In this talk, we delve into the promise of Dynamical Low-Rank Approximation (DLRA) as a model order reduction technique for addressing large-scale, time-dependent problems. By dynamically projecting the dynamics of high-dimensional systems onto a manifold of reduced complexity, DLRA reduces computational costs while preserving essential structural properties. Key to this approach are advanced tensor decompositions, such as the tensor train and hierarchical Tucker formats, which provide a foundation for efficient representation and computation. We will highlight recent advances, including the Basis-Update and Galerkin (BUG) integrator and its extensions, which enhance DLRA’s capabilities for preserving system dynamics and reducing memory requirements. Applications will illustrate DLRA’s versatility, from solving time-dependent partial differential equations to enhancing machine learning inference efficiency and reducing training costs in computer vision.

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