Date/Time : 24 Oct 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Michael Pevzner, Université de Reims-Champagne-Ardenne

Inspired by classical concepts of generating functions for orthogonal polynomials, this talk will explore “generating operators” for a family of differential operators acting between two manifolds.

We will present a formula for these generating operators associated with Rankin–Cohen brackets and discuss their connection to infinite-dimensional representation theory. Joint work with T. Kobayashi.

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Date/Time : 23 Oct 2024 16:00
Venue : S17 #04-05 (SR2)
Speaker : Jimmy He, Ohio State University

The study of quantum groups and integrable systems have long been connected through the Yang-Baxter equation. Recently, these ideas have been introduced in probability to study integrable stochastic models. I will discuss a similar story for models with boundary, which are connected to quantum symmetric spaces through the reflection equation, as well as some applications to probability. The focus will be on illustrating the connections through illustrative examples and applications for a specific model called the six vertex model.

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Date/Time : 23 Oct 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Wei Xin, University of Science and Technology of China

Separating hash families (SHFs) are useful combinatorial structures which generalize several well-studied objects in cryptography and coding theory, including perfect hash families, frameproof codes and parent-identifying codes. The classical upper and lower bounds for the extremum size of SHFs as a function of the alphabet size do not match. In 2008, Blackburn et al. conjectured that the upper bound has the correct exponent.Before our work, the only nontrivial known cases of this conjecture was by Shangguan and Ge in 2016, when the type sum of the SHF goes no larger than 4.

In this talk, I will discuss our recent proof of all the cases of Blackburn’s conjecture when the upper bound exponent is equal to 2. We do this by transforming the problem into finding a large common solution-free set of a large collection of related linear equations. The new idea is to use an algorithm to create a link from each equation to a class of equations with some good quality in finite many steps, while the good quality can be passed backwards through the link.

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Date/Time : 23 Oct 2024 16:00
Venue : S17 #04-04 (SR3)
Speaker : Zhong Zixin, The Hong Kong University of Science and Technology (Guangzhou)

We propose a novel piecewise stationary linear bandit (PSLB) model, where the environment randomly samples a context from an unknown probability distribution at each changepoint, and the quality of an arm is measured by its return averaged over all contexts. The contexts and their distribution, as well as the changepoints are unknown to the agent. We design Piecewise-Stationary ε-Best Arm Identification+ (PSεBAI+), an algorithm that is guaranteed to identify an ε-optimal arm with probability ≥1−δ and with a minimal number of samples. PSεBAI+ consists of two subroutines, PSεBAI and Naïve ε-BAI (NεBAI), which are executed in parallel. PSεBAI actively detects changepoints and aligns contexts to facilitate the arm identification process. When PSεBAI and NεBAI are utilized judiciously in parallel, PSεBAI+ is shown to have a finite expected sample complexity. By proving a lower bound, we show the expected sample complexity of PSεBAI+ is optimal up to a logarithmic factor. We compare PSεBAI+ to baseline algorithms using numerical experiments which demonstrate its efficiency. Both our analytical and numerical results corroborate that the efficacy of PSεBAI+ is due to the delicate change detection and context alignment procedures embedded in PSεBAI.

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Date/Time : 23 Oct 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Ellen Hammatt, Victoria University of Wellington

In this talk we investigate what happens when we take concepts from computable structure theory and forbid the use of unbounded search. In other words, we discuss the primitive recursive content of structure theory. The central definition is that of punctual structures, introduced by Kalimullin, Melnikov and Ng in 2017. We investigate various concepts from computable structure theory in the primitive recursive case. A common theme is that new techniques are required in the primitive recursive case. In particular we will focus on topics such as finite punctual dimension, punctual 1-decidability and the punctual degrees. Where the punctual degrees is a degree structure within punctual presentations of a fixed structure which is induced by primitive recursive isomorphisms. I will present various results from my PhD thesis as well as pose some open questions in the area.

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Date/Time : 23 Oct 2024 15:00
Venue : S17 #05-11 (SR5)
Speaker : Zhou Tao, National University of Singapore

While finite-time blowup solutions have been studied in depth for the Keller-Segel equation, a fundamental model describing chemotaxis, the existence of finite-time blowup solutions to chemotaxis-fluid models remains largely unexplored. To fill this gap in the literature, we use a quantitative method to directly construct a smooth finite-time blowup solution for the Keller-Segel-Navier-Stokes system with buoyancy in 3D. The heart of the proof is to establish the non-radial finite-codimensional stability of an explicit self-similar blowup solution to 3D Keller-Segel equation with the abstract semigroup tool from [Merle-Raphaël-Rodnianski-Szeftel, 2022], which partially generalizes the radial stability result [Glogić-Schörkhuber, 2024] to the non-radial setting. Additionally, we introduce a robust localization argument to find blowup solutions with non-negative density and finite mass. This is talk is based on the joint work with Zexing Li.

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Date/Time : 17 Oct 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Wang Yutong, National University of Singapore

The growing of verifiable formal languages, particularly interactive proof assistants like Lean 4, is shaping the future of mathematics and transforming how large language models (LLMs) can assist in automated reasoning. However, navigating the vast and complex Mathlib and composing reliable formal statements remains a challenge, with arcane syntax and scattered lemmas.

In this talk, I will present our recent developments: the LeanSearch platform V2 and the Herald translator, which features Lean-aware LLMs to streamline proof writing and auto-formalization of statements. We will explore specialized model training and data augmentation techniques that enhance LLM-accelerated formalization. I will demonstrate how these tools can assist both mathematicians and Lean developers.

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Date/Time : 16 Oct 2024 15:00
Venue : S17 #05-11 (SR5)
Speaker : Raphael Winter, Cardiff University

We present new results on the kinetic Balescu-Lenard and Landau equations, which describe the collisional relaxation of plasma. The first result establishes global existence of solutions to the Balescu-Lenard equation for initial data close to equilibrium, while the second result shows a new blow-down mechanism for the Landau equation for a family of super-critical initial data. We also discuss the breakthrough result of Guillen and Silvestre who identify the Fisher information as a Lyapunov functional of the spatially homogeneous Landau equation. Joint work with Mitia Duerinckx and Maria Gualdani.

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Date/Time : 16 Oct 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Zhang Xin, The University of Hong Kong

It is a discovery of Margulis in 1970s that congruence quotients of SL_2(Z) can be used to construct expanders, which are certain sparse but highly connected graphs and are ideal models for network building. The Super Approximation Conjec- ture of Salehi-Golsefidy and Varju gives a precise prediction on which more general subgroups of SL_d(Z) have this property.

In this talk, I will survey the history of this conjecture, and describe a recent progress by Tang Jincheng and myself that all Zariski dense subgroups of SL_2(Z) × SL_2(Z) or SL_2(Z) w Z_2 have this property. This progress relies on the development of a key tool in arithmetic combinatorics conjectured by Salehi-Golsefidy.

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