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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Date/Time : 16 Oct 2024 16:30
Venue : S17 #05-11 (SR5)
Speaker : Gu Yejun, Institute of High Performance Computing

Innovations in automated material design demand models that predict both mechanical properties and inherent uncertainties. Traditional deterministic models, reliant on fitting parameters subject to experimental conditions, fall short in addressing these needs. This study introduces a parameter-free statistical methodology tailored for alloy prediction and design. Centered on a data-driven mixed density neural network, our approach uses existing experimental data from traditional alloys to develop a new constitutive rule for the evolution of dislocation density. This methodology allows for predicting the uncertainty in the resulting stress-strain curves and quantifies the relevant microstructures that must be measured experimentally for more of a deterministic prediction. Further, we delve into the origins of uncertainty, identifying and narrowing the gap between experiments and simulations. Our findings highlight potential solutions for bridging experimental-simulation discrepancies and offer a reliable yet efficient path to developing new alloys.

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Date/Time : 16 Oct 2024 14:00
Venue : S17 #05-11 (SR5)
Speaker : Havva Yoldas, Delft University of Technology

After a short introduction on kinetic equations and classical L^2/H^1 hypocoercivity techniques due to Dolbeault, Mouhot, Schmeiser AMS 2015, I will talk about Harris-type theorems that is an alternative method for obtaining quantitative convergence rates. I will discuss some applications of these theorems summarising some recent results obtained jointly with Jo Evans (Warwick) on the run and tumble equations that is a kinetic-transport equation modelling the bacterial movement under the effect of a chemoattractant.

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Date/Time : 09 Oct 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Ho Boon Suan, National University of Singapore

When traversing a graph, each subsequent node is typically chosen from the set of unvisited nodes adjacent to the already visited ones. In this talk, I will explore how to count the number of possible traversals, particularly in graphs with a lot of symmetry. We will see how these counting problems lead to interesting hypergeometric identities and how computational methods can help in establishing such identities. The talk is intended for a general mathematical audience.

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Date/Time : 09 Oct 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Athipat Thamrongthanyalak, Chulalongkorn University

In this talk, we study a converse of the Banach fixed point theorem and its connection to tameness in expansions of a real closed field. Let R be a definably complete expansion of a real closed field.

We say that R has the BFPP (short for, Banach fixed point property) when, for every locally closed definable set E, if every contraction on E has a fixed point, then E is closed. In this talk, we prove that if R has an o-minimal open core, then R has the BFPP; and if R has the BFPP, then R has a locally o-minimal open core.

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Date/Time : 09 Oct 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Pierre Alquier, ESSEC

The PAC-Bayesian theory provides tools to understand the accuracy of Bayes-inspired algorithms that learn probability distributions on parameters. This theory was initially developed by McAllester about 20 years ago, and applied successfully to various machine learning algorithms in various problems. Recently, it led to tight generalization bounds for deep neural networks, a task that could not be achieved by standard “worst-case” generalization bounds such as Vapnik-Chervonenkis bounds. In this talk, I will provide a brief introduction to PAC-Bayes bounds, and explain the core ideas of the theory. I will also provide an overview of the recent research directions. In particular, I will highlight the application of PAC-Bayes bounds to derive minimax-optimal rates of convergence in classification and in regression, and the connection to mutual-information bounds.

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Date/Time : 09 Oct 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Enrique ARTAL BARTOLO, Universidad de Zaragoza

In this talk we discuss how algebraic (or arithmetic) and geometric properties of plane curves affect the topology. There are some classical results starting from Zariski where the position of some singular points with respect to curves of some fixed degree has surprising consequences on the topology of these curves. We discuss the state of art of the subject, leading to cases where for curves defined over a number field, the action of the Galois group has also influence on the topology. We will present recent discoveries on the subject, specially with my collaborators and from the Hiro Tokunaga’s team, and also open problems.

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