Date/Time : 18 Feb 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : Bao Huanchen, National University of Singapore

Let G be a connected reductive group over an algebraically closed field, e.g. the general linear group. Such groups are classified via root data and can be parameterised via Chevalley group schemes over integers. Let K be the fixed point subgroup of an involution of G. For example, when G is the general linear group, then K could be the group of orthogonal matrices.

In this talk, I shall discuss the construction of integral models of the symmetric pair (G, K) and the symmetric space G/K. I will also mention various applications of our new constructions, e.g., Frobenius splittings, canonical bases, spherical embeddings, etc.. This is based on several joint works with Jinfeng Song (NUS).

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Date/Time : 03 Dec 2024 14:00
Venue : S17 #04-06 (SR1)
Speaker : An Xinliang, National University of Singapore

In this talk, I will give an overview of my recent research. We prove a series of mathematical theorems that rigorously describe the entire process of black hole formation. With scale-critical short-pulse initial data, we solve Einstein’s equations and show that: “in the course of gravitational collapse, a whole trapped region and its boundary (apparent horizon) emerge from a singular spacetime point. The apparent horizon then expands as the spacetime evolves. Passing through a transition period, the spacetime later enters a regime, close to a slowly-rotating Kerr black hole solution. Nonlinear Kerr stability then comes into play and the spacetime metric eventually settles into the Kerr solution with small angular momentum.”

Meanwhile, I also introduce a new framework for proving the high co-dimensional nonlinear instability of naked-singularities. Specifically, I study the censorship of Christodoulou’s naked singularity and show that even a tiny anisotropic perturbation can trigger the formation of an anisotropic apparent horizon, thereby covering and censoring the naked singularity.

If time permits, sharp low-regularity illposedness and wellposedness results for nonlinear wave systems will also be presented.

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Date/Time : 14 Aug 2024 14:00
Venue : S17 #04-06 (SR1)
Speaker : Zhang Tengren, National University of Singapore

Anosov subgroups are a special class of discrete subgroups of a non-compact semisimple Lie group. This notion was first discovered by Labourie, who used it to prove the discreteness of Hitchin representations. In this talk, I will discuss my recent contributions to our understanding of Anosov subgroups and their various generalizations, as well as some applications to studying character varieties.

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Date/Time : 11 Jun 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Subhroshekhar Ghosh, National University of Singapore

The goal of this talk is to give an overview of my research over the past few years. Much of this talk is going to be devoted to the theory and applications of negative association. Two random variables are said to be negatively associated if an increase in one leads to a decrease in the other, in an appropriate stochastic sense. In statistical physics terms, it is a mathematical model that captures the phenomenon of repulsion between charged particles such as electrons. On the theoretical side, we will discuss the phenomena of rigidity and hyperuniformity, and their consequences for the behaviour of large scale particle systems. We will also discuss applications of such strongly correlated stochastic models in machine learning, and how these applications in turn naturally lead to phenomena and questions of interest in theoretical probability. Time permitting, we will also explore other recent results that, strictly speaking, fall outside the ambit of negative associations; a key direction being that of learning under latent group actions.

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Date/Time : 17 Nov 2023 16:15
Venue : via Zoom
Speaker : Chen Ying, National University of Singapore

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https://nus-sg.zoom.us/j/83899499647?pwd=U2EvRVBPZXpxeXNmV3lsVi95Z2FTQT09

Meeting ID: 838 9949 9647
Passcode: math

In the digital era, complex data has emerged across various disciplines, characterized by high dimensions, nonlinear temporal and spatial dependencies, and irregular dynamic changes (non-stationarity). Understanding and effectively analyzing such complex temporal structures and achieving high-precision forecasting have become pressing challenges. Additionally, comprehending the stochastic behaviors of forecasting is paramount for executives and managers, aiding them in making resilient and robust decisions in real-life scenarios. In my presentation, I will introduce my research on advancing time series modeling and methodologies. These approaches leverage the power of data science, machine learning, and deep neural networks, all tailored to address the complexities of modern data. Some of the innovative techniques I will discuss include functional neural tangent kernel estimators, mixed-frequency spatio-temporal models, and deep switching state space models. Together, these methods offer insights and tools for tackling the challenges posed by high-dimensional, nonlinear, and nonstationary data in the contemporary landscape.

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Date/Time : 22 Feb 2023 15:00
Venue : S17 #04-05 (SR2)
Speaker : Tan Yan Fu, Vincent, National University of Singapore

Join Zoom Meeting : https://nus-sg.zoom.us/s/9839098564

In this seminar, I will discuss two works of mine post-tenure, both in the study of multi-armed bandits (MABs), a popular model in online decision making.

In the first problem, we consider the cascading bandits model in the regret minimization setting. There are two major classes of learning algorithms for minimizing the regret in MABs, namely the upper confidence bound (UCB) and Thompson sampling (TS) algorithms. On one hand, the analysis of TS usually involves additional analytical challenges compared to UCB. On the other hand, the empirical performance of TS is usually better than UCB. Consequently, for MAB problems, it is desirable to derive a theoretical guarantee for the performance of TS, but this was unavailable for the cascading bandits model prior to our work. We prove a theoretical performance guarantee for a TS-based algorithm TS-Cascade that matches that of UCB.

In the second problem, we consider the best arm identification problem in linear MABs in the so-called fixed-horizon setting. By leveraging properties of the G-optimal design and incorporating it into the arm allocation rule, we design a parameter-free algorithm, Optimal Design-based Linear Best Arm Identification (OD-LinBAI). We provide a theoretical analysis of the failure probability of OD-LinBAI. Instead of all the optimality gaps, the performance of OD-LinBAI depends only on the gaps of the top d arms, where d is the effective dimension of the linear bandit instance. Complementarily, we present a minimax lower bound for this problem. The upper and lower bounds show that OD-LinBAI is minimax optimal up to constant multiplicative factors in the exponent, which is a significant improvement over existing methods (e.g., BayesGap, Peace, LinearExploration, and GSE), and settles the question of ascertaining the difficulty of learning the best arm in the fixed-horizon setting.

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Date/Time : 30 Nov 2022 14:00
Venue : S17 #04-06 (SR1)
Speaker : Ji Hui, National University of Singapore

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https://nus-sg.zoom.us/j/85864142504?pwd=aUUyaFRETE5hYit6WlYyMGhhdDZYUT09

Meeting ID: 858 6414 2504
Passcode: 764936

Imaging has been vital in advancing science, medicine, engineering, and industrial applications. Today’s imaging systems increasingly rely on computing power and algorithms to indirectly form images (visual interpretation) from only partial and noisy measurements. Most image reconstruction tasks are about solving a challenging linear or non-linear ill-posed inverse problem. For the past few decades, we have witnessed phenomenal developments in mathematical theories, models, and algorithms for solving inverse imaging problems. This talk will start by reviewing our works on prevailing model-driven mathematical approaches in computational imaging with manually-crafted model/transforms, including wavelets and sparsity-based regularization. Then we will describe how to advance these studies into the era of data-driven approaches that directly learn intrinsic structures from image data. Finally, we will discuss the connection between these data-driven models/algorithms and deep convolutional neural networks, which led to further advances in self-supervised deep learning for computational imaging.

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Date/Time : 26 Oct 2022 14:00
Venue : S17 #04-06 (SR1)
Speaker : Cai Zhenning, National University of Singapore

Abstract: The gas kinetic theory is usually employed to solve rarefied gas dynamics. The fundamental mathematical model for the gas kinetic theory is the Boltzmann equation, which describes the fluid states using the distribution function defined over the joint position-velocity space. Due to its high dimensionality and a complicated integral operator, straightforward discretization of the Boltzmann equation will lead to high computational cost. Therefore, we need either highly effective numerical schemes or reduced kinetic models to make simulations feasible. In this talk, I will introduce our improvement on the classical Fourier method to preserve all physical properties including the conservation laws and the entropy dissipation. Also, we will discuss methods for kinetic model reductions including the simplification of the collision operator and the moment methods. Our simplified collision operator includes a parameter that can be adjusted to balance the computational cost and the accuracy, and this parameter can be tuned adaptively during the simulation of the dynamics. As for the moment methods, we study the linear Boltzmann equation and introduce the regularized 13-moment equations with stable boundary conditions, which can be applied to solve slow-moving gases with moderate rarefaction effects.

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Date/Time : 04 Oct 2022 17:00
Venue : S17 #04-06 (SR1)
Speaker : Sun Rongfeng, National University of Singapore

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https://nus-sg.zoom.us/j/85460997100?pwd=S2FVVXUzSGU0U0tSN3Z4MURVSGhVUT09

Meeting ID: 854 6099 7100
Passcode: 504631

Abstract: The stochastic heat equation (SHE) and the Kardar-Parisi-Zhang (KPZ) equation are singular stochastic partial differential equations, for which robust solution theories have been developed in dimension 1. But in the critical dimension 2 for SHE and KPZ, no robust solution theory exists. I will review joint work with Caravenna and Zygouras on the scaling limit of the 2-dimensional directed polymer model, a space-time discretisation of the SHE, which implies that the 2-dimensional SHE and KPZ undergo a phase transition. Below the critical point, both SHE and KPZ have Gaussian fluctuations. At the critical point, we recently showed that the directed polymer model has a unique limit, which can be interpreted as the solution of the critical 2-dimensional SHE.

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