Date/Time : 19 Feb 2025 10:00
Venue : S17 #05-11 (SR5)
Speaker : Ma Jiajun, Xiamen University and Xiamen University Malaysia

This mini-course explores the fundamental aspects of local theta correspondence. The series consists of 5 lectures covering the following topics:

1. Models of the Heisenberg-Weil representation and the Metaplectic group
2. Dual pairs and general properties of theta correspondences
3. Theta correspondences over real fields
4. Theta correspondences over finite fields and p-adic fields
5. Geometric aspects of 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 : 19 Feb 2025 15:00
Venue : S17 #05-12 (SR4)
Speaker : Yue Xie, University of Hong Kong

The two-metric projection method is a simple yet elegant algorithm proposed by Bertsekas to address bound/box-constrained optimization problems. The algorithm’s low per-iteration cost and potential for using Hessian information make it a favorable computation method for this problem class. Inspired by this algorithm, we propose a two-metric adaptive projection method for solving the $\ell$_1-norm minimization problem that inherit these advantages. We demonstrate that the method is theoretically sound – it has global convergence. Furthermore, it is capable of manifold identification and has superlinear convergence rate under the error bound condition and strict complementarity. Therefore, given sparsity in the solution, the method enjoys superfast convergence in iteration while maintaining scalability, making it desirable for large-scale problems. We also equip the algorithm with competitive complexity to solve nonconvex problems. Numerical experiments are conducted to illustrate the advantages of this algorithm implied by the theory compared over other competitive methods, especially in large-scale scenarios. Compared with the two-metric projection method, our algorithm directly solves the l1-norm minimization problem without resorting to the intermediate reformulation as a bound-constrained problem, so it circumvents the issue of numerical instability.

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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 : 18 Feb 2025 10:00
Venue : S17 Level 3 (IMS executive seminar room)
Speaker : Qiang Du, Columbia University

Recent applications and theoretical developments of models of integral equations using nonlocal operators have shown promise as effective alternatives to local models, especially in the presence of singularities and anomalies. These models also serve as continuum limits for large-scale discrete models used in data learning and network analysis. We present these models on bounded spatial domains and discuss related modeling, analysis, and computational issues.

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Date/Time : 17 Feb 2025 15:00
Venue : S17 #05-11 (SR5)
Speaker : Li Mengni, Southeast University

In this talk we focus on the inverse scattering problem of nonlinear Alfvén waves governed by the three dimensional ideal incompressible magnetohydrodynamics (MHD) system. By bridging together geometric methods and weighted energy estimates, we establish a set of scattering isomorphisms to substantially strengthen our previous rigidity results. This answer is consistent with the physical intuition that Alfvén waves behave exactly in the same manner as their scattering fields that are detected by faraway observers. The novelty of the present work is twofold: for one thing, the relationship between Alfvén waves emanating from the plasma and their scattering fields at infinities is explored to the best; for another thing, the null structure inherent in MHD equations is thoroughly examined, especially when we estimate the pressure term.

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Date/Time : 14 Feb 2025 11:00
Venue : S17 #04-04 (SR3)
Speaker : Wang Yutong, National University of Singapore

When formalizing problems in high-school and undergraduate level, we encounters different types of formulas and equalities from time to time, and one of the most critical steps is to simplify and solve them in efficient ways. This is also the step where most beginners feel tedious and challenging. In this course, we will dive into tactics like simp, linarith, norm_num, group, ring, etc to make the process easier. If time permitted, we will also briefly explain how you can design your own tactics to provide further simplifications. Various examples and practices will be given in this course.

This mini-course will live stream. Here are the zoom ID and the link:
https://nus-sg.zoom.us/j/81322620719?pwd=Wg8FVTZbfX4QpZ7vbiYwaBN3n1MO7A.1

Meeting ID: 813 2262 0719
Passcode: 359631

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Date/Time : 14 Feb 2025 10:00
Venue : S17 #05-11 (SR5)
Speaker : Luca Makowiec, National University of Singapore

We will introduce Random Spanning Trees in Random Environment (RSTRE), a disordered system on spanning trees that interpolates between the Uniform Spanning Tree (UST) and Minimum Spanning Tree (MST) measures. Our primary goal is to study how local and global observables of the RSTRE depend on the inverse temperature beta. Of particular interest is the diameter of a (typical) spanning tree, as it is the first step towards establishing convergence to a non-trivial scaling limit. Furthermore, for the RSTRE on the complete graph with uniform disorder, we discuss a sharp transition of the local limit of the RSTRE to either the UST or MST local limit. This stands in contrast to our conjectured smooth transition of the diameter when beta is inside the so-called intermediate regime.

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Date/Time : 14 Feb 2025 11:00
Venue : S17 #05-11 (SR5)
Speaker : Benoît Corsini,

In this talk, I will present an upcoming result regarding the local evolution of Prim’s algorithm seen as a random process on graphs. To do so, I will start by introducing the concept of local limit for a sequence of graphs, before defining Prim’s algorithm and showing its relation to the minimum spanning tree. Finally, I will combine these two concepts to provide a representation of the limit, in particular its relation to invasion percolation and the percolation levels of a graph.

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Date/Time : 14 Feb 2025 15:00
Venue : S16 #03-06
Speaker : Xiong Jingang, Beijing Normal University

I will talk about the extinction dynamics of nonnegative solutions to the fast diffusion equation within bounded, smooth domains subject to the Dirichlet zero boundary condition. Initially, I will provide a concise overview of the comprehensive understanding achieved for the subcritical regime of this equation. Subsequently, I will shift my focus to the critical case, which remains an active area of investigation. Under the two-bubble energy threshold assumption on the initial data, we prove the dichotomy that every solution converges uniformly, in terms of relative error, to either a steady state or a blowing-up bubble. This is joint with Tianling Jin at HKUST.

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