陽明交通大學應用數學系

摘要
根據基督教的傳統: 在(舊約)先知瑪拉基之後，「上帝的聲音」靜默了四百多年。沉寂被曠野的呼聲所打破，這個人就是出現在舊約和新約中間在耶穌基督誕生之前的關鍵人物。他被上帝揀選呼召成為先知，他是耶穌的先鋒，為將要來臨的耶穌鋪下救恩之路，這人就是 --- 施洗約翰。類比於此我把格林函數稱之為微分方程的施洗約翰，因為格林函數的目的是得到微分方程的解。
這個演講我將從歷史的觀點特別是 J. Green 的思想介紹格林函數(Green function)。基本上這是一個Poisson方程的問題，需要處理Dirac-delta 函數、基本解(fundamental solution)、單層位勢(single layer potential)、雙層位勢(double layer potential)與牛頓位(Newtonian potential)。除了物理的詮釋之外我還從量綱分析(Dimensional Analysis)的角度引導聽眾如何簡單且直觀地猜出這些量。另外這些量也牽涉到數學分析的核心問題: 連續性(continuity)、可微性(differentiability)與可積性(integrability)。最後我也是透過量綱分析來解釋　Holder 空間為何會出現在(橢圓)偏微分方程的研究。

Abstract. Transformer-based machine learning models have achieved state-of-the-art performance in many areas. However, the quadratic complexity of the self-attention mechanism in Transformer models with respect to the input length hinders the applicability of Transformer-based models to long sequences or large images. To address this, we present Fast Multipole Attention (FMA), a new attention mechanism that uses a divide-and-conquer strategy to reduce the time and memory complexity of attention from $O(n^2)$ to $O(n \log n)$ or $O(n)$, while retaining a global receptive field. The hierarchical approach groups queries, keys, and values into $O(\log n)$ levels of resolution, where groups at greater distances are increasingly larger in size and the weights to compute group quantities are learned. As such, the interaction between tokens far from each other is considered in lower resolution in an efficient hierarchical manner. This multi-level divide-and-conquer strategy is inspired by fast summation methods from n-body physics and the Fast Multipole Method. We perform evaluation on language modeling and image processing tasks and compare our FMA model with other efficient attention variants on medium-size datasets. We find empirically that the Fast Multipole Transformer outperforms other efficient transformers in terms of memory size and accuracy. For large language models, the FMA mechanism has the potential to enable greater sequence lengths, taking the full context into account in an efficient, naturally hierarchical manner during training and when generating long sequences.

Abstract

A domino (resp., lozenge) tiling is a covering of a region on the square (resp., triangular) lattice using dominoes (resp., lozenges) without gaps or overlaps. In this talk, I will introduce the physical background of these objects and several classical techniques for enumerating tilings of specific regions. Finally, I will present recent results regarding the symmetry classes of domino tilings of the Aztec diamonds. This talk does not assume any prior background; undergraduate students are welcome.

Abstract. The heat equation with a random potential driven by space-time white noise serves as a basic model for continuum directed polymers in random media, a topic of longstanding interest in the physics literature. While this model generates systems exhibiting scale invariance and universality, establishing their fundamental properties has posed deep mathematical challenges. In particular, although classical stochastic analysis can construct the corresponding Gibbs measures for polymers as two-dimensional curves, extending these results to three dimensions introduces substantial complexity due to the critical, non-Gaussian behaviour.

This talk will present the basic ideas of continuum directed polymers in random media, focusing on their behaviour as three-dimensional curves.

Abstract. Newton-type methods are among the most effective tools for solving large-scale nonlinear systems arising in scientific computing. Despite their fast local convergence, their global behavior can be unpredictable, with common issues such as overshooting, stagnation, and sensitivity to problem scaling—especially in stiff or highly unbalanced systems.

In this talk, we present a dynamical systems perspective for understanding these behaviors by interpreting Newton iterations as discrete approximations of an underlying continuous-time flow. This viewpoint provides an intuitive characterization of nonlinear imbalance in terms of stiffness, offering insight into why classical globalization strategies, particularly line search, may become ineffective or overly restrictive.

Motivated by this perspective, we revisit line search methods and introduce improved strategies, including curve search techniques, that better align with the intrinsic dynamics of the nonlinear system. We further show that nonlinear preconditioning can be naturally interpreted as a transformation that reduces stiffness and restores balance, leading to improved robustness and convergence.

Numerical examples from nonlinear PDEs illustrate how this framework not only enhances performance, but also provides a unified viewpoint for understanding globalization, acceleration, and stabilization in Newton-type methods.

Abstract

The Kadomtsev-Petviashvili II (KPII) equation is one of the few physically relevant integrable systems in more than one spatial dimension. In this talk, we present an overview of the inverse scattering theory and the stationary phase method, and explain how these tools are used to derive the long-time asymptotic behavior of solutions.