陽明交通大學應用數學系

Abstract
Machine learning often asks how two representations of the same data relate: an image encoder and a text encoder, a teacher and a student, or the observed signals of two coupled dynamical systems. Persistent homology offers a multiscale summary of such representations. The usual practice, however, is to match persistence diagrams with a symmetric distance, which leaves two questions open: what should align, and in which direction?
The first part of the talk asks what should align. The death edges of zero-dimensional persistent homology are exactly the edges of a minimum spanning tree, so topology records which samples are linked, not only when components merge. Using this identity, our recent work revisits the Platonic Representation Hypothesis across 204 vision–language model pairs. After calibration, the relational structure of representations converges at both local and global scales. Agreement on exact distances, by contrast, weakens quickly and loses its dependence on model capacity.
The second part asks in which direction alignment runs. We introdcue persistent cross entropy (PCE) that extends persistent entropy to pairs of diagrams. One diagram induces a probability on the points of the other, and an explicit event collects what it fails to explain. The excess over persistent entropy is a Kullback–Leibler divergence and is stable in the Wasserstein distance. We present several numerical examples that shows that PCE is directional and it recovers causal direction in dynamical systems without a joint attractor, and in a first distillation experiment it serves as a teacher-to-student topology loss.

Graphs can be viewed as (non-archimedean) analogs of Riemann surfaces. For example, there is a notion of Jacobians for graphs. More classically, graphs can be viewed as electrical networks. I will explain the interplay between these points of view and some recent applications in algebraic and arithmetic geometry.

In living cells, biochemical reactions form intricate networks, such as metabolic pathways, that give rise to essential cellular functions. However, the complexity of these network structures, together with our limited knowledge of their detailed kinetics, makes it challenging to understand how these functions emerge from network dynamics.
In this presentation, we introduce a method that enables us to determine system responses based solely on network structure. Using this method, we show that the effects of parameter perturbations on the steady state are confined to subnetworks satisfying specific topological conditions. Furthermore, we show that these subnetworks govern the steady-state bifurcations of reaction systems, including which parameters control the onset of bifurcations and which chemical species exhibit bifurcating behavior. Our findings demonstrate that network topology can underlie biological robustness and plasticity, two fundamental characteristics of living systems.

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。

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Abstract
Semiclassical analysis involves the study of the solutions of singularly-perturbed partial differential equations with given initial data or boundary conditions. The singular perturbation enters via small multipliers on derivatives, and the aim is to describe how the given data taken independent of the small parameter evolves over space and time in the asymptotic limit that the parameter tends to zero. This lecture will introduce the topic, present illustrations from the theory of linear and integrable nonlinear equations, and then embark on a detailed analysis of the Cauchy problem for the defocusing nonlinear Schrödinger equation in the semiclassical limit. The latter problem is solved via an inverse-scattering transform and here we focus on the computation of scattering data in the semiclassical limit, which employs the WKB method.

Abstract
Ice sheet flow laws currently use coefficients that depend on their respective power-law exponents, complicating systematic parameter exploration in models. This study proposes dimensionless formulations for both basal sliding and internal deformation laws, which decouple coefficients from exponents, simplifying sensitivity studies and making independent variation of these parameters feasible in ice-sheet models.