歷史演講 PAST TALKS

Abstract：

We establish Hoeffding inequalities for the summation of atomic functions possessing suitable almost-orthogonality via introducing a new square function and by constructing an iterative algorithm on the scales of this new square function. The main result yields new multi-parameter Hoeffding inequalities, and generalizes those for dyadic martingales. Our square function and the associated algorithm represent a refined design—contrasting with the classical dyadic martingale framework—enabling a layer-by-layer exploitation of multi-scale cancellation phenomena. 

As an application, we obtain several equivalent characterizations of our Hoeffding inequalities, including the sharp order of local integrability for our generalized multi-parameter square functions. Our work extends the classical result of Chang–Wilson–Wolff (1985) in one-parameter setting and bridges the gap by resolving the multi-dimensional counterpart of Pipher’s work on the bi-disc (1986).

This talk is mainly based on the recent progress: 

Ji Li, Jill Pipher and Liangchuan Wu, “Chernoff–Hoeffding inequalities, almost orthogonality and bounded multi-parameter square functions”, submitted.

20250623演講_李冀教授 (Macquarie University, Australia)

摘要詳如附件

20250623演講_張德健教授 (GEORGETOWN UNIVERSITY)

Abstract：

有幸自1983年在台大土木接觸結構學中的影響線，後經修習物理數學方知那是格林函數的一種。1984年第一屆應力所林紹雄教授開授的應用數學課程讓本人對於勢能理論有了更深入的了解，於碩(1984-1986)博士(1990-1994)論文議題也是圍繞著邊界元與積分方程。講者有幸在1986-1990年間到中山科學研究院二所三組服務，有機會將所發展的對偶邊界元素法應用在火箭、飛彈與飛機的氣動力與結構應力分析方面，讓超奇異積分方程與Hadamard有限部(finite part)不再那麼不可親近。

2003年李明恭主任所主辦的台灣計算數學研討會在林文偉教授與劉晉良教授的推薦下，本人得以Plenary lecture進行大會報告，將我們在工程應用時所需要學習與用到的數學進行介紹並得以解決實際工程問題分享給數學界的朋友，交流過程中個人收穫良多並受到啟發。本演講擬從數學力學與計算的角度檢視四十年來邊界元所呈現的Paradise, Parasite與Paradox的三種面相和個人感受與體會，並以蘇東坡的一首詩〈觀潮〉

廬山煙雨浙江潮，未到千般恨不消。

到得還來別無事，廬山煙雨浙江潮。

作為結語，從看山是山，進而看山不是山到終究看山又是山的心境分享，野人獻曝與數學力學界的朋友們交流與切磋。

20250605演講_陳正宗教授 (國立臺灣海洋大學河海工程學系)

Abstract：

In this presentation, I will introduce Neural ODEs—a framework that integrates traditional numerical methods into the design of its loss function. A primary application of Neural ODEs is to uncover the governing dynamics of a system from time-series snapshots of observable data. For an autonomous system described by the differential equation dx/dt = f(x), the ODEnet uses a trainable neural network to approximate the unknown function f(x), minimizing the error between the numerically simulated states and the observed data sampled at fixed time intervals. My talk will focus on the specific case of learning linear dynamical systems, both conservative and dissipative.

In the first part, I’ll address structural challenges—such as rotational dynamics and long-term trends—associated with using one-step versus multi-step integration schemes. Even when the loss function is minimized effectively, the learned dynamics may fail to reflect the true behavior of the system. I will share our findings and conclude this section with practical recommendations for selecting numerical integrators that better preserve the underlying structure of the system.

In the second part, I’ll explore the effects of observational noise. By comparing results obtained from clean data with those from noisy inputs, I’ll demonstrate how different noise levels can impact the accuracy and stability of the learned dynamical model.

20250515演講_呂秉澤博士 (國立成功大學數學系)

Abstract：

According to Hadamard's definition, a well-posed problem satisfies three criteria: existence, uniqueness, and continuous dependence on the data. Most of forward problems (e.g., the boundary value problem or Calder\'on's problem) can be proved to be well-posed. However, many inverse problems are known to be ill-posed, for example, the inverse boundary value problem in which one would like to determine unknown parameters from the boundary measurements. The failure of the continuous dependence on the data in Hadamard's sense makes the feasible determination of unknown parameters rather difficult in practice. However, if one restricts the unknown parameters in a suitable subspace, one can restore the continuous dependence or stability. Nonetheless, the ill-posedness nature of the inverse problem may give rise a logarithmic type modulus of continuity. For Calder\'on's problem, such logarithmic stability estimate was derived by Alessandrini and Mandache showed that this estimate is optimal by proving an instability estimate of exponential type. When we consider the time-harmonic equation, it was first proved by Isakov that the stability increases as the frequency increases. In this talk, I would like to discuss a refinement of Mandache's idea aiming to derive explicitly the dependence of the instability estimate on the frequency. If time allows, I also want to discuss the stability from the statistical viewpoint based on the Bayes approach. The aim is to show that the posterior distribution contracts around the true parameter at a rate closely related to the stability estimate derived above.

20250508演講_王振男教授 (國立臺灣大學數學系)

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20250410演講_余冠儒教授 (國立中山大學應用數學系)
 

摘要詳如附件

20250327演講_李渭天教授 (國立中興大學應用數學系)

Abstract：

A defining feature of nonlinear PDEs is the formation of singularities, where certain quantities diverge to infinity, causing the PDE to break down in the classical sense. These singularities often indicate extreme physical phenomena, such as shock waves or the formation of black holes. Understanding how and when singularities form is therefore essential.

Since singularities typically occur on very small scales, one can "zoom in" around the singular region to study their structure more closely. The resulting limiting solutions are known as singularity models. Classifying these models reveals potential singular structures, while conversely, one can investigate whether a given model can arise as the blow-up limit of a solution.

In this talk, I will explore both perspectives, focusing on Lagrangian mean curvature flow (LMCF) and curve shortening flow (CSF). I will present our construction of solutions to LMCF that develop finite-time / infinite-time singularities, with prescribed singularity models and precise blow-up rates for the curvature. Additionally, I will discuss the construction of new singularity models in LMCF and partial classifications in CSF.

20250306演講_蘇瑋栢博士 (國家理論科學研究中心)

Abstract：

This study addresses a cooling system design problem reformulated as a black-box optimization task, where the objective function is unknown and evaluated only via a numerical simulator. The main challenge lies in building a surrogate model to approximate the objective function. Gaussian processes (GPs) are commonly used for this due to their flexibility and uncertainty quantification, but they struggle with datasets involving both qualitative and quantitative variables, especially when managing many categorical combinations. To tackle this, we propose a tree-based GP method called the category tree GP (ctGP), designed for models with qualitative factors having numerous categories. It uses a tree structure to split qualitative categories, with GP or mixed-input GP models at the leaf nodes. The splitting is guided by category cross-correlations, which improve prediction accuracy, while a cross-validation-based pruning step refines the model. Results from numerical experiments and the cooling system design case show that ctGP enhances computational efficiency, prediction accuracy, and insights into the cooling system via the tree structure.

20250227演講_林蔚安博士 (國立清華大學統計與數據科學研究所)

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20250220演講_李金彥先生 (國立中央大學數學系)

Abstract：

In this talk, we provide an accessible introduction to Lie Theory, requiring minimal prior knowledge, and highlight its foundational role in advanced robotic applications. We begin with intuitive examples of Lie groups, such as unit complex numbers and 2D rotation matrices, illustrating their use in representing rotations and transformations. Building on these, we introduce unit quaternions and rigid transformation groups, emphasizing their significance in robotic motion representation. Key theoretical concepts—including groups, manifolds, tangent spaces, Lie algebras, exponential and logarithmic maps, and adjoint matrices—are explored, with a focus on their integration into calculus for operations like derivatives, Jacobians, and uncertainty management. The practical relevance of Lie Theory is demonstrated through topics such as perturbations, covariance handling, and integration methods in both continuous and discrete time. These techniques are illustrated with examples of EKF-based localization and smoothing for mobile robots, where motion and measurement models are combined with Jacobian computations.

20250220演講_張逸塵博士 (Isuzu – Technical Center of America, Inc. (ITCA))