Báo cáo khoa học 演講訊息

Abstract：

傅立葉分析起源於兩百多年前對熱傳導問題的研究，如今已成為現代數學、物理及工程科學中最廣泛而深刻的數學工具之一。其核心思想是將複雜的函數與現象分解為基本的振盪成分，藉此揭示其內在結構與規律。

本次演講將從傅立葉級數與傅立葉變換在微分方程中的經典應用出發，逐步介紹傅立葉分析在波動傳播、光學繞射與成像、現代訊號處理，以及量子計算中的量子傅立葉變換與量子相位估計。

演講的一個核心主題是從分解到重建（From Decomposition to Reconstruction）：傳統傅立葉分析著重於將函數分解為基本振盪模式，而許多現代逆問題則試圖從不完整、間接或缺乏相位資訊的測量資料中，重建隱藏的幾何與物理結構。

沿著這條思路，我們將探討逆散射（Inverse Scattering）、相位恢復（Phase Retrieval），以及 MUSIC、ESPRIT 等高解析度次空間估計方法（Subspace Estimation）。

最後，結合我們近期的研究成果，說明如何運用傅立葉分析、球諧函數（Spherical Harmonics）與次空間結構，研究逆問題中的唯一性、穩定性與解析度。

本演講將著重於不同領域之間的數學思想與內在聯繫，而非繁複的技術細節，適合不同專業背景的數學研究者與學生參與。

20261022演講_范姜群健教授 (Department of Mathematics, UC Davis)

Abstract：

Modern generative models can capture rich and complex data distributions, but practical applications often require more than unconditional generation: we may want samples that satisfy a desired property, achieve high reward, or reach rare regions of the model distribution. Retraining or fine-tuning the model for every new objective can be expensive, while gradient-based guidance may be unavailable or ineffective when rewards are black-box, non-differentiable, or highly multimodal.

In this talk, I will present our recent work on test-time steering, where a pretrained generative model is kept fixed and control is performed entirely during inference. I will first introduce Source Parallel Tempering, which moves the steering problem to the simple source distribution of continuous generative models and uses parallel tempering to explore multimodal reward landscapes without requiring reward gradients. I will then discuss how similar ideas extend to masked discrete diffusion models, where reward-aware remask-and-denoise updates are combined with parallel tempering to efficiently explore separated high-reward modes in applications such as biological sequence design. Together, these works suggest a broader perspective: test-time steering can be viewed as a sampling problem over reward-tilted generative distributions, where effective control requires balancing reward optimization, model fidelity, and exploration. I will conclude with some open directions toward test-time control for generative models.

20261015演講_王士欣助理教授 (國立臺灣大學資訊工程學系)

Abstract：

In this talk, “Inspiration and Discovery : Some Professors I Met at Princeton University”, the speaker will share his experiences and insights gained from the remarkable mathematics professors at Princeton. These educators not only shaped his academic journey but also inspired a profound appreciation for the beauty and complexity of mathematics.

Through personal anecdotes and stories of innovative research, the speaker will highlight how their unique teaching styles and groundbreaking work influenced his understanding of the subject. From their passion for problem-solving to their dedication to mentoring students, each professor offered a distinct perspective that illuminated the path of discovery in mathematics.

Join the speaker as he explores the intersection of inspiration and discovery in academia, illustrating how the spirit of inquiry and collaboration at Princeton fosters a vibrant intellectual community.Whether you are a student, educator, or simply curious about the world of mathematics, this talk aims to ignite a spark of curiosity and appreciation for the transformative power of education.

20260623演講_張德健教授(Georgetown University)

Abstract：

Immersed interface problems in Stokes flow arise in fluid–structure interaction,where an internal interface is coupled to a viscous incompressible fluid. One of the simplest of such problems is the 2D Peskin problem, in which a 1D closed elastic structure is immersed in a 2D Stokes fluid. This has been extensively studied computationally and analytically. We extend the 2D Peskin problem into two in two directions:

(1) 2D inextensible interface problem.

(2) 3D Peskin problem.

For the 2D inextensible interface problem, we impose an inextensibility constraint on the interface. Through the boundary integral method, we reformulate the problem as two coupled contour equations, an evolution equation and a tension determination equation. We first study the well-posedness and the regularity of the generalized tension determination problem in Hölder spaces. Next, we use an appropriate time-weighted Hölder space to study the well-posedness and regularity of the full dynamical problem.

We also study the Peskin problem in the 3D case. With the boundary integral

method, the 3D Peskin is reformulated as an evolution equation on a unit sphere for the elastic interface. We use multiple local charts to prove that the problem is well-posed in low-regularity Hölder spaces. Moreover, we prove that the elastic interface becomes smooth instantly in time.

20260611演講_郭柏均助理教授(Purdue University)

Abstract：

Orthogonal polynomials arise in many areas of mathematics, including differential equations, probability theory, and numerical analysis. In this talk, I will present several methods for constructing orthogonal polynomials, and illustrate their rich connections with combinatorial structures, including lattice paths and continued fractions.

20260601演講_鄭硯仁助理教授(國立屏東大學應用數學系)

Abstract：

A countable sequence {x_n} in a separable Banach space X is said to admit a reconstruction formula if every element in the Banach space can be represented as an "infinite linear combination" of x_n, with the convergence of the series in the norm of X. Over the past several decades, the problem of constructing sequences that admit reconstruction formulas with prescribed forms and desirable convergence properties has become a central topic in Banach space theory, approximation theory, and some other related areas.

In this talk, we will provide an introduction to the notion of reconstruction formulas and discuss several related research directions, with particular emphasis on some open problems. The presentation is intended to be introductory in nature, only a basic background in real analysis will be assumed.

20260528演講_游步庭博士後研究員(University of Oregon)

Abstract：

摘要詳如附件。

20260526演講_Kasso Okoudjou教授(Tufts University)

Abstract：

Let f be a polynomial with coefficients in a field K and let t be a variable. The Galois group of the extension of K(t) formed by adjoining all iterated f pre-images of t is called the arithmetic profinite iterated monodromy group (pIMG) of f. I'll discuss some recent joint work with Ophelia Adams on the structure of the pIMG of when f is a unicritical polynomial with degree coprime to char(K).

20260521演講_Trevor Hyde 助理教授 (Mathematics and Statistics, Vassar College)

線上演講連結：

1.https://meet.google.com/hpq-hhwo-xoe

2.開啟 google meet 並輸入下列代碼：hpq-hhwo-xoe

Abstract：

未來的 6G 無線網路將在高度動態、高維度且硬體資源受限的環境中運行，傳統的最佳化和短期學習方法已不再適用。本次演講將探討向原生 AI 無線系統轉變的範式，該系統融合了數據驅動的智慧和推理能力。

在物理層，深度學習將取代迭代最佳化和明確通道估計，從而實現可擴展且節能的 RIS 輔助毫米波 MIMO 系統設計。在網路層，我們將引入雙思維世界模型 (DMWM)，將快速模式識別與緩慢的邏輯推理相結合，從而支持長期規劃、穩健泛化以及在不確定性下的適應性。

除了技術發展之外，本次演講還將探討攻讀博士學位的更深層意義。正如下一代無線系統從被動優化演進到預測智能一樣，攻讀博士學位的過程也代表著在複雜世界中駕馭不確定性、做出基於價值的決策並不斷重新定義自身方向的過程。

20260514線上演講_周伯衡博士後研究員(美國Virginia Tech)

Abstract：

Free probability, initiated in the 1980s by Dan Voiculescu, has become a fundamental tool for studying the asymptotic behavior of random matrices. In general, there are two main approaches in free probability: the analytic approach and the combinatorial approach.

In this talk, we will first review the random matrix problems that motivate our study. We then demonstrate how the combinatorial approach can be used to address these questions effectively. If time permits, we will also introduce some generalizations of free probability and discuss their connections to random matrix theory.

20260507演講_曾培倫助理教授(中原大學應用數學系)

摘要詳如附件

20260430演講_鄭堯副教授(淡江大學應用數學與數據科學學系)

Abstract：

At the 1977 Corvallis conference, P.Deligne proposed a remarkable conjecture describing the algebraicity of critical L-values in terms of motivic periods. The conjecture has recently been settled for pure motives of rank one. In all other known cases in the literature, proofs typically depend on integral representations of the relevant L-functions. In the framework of the Langlands–Clozel correspondence, we introduce, on the automorphic side, a cross-ratio formula for the critical values of Rankin–Selberg L-functions. This formula provides a new method for addressing Deligne’s conjecture. In this talk, we will introduce Deligne’s conjecture and present the cross-ratio formula. If time permits, we will also outline a proof of Deligne’s conjecture for the symmetric power L-functions of modular forms.

20260423演講_陳昰宇助理教授(國立清華大學數學系)

線上演講連結：

1. https://meet.google.com/hpq-hhwo-xoe
2. 開啟 google meet 並輸入下列代碼：hpq-hhwo-xoe

Abstract：

Manifold learning aims to recover the intrinsic geometry of high-dimensional data through spectral analysis of graph operators. In this talk, I focus on scalable spectral methods for multi-sensor data and large datasets. In particular, I introduce Landmark Alternating Diffusion (LAD), a scalable spectral method for sensor fusion, which builds on the Alternating Diffusion (AD) framework to extract the common manifold structure shared by multiple observations. By approximating the alternating diffusion process using a small set of landmark points, LAD significantly reduces computational cost while preserving the geometry captured by AD. Under standard common manifold assumptions, we establish theoretical guarantees including consistency, convergence, and finite-sample error bounds. We also demonstrate the practical effectiveness of LAD on an EEG sleep stage annotation task using multi-channel signals. Besides, I will also briefly discuss related work on Landmark Vector Diffusion (LA-VDM), which extend the landmark framework to Vector Diffusion Maps in order to capture directional structures and connection information in data. Finally, I will mention current work on sensor fusion using interpolation on the manifold of symmetric positive definite (SPD) matrices.

20260326線上演講_葉行遠博士候選人 (國立台灣大學數學系)

Abstract：

A domino tiling is a covering of a region on the plane using dominoes without gaps or overlaps. I will begin by discussing its background from physics, called the dimer covering, which models how diatomic molecules stick on a crystal surface. This talk focuses on two aspects of domino tilings: in combinatorics, I will review their symmetry classes on the Aztec diamond and present new properties; in algebra, I will establish a connection between specific tiling models and symmetric functions from the viewpoint of Macdonald theory. Finally, I will introduce my recent research in dynamical combinatorics, a vibrant field at the intersection of discrete dynamical systems and combinatorics. This talk does not assume any prior background.

20260319演講_李宜霖博士 (國立台灣師範大學數學系)

Abstract：

In this talk, the speaker begins with the story of one of his teachers, Professor John Nash Jr., who was a faculty member in the Mathematics Department at Princeton University. He was awarded the Nobel Prize in Economics in 1994 and the Abel Prize in 2015 for his outstanding career achievements. The Abel Prize is one of the most prestigious awards in mathematics, honoring lifetime contributions to the field. Professor Nash made significant contributions to geometric analysis, partial differential equations, and game theory, which have profoundly influenced not only mathematics and physics but also the real world. In this presentation, the speaker aims to explain Nash's work at an expository level. Furthermore, the speaker seeks to use Nash's example to encourage students and young researchers to consider a deeper question: "Which is more important: a beautiful mind or a beautiful heart?"

20251226演講_張德健教授(Georgetown University)

Abstract：

In this talk, I will discuss the theoretical analysis of the sample complexity of kernel methods in machine learning, with applications to manifold learning and reinforcement learning. In manifold learning, we introduce scalable landmark-based spectral algorithms, Landmark Alternating Diffusion (LAD) and Landmark Vector Diffusion Maps (LA-VDM), designed for sensor fusion and for capturing complex geometric structures,respectively.Under standard manifold assumptions, we present theoretical guarantees on consistency, convergence, and finite-sample error. In reinforcement learning, we analyze kernel-based Q-learning and derive finite-sample complexity bounds for learning an ϵ-optimal policy in large state–action spaces, where the efficiency is characterized by the kernel’s information gain. Together, these results provide a unified sample-complexity perspective on kernel methods across different learning settings, with brief remarks on related work in topological data analysis and signal processing.

20251222演講_葉行遠博士(國立臺灣大學)

Abstract：

From a perspective of designing or engineering for opinion formation games in social networks, the opinion maximization (or minimization) problem has been studied mainly for designing seeding algorithms that aim at selecting a subset of nodes to control their opinions. We first define a two-player zero-sum Stackelberg game of competitive opinion optimization by letting the player under study as the leader minimize the sum of expressed opinions by doing so-called “internal opinion design”, knowing that the other adversarial player as the follower is to maximize the same objective by also conducting her own internal opinion design. We furthermore consider multiagent learning, specifically using the Optimistic Gradient Descent Ascent, and analyze its convergence to equilibria with a fast convergence rate in the simultaneous-game version of competitive opinion optimization.

In another line of previous work, designing learning dynamics has been considered for linear Arrow-Debreu market equilibria. Since Jain reduced equilibrium computation in linear Arrow-Debreu markets to the equilibrium computation in “bijective” markets, we have designed a partially-distributed alternating algorithm for reaching linear bijective market equilibrium, based on solving the rational convex program formulated by Devanur et al., with a convergence rate which can be improved. Thus, we adapt the Optimistic Gradient Descecnt Ascent for a modified program of Devanur et al. with Lagrange multipliers and show converges to market equilibria with bounds of O(1/T^0.5) on the last-iterate convergence rate, where T is the number of iterations, while yet another very recent related work enjoying its proposed distributed dynamics did not consider convergence rates.

20251218演講_陳柏安教授兼所長(國立陽明交通大學資訊管理研究所)

Abstract：

The study of Macdonald symmetric polynomials has produced many interesting combinatorial objects. Perhaps the most famous and well-studied such objects are the $q,t$-Catalan numbers, which can be defined combinatorially as the sum over Dyck paths weighted by the area and dinv statistics. Numerous generalizations of the $q,t$-Catalan numbers have been developed, including extensions to Schroder paths and to nested families of Dyck paths. All of these objects have natural interpretations in terms of the nabla operator $\nabla$ on symmetric functions.

In this talk, I will introduce the algebraic and combinatorial background of the nabla operator and present its new connections with domino tilings of a certain region on the square lattice. In particular, a product formula for the $q,t$-generalization of domino tilings of the Aztec diamond, together with a combinatorial proof of the joint symmetry of the area and dinv statistics on the Aztec diamond, is presented. If time permits, I will also outline some proof ideas and related results on tiling enumeration. This talk does not

assume any prior background.

20251127演講_李宜霖博士 (國立台灣師範大學數學系博士後研究員)

摘要詳如附件

20251120演講_鄭堯助理教授(淡江大學應用數學與數據科學學系)