歷史演講 PAST TALKS

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)

Abstract：

Birational geometry aims at classifying projective varieties according to certain invariants, and derived categories are a convenient framework to compute such invariants. At the heart of modern birational geometry lies the Minimal Model Program: a quasi algorithmic process that performs a series of basic operations on any given projective variety and produces a relatively simple representative of its birational equivalence class (a minimal variety). These operations induce semiorthogonal decompositions in the derived category. And conversely, it is expected that most derived categories of minimal varieties do not admit any semiorthogonal decomposition. In this talk, we will introduce some of these concepts and present some recent progress in the direction of this indecomposability conjecture. The results are based on joint work with Pieter Belmans and Andreas Demleitner, and joint work in progress with Jungkai Alfred Chen.

20251023演講_Pedro Núñez  紐培卓博士(台大數學系博士後研究員)

Abstract：

In the 1980s, David Goss introduced Drinfeld modular forms in the rank two case, which can be seen as a function field analogue of elliptic modular forms. Later on, their higher rank generalization was studied extensively by Basson, Breuer, Gekeler, and Pink. In this talk, we will discuss a family of Hecke eigenforms in the arbitrary rank setting and provide a recipe for how to construct them. This is a joint work with Ö. Ülkem.

20251016演講_Oguz Gezmis 博士 (清華數學系研究學者)

Abstract:
The Poisson--Boltzmann (PB) equation is a fundamental model for describing ionic distributions. When a charged surface (e.g., an electrode, membrane, or colloid) is in contact with an electrolyte, a structured layer of charge known as the electric double layer (EDL) forms. To understand the structure of the EDL, we study the boundary layer solutions to the singular perturbed problems for PB type equations with a small parameter $\epsilon$ in bounded smooth multiply connected domains under the Robin boundary conditions. The PB type equations include the classical PB, modified PB and charge-conserving PB (CCPB) equations. The CCPB equations present particular analytical challenges due to their nonlocal nonlinearity introduced through integral terms enforcing charge conservation. Using the principal coordinate system, exponential-type estimates and the method of moving planes, we rigorously prove asymptotic expansions of boundary layer solutions throughout the whole domain and show the effect of boundary mean curvature.

20251002演講_呂治鴻博士 (NCTS博士後研究員)

Abstract:
This talk explores several foundational and emerging concepts at the intersection of machine learning theory, optimization, and reinforcement learning. We begin by presenting a novel perspective that frames self-taught reasoning in large language models (LLMs) through the lens of Markov chains, offering a probabilistic
interpretation of multi-step in-context learning. We then investigate the convergence behavior of gradient descent under adversarial conditions, showing that with carefully controlled step sizes, gradient-based methods can still converge to the optimum even in the presence of perturbations, provided the objective satisfies smoothness and Polyak-Łojasiewicz (PL) conditions. Transitioning to sequential decision-making, we revisit the principle of Optimism in the Face of Uncertainty (OFU) and its implications for exploration strategies. We introduce a regret analysis of the Upper Confidence Bound (UCB) algorithm in the linear bandit setting, highlighting key theoretical guarantees. Finally, we examine the Least-Squares Value Iteration (LSVI) framework and its UCB-augmented variant (LSVI-UCB), which bridges supervised learning and reinforcement learning with provable exploration efficiency. Together, these topics emphasize the importance of structure, robustness, and principled exploration in modern learning systems.

20250925演講_吳沛遠副教授 (臺灣大學電機工程學系)

Abstract：
In an electrical network with multiple resistors, it is a classical result of Kirchhoff that the effective resistance between two points can be expressed as a ratio of spanning tree counts. More recently, effective resistances have been used to define a notion of curvature on a graph, which induces a curvature flow, analogous to Ricci flow on manifolds. We will describe some results about this flow, joint with several collaborators, including examples and applications. This notion of graph curvature was motivated by some earlier ideas from arithmetic geometry via tropicalization. 
 

20250911演講_Harry Richman博士 (NCTS博士後研究員)