陽明交通大學應用數學系

Abstract. Branching processes are stochastic processes that are often used to model the evolution of populations. They are also widely used in probability theory, biology (e.g., population genetics, epidemiology), and other fields. In this talk, we will introduce the most fundamental type of branching process, the Galton-Watson branching process, and review some classical limit theorems that describe the long-term behavior of populations. On the other hand, we will also discuss the coalescence problem, which provides us with a different perspective on investigating the history of the population. In addition, we will apply the results of the coalescence problem to explore the limiting behavior of the distribution of individuals’ positions.

Abstract. Recently in the area of enumerative combinatorics, the $q$-analogue of combinatorial structures have been studied widely. Specifically, the aim of this work is to identify two statistics over combinatorial structure $S$ and a proper subset $S'$ of $S$ such that the$q$-analogue represents the $q$-$(1+q)$-expansion over $S'$, and to explore the poset and topological interpretations of this expansion. People provided comprehensive profiles for several integer sequences like classical Stirling numbers of both kinds and Catalan number within this framework. In this work, we extend the work on colored $q$-Stirling numbers of both kinds, as well as colored $q$-Lah numbers. We also briefly discuss $q$-Stirling and $q$-Lah numbers of type $D$.

Abstract.
Wetting phenomena are common in daily life, such as water droplets sliding on a window during rainy days. Recent research has focused on wetting behavior on soft solid materials such as elastomers and polymeric gels, which have promising applications, for example in biomedicine, tissue engineering, and soft robotics. Studies have shown that droplets move significantly slower on soft solids than on rigid surfaces—by orders of magnitude—due to dominant viscoelastic dissipation within the material. In this talk, I will introduce a model that explores contact line motion and the dissipation mechanisms arising from both viscoelasticity and poroelasticity in soft solids. Understanding these dissipation processes is essential for accurately describing droplet motion across different solid materials.

Abstract
We describe a new mechanism that triggers periodic orbits in smooth dynamical systems. To this end, we introduce the concept of hybrid bifurcations, which consists of a bifurcation without parameters and a classical bifurcation. Our main result classifies the hybrid bifurcation when a line of equilibria with an exchange point of normal stability vanishes. We showcase the efficacy of our approach by proving stable periodic coexistent solutions in an ecosystem of two competing predators with Holling’s type II functional response. This is a joint work with Alejandro López Nieto, Phillipo Lappicy, Nicola Vassena, and Hannes Stuke.

Abstract
This talk considers the 3-D guidance law based on target lead angle information and the state-dependent differential Riccati equation (SDDRE) scheme. In an application-oriented manner, it presents theories to significantly improve critical computational performance and thus aims at a fast implementation for impact-angle-constrained interception of agile maneuvering targets. More specifically, regarding the two major computational burdens using SDDRE, we have replaced the burden in numerical applicability checking by a simple, equivalent, and closed-form condition for the entire state space, which is actually the dominant burden as supported by complexity analysis and extensive validations. Notably, the proposed analysis not only complements the early findings of applicability guarantee in literature, but also promotes the efficiency of the proposed philosophy when compared to the classic method, where the latter has caused concerns/reservations due to its feasibility and difficulty. On the other hand, we have largely mitigated the second major burden of SDDRE by—after exhaustive trials—selecting the most efficient Riccati-equation solver until the latest benchmarks. Such evaluations are: 1) in favor of a much-less-known achievement, rather than the common QR-based benchmark and 2) subject to both numerical and hardware experiments including, notably, implementations on microcontrollers and field-programmable gate arrays.

Abstract
In this talk, I will share my journey from being an uncertain mathematics undergraduate to becoming a professional researcher in quantitative finance. I will recount the pivotal moments in my academic path and discuss how I transitioned into the field of finance. Additionally, I will introduce my research areas, including Bayesian statistics, statistical computing (with applications such as Greeks calculation, importance sampling, and spherical Monte Carlo methods), machine learning, and FinTech. By bridging financial applications and statistical methodologies, I aim to provide insights into how mathematics can be a powerful foundation for a career in finance. This talk is designed to inspire and guide math students who are curious about pursuing opportunities in the finance industry.

Abstract
The group of algebraic cycles, i.e, formal sums of algebraic subvarieties of a fixed algebraic variety, is a fundamental invariant in algebraic geometry. Since the group is too large, we usually consider its quotients by adequate equivalence relations. Numerical equivalence is one of such equivalence relations given abstractly. In this talk, I will explain a description of the quotient by numerical equivalence (with rational coefficients) as cohomology of some geometric object under suitable assumption.

Abstract.
A statistical model is said to be calibrated if it has the appropriate level of confidence in its own predictions: that is, the confidence that it assigns to a predicted outcome should accurately reflect that outcome's likelihood. For example, if a weather model is calibrated, then out of all of the days when the model predicts a 30% chance of rain, we should expect that it actually will rain on 30% of them. Calibration is crucial for managing the risks associated with incorrect predictions, but modern deep learning models are systematically miscalibrated: they are overconfident when they are incorrect. To make matters worse, theorists can't agree about how miscalibration should be quantified! The prevailing miscalibration metric in engineering applications is the expected calibration error (ECE), which has been widely criticized because it is discontinuous: a tiny change in the model can lead to a large change in the error. In this talk, I'll try to convince you that this problem isn't really a problem, that ECE was fine all along, and that engineers should feel free to keep using it the way they always have (at least for binary classification tasks). The argument will require us to answer a strange but fundamental question about the topological properties of the conditional expectation operator.

Abstract.
The study of mathematics in China has often focused on the ‘procedure of the Celestial Source’ (天元術tian yuan shu), which is used to set up polynomial equations. The geometrical ancestors of this procedure are less known. The 益古演段Yigu yanduan, authored by the 13 th century mathematician, 李冶Li Ye (1192-1279), however, presents the procedure alongside its two geometrical counterparts, the ‘Section of Pieces [of Areas]’ (條段tiao duan) and the ‘Old Procedure’ (舊術jiu shu). The three procedures are known to represent three generations of algorithms to set up quadratic equations. This presentation aims to make the geometrical procedure ‘speak’ about its genealogy. That is to say, to attempt the reconstruction of the evolution of the geometrical roots of the famous procedure of the Celestial Source. The construction of negative coefficients plays a pivotal role in this evolution. It also is possible to distinguish several layers of composition that reflect several episodes in the development of the quadratic equation with negative coefficients. In other words, this analysis raises a philological problem pertaining to the question of textual transmission and the nature of authorship. This is the opportunity to show a landscape of history of mathematics in China, from its sources to its connection to Taoism.

Abstract
Tropical geometry is a way to approach geometry at "extreme magnitudes", and to work with "combinatorial shadows" of algebraic varieties. It is a recent branch of mathematics, but its central ideas can be traced back much farther. I will present some ideas underlying tropical mathematics, starting with two tools for solving for roots of a single-variable polynomial equation: Descartes's "rule of signs" and Newton polygons. In more modern mathematics, these tools are extended to "higher dimension" versions and used for understanding moduli spaces of varieties and their degenerations. I will describe these ideas through examples and cover some current developments.

Abstract
The classical isoperimetric inequality asserts that the sphere encloses the largest volume with a given surface area, but we can pretty much ask the problem of optimising any quantity with a given geometrical constraint. We will review existing results for the problem of optimising the kth-eigenvalue of the Laplacian with various boundary conditions subject to volume constraint. We will then discuss the problem of optimising the first nonzero eigenvalue of a certain coupled ODE that is motivated by the sloshing problem in fluid dynamics.

Abstract
We will go over many different flavours of the basics of minimal surfaces and see how it connects to calculus of variations. In doing so we will touch on Morse theory of infinite dimensional spaces and the relation between topology and critical points of geometric functionals. We will present many examples and eventually reach some of the modern open questions in the theory.

數學在科學、工程、醫學、經濟和金融等領域有著廣泛的應用，這些應用通常被歸類為應用數學。它不僅能幫助其他學科解決實際問題，有時還會促成新的數學分支或成果的誕生。本次演講將分享個人的應用經驗，涵蓋如何將凸幾何的重要定理結合影像視覺辨識技術，用於產業中的物件量測，為業主需求提供理論基礎。同時，也會討論利用變分法（variational method）解決影像邊框勾勒的實際案例，並分享與產業合作中面對的技術挑戰及解決方法的經驗。

Abstract
Ricci flow was introduced by Hamilton in 1982 to study the topology of 3-manifolds. It can be viewed as a heat flow of the Riemannian metric which evolves in the direction of its Ricci curvature. Ricci soliton is a self similar solution to the Ricci flow and often models the singularity of the flow. In this talk, we will discuss some basic concepts on the Ricci flow and Ricci soliton. Some of the results are based on joint works with Zilu Ma and Yongjia Zhang, also with Man-Chun Lee and Luke Peachey.

Abstract
Mirror symmetry is a duality between complex and symplectic geometry. In 1994, Kontsevich proposed a mathematical definition for mirror symmetry which is now known as homological mirror symmetry (HMS). HMS predicts that the Fukaya category of a symplectic manifold is quasi-equivalent to the derived category of its mirror complex manifold. Despite HMS has been proven in many interesting cases, it's usually hard to give an exact geometric correspondence between objects due to its homological nature. Two years after Kontsevich's proposal, Strominger-Yau-Zaslow introduced an entirely geometric approach to mirror symmetry, which is now known as the SYZ proposal. SYZ suggested that mirror pairs can be obtained by taking dual torus fibration and the mirror functor in HMS can be obtained by a Fourier-Mukai-type transform. In this talk, I would like to introduce mirror symmetry from the SYZ perspective. If time permits, I will talk about realization problems in tropical geometry.

--今日演講者因身體微恙，臨時取消今天演講，造成不便敬請見諒。--

Abstract. The study of mathematics in China has often focused on the ‘procedure of the Celestial Source’ (天元術tian yuan shu), which is used to set up polynomial equations. The geometrical ancestors of this procedure are less known. The 益古演段Yigu yanduan, authored by the 13 th century mathematician, 李冶Li Ye (1192-1279), however, presents the procedure alongside its two geometrical counterparts, the ‘Section of Pieces [of Areas]’ (條段tiao duan) and the ‘Old Procedure’ (舊術jiu shu). The three procedures are known to represent three generations of algorithms to set up quadratic equations. This presentation aims to make the geometrical procedure ‘speak’ about its genealogy. That is to say, to attempt the reconstruction of the evolution of the geometrical roots of the famous procedure of the Celestial Source. The construction of negative coefficients plays a pivotal role in this evolution. It also is possible to distinguish several layers of composition that reflect several episodes in the development of the quadratic equation with negative coefficients. In other words, this analysis raises a philological problem pertaining to the question of textual transmission and the nature of authorship. This is the opportunity to show a landscape of history of mathematics in China, from its sources to its connection to Taoism.

Abstract: In a series of influential papers in the 1980s, DiPerna and Majda introduced a rigorous framework of approximate solutions of the Euler equations and proved several results concerning concentration of solutions related to hypothesis on $\omega:= \operatorname*{curl}u:\mathbb{R}^2 \times [0, T] \to \mathbb{R}$ the \emph{vorticity} and $\omega_0:= \operatorname*{curl}u_0:\mathbb{R}^2 \to \mathbb{R}$ the \emph{initial vorticity}. Briefly, they proved that for vorticities bounded in an $\alpha$ log-Morrey space one does not have any concentration for $\alpha>1$, while for $\alpha\leq 1/2$ one may have \emph{concentration-cancellation}. The interval $\alpha \in (1/2,1]$ remained an open question in their paper and subsequent papers, whether one can rule out concentration or find sequences which admit concentration-cancellation. In this talk I discuss a recent result in collaboration with Oscar Dominguez in which we resolve this question, closing the gap, showing in particular that one may have concentration-cancellation up to $\alpha=1$.

Abstract
RSA signature is a cornerstone of network security, widely used by different systems and applications as the means of achieving cryptographic authentication guarantees. In this talk, we will revisit the problem of verifying RSA signatures. Using different techniques, our recent research revealed many instances of unwarranted leniency in implementations of RSA signature verifiers. Critically, our findings suggest that many systems are susceptible to variants of the Bleichenbacher-style RSA signature forgery attack. We will look at how this attack, enabled by weak implementations, can nullify the security guarantees promised by the underlying cryptography, and discuss how this threat can be mitigated in practice.

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 increasing stability phenomenon 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 decreasing instability estimate derived above.

Abstract
Many central problems in geometry, mathematical physics, and biology are reduced to questions regarding the behavior of solutions of nonlinear evolution equations. The global dynamical behavior of bounded solutions for large times is of significant interest. However, in many real situations, solutions develop singularities in finite time. The singularities must be analyzed in detail before attempting to extend solutions beyond their singularities or to understand their geometry in conjunction with globally bounded solutions. In this context, we have been particularly interested in qualitative descriptions of blowup. This talk considers the complex Ginzburg-Landau equation as a particular model for which I will present constructive methods based on spectral analysis and energy-type estimates for the existence and stability of blowup solutions.

Abstract. Nonlinear Schrödinger equations are a widely studied topic due to their various applications in nonlinear optics and Bose–Einstein condensates. The semiclassical analysis serves as a link between quantum mechanics and classical mechanics, providing insight into how classical physics emerges from quantum mechanical principles, with Planck's constant playing a crucial role.

In this talk, I will first apply Madelung's transform to the Schrödinger equation and formally derive the hydrodynamic equations. Then, I will introduce a mathematical method for investigating the semiclassical limits of Schrödinger equations. The focus will be on introducing a modulated energy functional to govern mass density and linear momentum, particularly in relation to the compressible Euler equations. Finally, I will introduce my ongoing work about a nonlocal Schrödinger equation, drawing connections to a nonlocal compressible Euler equation.

Abstract
Pressure-induced drag and lift are key to the performance of wings, rotors and propellers; undulating fins and flapping wings generate forces that are key to locomotion in fish, birds and insects; time-varying fluid dynamic forces drive flutter and flow-induced vibrations of flexible structures in engineering and biology, and these same forces enable the extraction of energy from flow via devices such as wind-turbines. Pressure on a body immersed in a flow is however induced simultaneously by vortices, acceleration reaction (a.k.a. added mass) effects associated with body and/or flow acceleration, and viscous diffusion of momentum, and determining the relative contribution of these different mechanisms on surface pressure remains one of the most important and fundamental issues in fluid dynamics. I will describe the force partitioning method (FPM), a new data-enabled method that partitions pressure forces into components due to vorticity, acceleration reaction and viscous diffusion. FPM has been used to gain new insights into a variety of vortex dominated flows including dynamic stall in pitching foils, vortex-induced vibration of bluff-bodies, hydrodynamics of schooling fish and rough-wall boundary layers, and results from these analyses will be presented. Application of FPM to data generated from experiments will also be described. Finally, FPM has been extended to aeroacoustics, and applications of the aeroacoustic partitioning method (APM) to dissect aeroacoustic noise in engineering and biological flows will be presented.

Abstract. We consider the hard-capacitated facility location problem with uniform facility cost (CFL-UFC). This problem arises as an indicator variation between the general CFL problem and the uncapacitated facility location (UFL) problem, and is related to the profound capacitated k-median problem (CKM). In this work, we present a rounding-based 4-approximation algorithm for this problem, built on a two-staged rounding scheme that incorporates a set of novel ideas and also techniques developed in the past for both facility location and capacitated covering problems. Our result
improves the decades-old LP-based ratio of 5 for this problem due to Levi et al. since 2004. We believe that the techniques developed in this work are of independent interests and may further lead to insights and implications for related problems.

Abstract. In this talk, we consider some nonlinear hyperbolic systems of balance laws near resonance. We focus on the classical solutions near or away from vacuum state (resonance ) to the compressible Euler equations for either isentropic or non-isentropic flows through non-uniform ducts. We establish the global existence of classical solutions for both cases. By the normalization of systems, maximum principle, generalized Lax transformation and the detail study of Riccati equations, we are able to obtain uniformly bounded C^1 norms of those solutions . The asymptotic behavior of solutions is also studied.

This series of projects are the joint works with Hsin-Yi Lee, Jay Chu, Ying-Chieh Lin, Shih-Wei Chou and Shih-Ming Wang.