陽明交通大學應用數學系

Abstract
We continue the analysis of the defocusing nonlinear Schrödinger equation in the semiclassical limit. With the scattering data approximated using the WKB method, we turn to the inverse-scattering problem which is formulated as a matrix Riemann-Hilbert problem, here also involving the small parameter in a singular fashion. We develop the key ideas of the Deift-Zhou steepest descent method, and show how it leads to phenomena such as wave breaking.

Abstract. Boundary integral equations (BIEs) efficiently reduce elliptic and wave problems to the boundary, but standard implementations require explicit surface parametrizations and produce fully dense matrices. The Implicit Boundary Integral Method (IBIM) avoids parametrization by using a level-set representation and evaluating layer potentials in a tubular neighborhood of a Cartesian grid, at the cost of dense extended operators and high computational expense.

I will present a complementary approach based on spectral-bias-aided multilevel training of neural-network surrogates for IBIM operators. Exploiting the tendency of neural networks to learn low frequencies first, we design a coarse-to-fine training strategy aligned with the IBIM grid hierarchy. This allows information from coarse levels to accelerate training and inference on finer grids, yielding speedups of about 40–600×. I will show results for Laplace and Poisson problems, and briefly discuss extensions to Helmholtz equations and “numerically consistent” machine learning for scientific computing.

Abstract. The global navigation satellite system (GNSS) has been widely used in positioning, navigation, and timing. The GNSS satellite orbit serves as a reference datum in connection to the International GNSS Service (IGS)-defined reference frame, not only for GNSS ranging measurements but also for the so-called precise point positioning (PPP) technique. Therefore, the accuracy of the reference orbit is crucial for precise geodetic applications. On the other hand, in recent years, with the rapid development of the space satellite industry, countries around the world have been actively promoting the research, development, and application of Low Earth Orbit (LEO) satellites. These satellites have been widely utilized in scientific research and commercial sectors and are quickly advancing toward commercialization. LEO satellites refer to satellites operating at altitudes of approximately 200 to 2,000 kilometers above the Earth’s surface. Compared to medium- and high-orbit satellites, LEO satellites offer advantages such as lower orbital altitude, reduced latency, and higher coverage accuracy. As technology progresses and demand increases, the application scope of LEO satellites continues to expand, particularly playing a critical role in Positioning, Navigation, and Timing (PNT) as well as in communication services.

Abstract. Exploring the mathematical principles behind remote sensing, starting with how to define the location using Earth Coordinate Systems and Geodetic Coordinate Systems, and demonstrate how to map satellite images to a Ground Coordinate System. The presentation will also introduce image processing techniques for both optical and Synthetic-aperture radar (SAR) images, including radiometric calibration and geometric correction. Finally, I'll showcase specific applications such as Total Variation (TV) models, image fusion using neural networks and SAR image denoising.

Abstract. Computational galaxy formation has been remarkably successful in recreating realistic galaxies on a supercomputer using the so-called “cosmological simulations”, where a smooth mixture of gas and dark matter in the early Universe gradually evolves into thousands of galaxies similar to those observed today. The key to this success lies in the physical processes (collectively referred to as "feedback") that drive the cycling of gas in and around galaxies. However, all existing cosmological simulations face a fundamental limitation due to their empirical "sub-resolution" models. In this talk, I will introduce the successes and challenges in this field and discuss exciting recent progress on high-resolution, small-scale simulations that aim to tackle the problem by directly modeling the physics in the interstellar medium. I will also discuss the critical role of innovative numerical algorithms in advancing our field.

Abstract.
Mean field games (MFGs) are formulated as nonlinear coupled systems of partial differential equations, consisting of the Fokker–Planck equation, which governs the density distribution of agents, and the Hamilton–Jacobi–Bellman equation, which describes the temporal evolution of their control inputs. Such systems arise in a broad range of applications, including crowd dynamics, control of autonomous vehicle fleets, mathematical biology, engineering, and economics. Since MFGs are typically posed as space–time boundary value problems, numerical schemes designed for standard initial value problems cannot be directly applied. This motivates the development of new computational methods together with a rigorous mathematical foundation. In this talk, I present an implementation-friendly approach based on a generalized conditional gradient (GCG) method and discuss its convergence properties. In particular, I report recent results, obtained in collaboration with H. Nakamura, for MFGs with local coupling terms.

Abstract
Given a graph and a function on its vertices, how do we partition the vertices into clusters so that (1) vertices with similar function values are in the same cluster and (2) the induced subgraph on each cluster is connected as much as possible? Such a problem has applications in detecting the sources of air pollution, image segmentation, and so on. We will go through the theoretical background of this algorithm and demonstrate some of its applications.

Abstract
The Quantic Tensor Train (QTT) is a tensor network framework that provides highly compact representations of high-dimensional functions and operators. This makes it a promising tool for tackling nonlinear partial differential equations that are otherwise computationally demanding. In this work, we explore the use of QTT for solving the Gross–Pitaevskii equation (GPE), a fundamental nonlinear model describing Bose–Einstein condensates and related quantum systems. By using QTT, we significantly reduce the computational complexity compared to conventional discretization methods, achieving efficient and scalable solutions. Our results demonstrate the potential of QTT to extend tensor network techniques beyond linear problems, opening the door to new applications in nonlinear quantum dynamics and many-body physics.

Abstract. Topological entropy is often regarded as an indicator of complexity. However, there is no finite algorithm to determine whether a d-dimensional (d>1) shift of finite type has positive entropy. In fact, this property is recursively enumerable. In this presentation, I will characterize positive entropy in Markov tree-shifts using adjacency matrices. Additionally, I will explore the relationships among positive entropy, topological properties, and chaotic behavior.

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
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
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.