陽明交通大學應用數學系

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：
1. Operation method of MOSFET and memory devices
2. Recent development of 3-Dimensional devices (Flash memory & DRAM)
3. Reliability issues of Flash memory
(limitation of number of rewrite cycles and the difficulty to analyze)
4. Short review of my reliability studies

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. We consider nearest-neighbor oriented percolation defined on the product space of a multi-dimensional integer lattice and the set of positive integers.

A point in the product space is described by a vector, where the first component (space component) is a point of the lattice and the second component (time component) is a positive integer.

For a pair of points, where the space components are neighbors and the difference in their time component is 1, we can define a bond, which is independently open with probability p/2d with 0 ≤ p ≤ 2d, regardless of the other bonds. It is well known that oriented percolation exhibits a phase transition as the parameter p varies around a critical point pc which is model-dependent. As the dimension tends to infinity, pc coverges to 1.

However, the best estimate for pc provided by Cox and Durret (Math. Proc. Camb. Phil. Soc. (1983)) give upper and lower bounds, but do not yield an explicit expression for pc.

In this talk, we investigate the explicit expression for pc when d > 4, in a way that pc = 1 + C1d^{-2} + C2d^{-3} + C3d^{-4}+ O(d−5), where C1 to C3 are constants. The proof relies on the lace expansion, which is one of the most powerful tool to analyze the mean-field behavior of statistical-mechanical models in high dimensions. We focus less on the details of the proof and more on the background related to the topic.