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. 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.

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. The study of reaction-diffusion equations on metric graphs has been drawing attention recently. Two different research directions will be introduced. First topic is related to the question of whether excitation waves propagate along the branching of axons of nerve cells or not. Namely, we are going to consider a scaler reaction diffusion equation on a star-shaped metric graph. We can observe propagation blocking depending on the numbers of input and output edges. We also discuss the related problem. Second, we study pattern dynamics on compact metric graphs. We consider systems of reaction-diffusion equations on compact metric graphs with Turing or Wave instability. We construct eigenfunctions of Laplacian on specific metric graphs to see pattern onsets depending on the lengths of the edges. By using the normal form analysis and symmetry arguments we study the local bifurcation structures around the bifurcation points. In both cases, we impose natural boundary conditions, namely, Neumann‒Kirchhoff conditions at the junction.

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.