陽明交通大學應用數學系

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. 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
Given a set of lines in R^d, a joint is a point contained in d linearly independent lines. Guth and Katz showed that N lines can determine at most O(N^{3/2}) joints in R^3 via the polynomial method.
Yu and I proved a tight bound on this problem, which also solves a conjecture proposed by Bollob\'as and Eccles on the partial shadow problem. It is surprising to us that the only known proof to this purely extremal graph theoretic problem uses incidence geometry and the polynomial method.