Algebraic Dynamics

我的研究方向是数论，这是自古希腊时代以来最古老的数学分支。数论起源于对整数离散性质的研究，在过去的2500年里演变成了数学中最困难的领域。我的实际研究研究了数论中的以下普遍现象：假设一个“不太可能”的离散事件无限频繁地发生，人们就会推断出该事件无限频繁发生的环境空间的一种“全局的、刚性的”性质。 例如，设f是具有有理系数的两个变量的“一般”多项式。然后，Mordell猜想预言，如果存在无穷多对(x，y)，其中x和y都是有理数，使得f(x，y)=0，则f的次数至多为3。因此，在这种情况下，环境空间是方程f(x，y)=0(这只是一条平面曲线)的复数中所有解(x，y)的集合，而离散事件是解f(x，y)=0的每对(x，y)有理数的出现。因此，一旦有无限多这样的离散事件发生，这就迫使环境空间变得“非常特殊”，这是对f的次数至多为3的限制。数论的这个特殊部分被称为算术几何。 我在一个相对较新的算术几何领域工作，该领域基于f上给定的离散信息来研究多项式f的全局性质，例如列出给定数字x在f下的所有迭代。例如，我与Thomas Tucker和Michel Zieve一起证明了，如果对两个次数大于1的多项式f和g，以及对两个复数x和y，在f下x的轨道上存在无穷多个公共数，在g下y的轨道上分别存在无穷多个公共数，那么f和g肯定共享一个公共迭代。这一特殊的工作称为代数动力学。

My research is in number theory, which is the oldest branch of mathematics since the times of the ancient Greeks. Number theory originated from the study of the discrete properties of integers, and over the past 2,500 years evolved into the most difficult area of mathematics. My actual research studies the following widespread phenomenon in number theory: given the occurence of an "unlikely" discrete event infinitely often, one deduces a "global, rigid" property of the ambient space where that event takes place infinitely often. For example, let f be a "generic" polynomial of two variables with rational coefficients. Then Mordell's Conjecture predicted that if there exist infinitely many pairs (x,y) where both x and y are rational numbers such that f(x,y)=0, then the degree of f must be at most 3. So, in this case, the ambient space is the set of all solutions (x,y) in complex numbers of the equation f(x,y)=0 (this is simply a plane curve), while the discrete event is the occurence of each pair (x,y) of rational numbers which solves the equation f(x,y)=0. Thus, once there exist infinitely many such discrete events taking place, this forces the ambient space be "very special", which is the constraint on the degree of f being at most 3. This particular part of number theory is called arithmetic geometry. I work in a relatively new area of arithmetic geometry which studies the global properties of polynomials f based on a discrete information given on f, such as listing all the iterates of a given number x under f. For example, together with Thomas Tucker and Micheal Zieve, I proved that if for two polynomials f and g of degree larger than 1, and for two complex numbers x and y, there exist infinitely many numbers in common in the orbit of x under f and, respectively in the orbit of y under g, then f and g must share a common iterate. This particular line of work is called algebraic dynamics.