Algebraic Dynamics

我研究的是数论，这是自古希腊时代以来最古老的数学分支。数论起源于对整数的离散性质的研究，在过去的2,500年里，数论发展成为数学中最困难的领域。我的实际研究是研究数论中的以下普遍现象：给定一个“不太可能”的离散事件无限频繁地发生，人们推断出该事件无限频繁地发生的环境空间的“全局，刚性”属性。例如，设f是具有有理系数的两个变量的“通用”多项式。然后莫德尔猜想预测，如果存在无限多对（x，y），其中x和y都是有理数，使得f（x，y）=0，那么f的次数必须至多为3。因此，在这种情况下，周围空间是方程f（x，y）=0的所有复数解（x，y）的集合（这只是一条平面曲线），而离散事件是解方程f（x，y）=0的每对有理数（x，y）的出现。因此，一旦存在无限多个这样的离散事件发生，这迫使周围空间是“非常特殊的”，这是对f的次数最多为3的约束。数论的这一特殊部分被称为算术几何。我在算术几何的一个相对较新的领域工作，该领域研究多项式f的全局性质，基于f上给定的离散信息，例如列出给定数x在f下的所有迭代。例如，我与托马斯·塔克和迈克尔·齐夫一起证明了，如果对于两个次数大于1的多项式f和g，以及两个复数x和y，在x在f下的轨道中以及在y在g下的轨道中分别存在无穷多个公共数，那么f和g必须共享一个公共的n。这种特殊的工作叫做代数动力学。

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.