Diophantine geometry and arithmetic dynamics

我在数论方面的研究主要涉及来自动力系统的问题。特别是，我的研究涉及以下形式的问题：给定一个有理函数和一些初始值（一个有理数），通过对初始值重复应用该函数产生的数列的算术性质是什么？例如，如果一个这种形式的序列落入一个重复的循环，人们想知道这个循环可能有多长，给出一些关于函数的信息。另一方面，如果数列从不重复，则项的“高度”（本质上是这些有理数的分子和分母上的位数）以指数速率增长。在这个领域中有一个重要的问题，我已经取得了一些进展，并计划进一步研究，那就是为某些有理函数族获得指数增长的确切速率的显式下界。这项研究涉及数论的各个领域（椭圆曲线、递归序列等），以及动力学、计算和数学逻辑方面的问题。

My research in number theory deals largely with problems coming from dynamical systems. In particular, my research deals with questions of the following form: given a rational function and some initial value (a rational number), what are the arithmetic properties of the sequence of numbers generated by repeatedly applying the function to the initial value. For example, if a sequence of this form falls into a repeating cycle, one would like to know how long such a cycle might be, given some information about the function. One the other hand, if the sequence never repeats, then the "heights" of the terms (essentially the number of digits in numerator and denominator of these rational numbers) grow at an exponential rate. One significant problem in the area, one which I have made some progress on, and plan to research further, is to obtain explicit lower bounds on the exact rate of this exponential growth, for certain families of rational functions. This research has implications in various areas of number theory (to elliptic curves, recurrence sequences, etc.), as well as to problems in dynamics, computation, and mathematical logic.