Algebraic Dynamics

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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.