The PI proposes to study several questions involving arithmetic aspects of maps between algebraic varieties. These questions include new far-reaching generalizations of major results about abelian varieties, including the Mordell--Lang conjecture (Faltings' theorem) and Merel's uniform boundedness theorem for rational torsion on elliptic curves over a number field. In the generalizations, the group structure in an abelian variety is replaced by objects which exist in any variety. The PI's study of maps between varieties will yield results about dynamical systems, Diophantine equations, automorphism groups of curves, and value distribution of meromorphic functions.
Polynomials whose coefficients are rational numbers are a fundamental tool in many aspect of modern life, ranging from information security to weather prediction. These basic objects have been intensively studied for over two thousand years, but recent years have seen the discovery of previously unexpected phenomena. The proposed research will examine these fundamental objects from several perspectives. This will likely yield immediate consequences in computer science, in addition to long-term consequences across many disciplines. Finally, much of the proposed research will be carried out jointly with graduate and undergraduate students, so that it will make an educational contribution in addition to its scientific impact.
