I will discuss recent work on Diophantine equations and on value distribution of meromorphic functions. One highlight on the number theory side will be a proof that, for any polynomial f(x) with rational coefficients, only finitely many rational numbers have more than six rational preimages under f(x). A highlight on the complex analysis side will be a determination of all solutions to the functional equation f o u = g o v in polynomials f,g and meromorphic functions u,v. I will explain how these different topics connect with one another and with several other areas of math. Among other things, the classification of finite simple groups plays a crucial role.

The work described is joint with several undergraduates, and the entire talk will be accessible to all faculty and graduate students.