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Thursday, March 5, 2015

**Abstract:** I will explain recent work with Danny Neftin which describes all complex rational functions of sufficiently large degree whose crucial invariants are not those of a random rational function. I will present several consequences of this result, including refinements of Hilbert's irreducibility theorem, a generalization of Mazur's theorem on rational torsion on elliptic curves, a description of the possible image sizes of rational functions over finite fields, and a result about reducibility of polynomials of the form $f(x)-g(y)$.