Title: Arithmetic Dynamics, the geometric Frey-Mazur conjecture, and monodromy representations

Abstract: Which representations of the fundamental group of a complex quasiprojective variety X arise from geometry, i.e. appear inside of the monodromy representation on the cohomology of a family of varieties over X? I'll discuss new results on this topic arising from arithmetic dynamics, p-adic transcendence theory, and the geometric Langlands program. The talk will not assume familiarity with any of these topics.

Abstract: A polynomial can be viewed as a branched cover of the sphere over itself that is compatible with a complex structure. If handed a topological branched cover of the sphere, we can ask whether it can arise from a polynomial—and if so, which one? In 2006, Bartholdi and Nekrashevych used group theoretic methods to explicitly solve this problem in special cases, including Hubbard’s twisted rabbit problem. We introduce a new topological approach that draws from the theory of mapping class groups of surfaces.

Title of talk: Acyclotomy of torsion on elliptic curves