Field extensions built from dynamical systems, ramification, and Julia sets (Michelle Manes, American Institute of Mathematics)

Given a polynomial defined over a field $K$, you can build an extension of $K_\infty / K$ by adjoining inverse images of some point $\alpha \in K$. The behavior of these extensions (including the degree [K_\infty : K], ramification, and associated Galois representations) are surprisingly different in the cases that $K$ is a number field versus a local field. In this talk, I’ll describe the development of these ideas over the past almost 20 years, culminating in some ongoing work that pushes our understanding of the local field case even further and connects these ideas to the (Berkovich) Julia set for the polynomials.