Perspectives on Hilbert's 13th Problem

 There are still completely open fundamental questions about polynomials in one variable. One example is Hilbert’s 13th Problem, a conjecture going back long before Hilbert. Indeed, the invention of algebraic topology grew out of an effort to understand how the roots of a polynomial depend on the coefficients. The goal of this talk is to explain part of the circle of ideas surrounding these questions, discuss partial progress and explore current obstructions. This is ongoing joint work with Benson Farb and Mark Kisin, as well as with Claudio Gomez-Gonzales, Sidhanth Raman and Alex Sutherland.