Leonidas Alaoglu Memorial Lecture

One of the most famous -- and still not fully understood -- objects in mathematics is the Mandelbrot set. By definition, it is the set of complex numbers c for which the recursive sequence defined by x_1 = c and x_{n+1} = (x_n)^2+c is bounded. This set turns out to be rich and complicated and connected to many different areas of mathematics. I will present an overview of what's known and what's not known about the Mandelbrot set, and I'll describe recent work that (perhaps surprisingly) employs tools from number theory and arithmetic geometry. The recent project is a special case of a broader conjecture in algebraic dynamical systems on the geometry of periodic points and joint work with Myrto Mavraki.