Monday, March 16 2026, 4:25 - 5:25pm Tuesday, March 17 2026, 4:25 - 5:25pm Wednesday, March 18 2026, 4:25 - 5:25pm MLC 0150, Boyd 0328 Laura DeMarco Radcliffe Alumnae Professor at the Radcliffe Institute for Advanced Study Harvard University Laura DeMarco, Professor of Mathematics 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 related to many different areas of mathematics. In the first talk, 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. In the second talk, I will explain historical connections to the study of elliptic curves and the geometry of their torsion points, and I will show how our analysis of bifurcations (for example, in studying the Mandelbrot set) allowed us to say new things. In the third talk, we will look at the bigger picture, as I present a conjecture on the geometry of periodic points for dynamical systems on P^N (and of torsion points in abelian varieties as a special case), encompassing the theorems presented in the first two talks and many well-known results from the last 50 years. The three talks are independent.
