Lecture 1: Monday, March 16, in 150 Miller Learning Center

The (algebraic) geometry of the Mandelbrot set

Lecture 2: Tuesday, March 17, in 328 Boyd

Elliptic curves, periodic points, and bifurcations

Lecture 3: Wednesday, March 18, in 328 Boyd

From abelian varieties to dynamical rigidity: a unifying conjecture  

Abstract: 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.

Laura DeMarco is the Hollis Professor of Mathematicks and Natural Philosophy at Harvard University and a Radcliffe Alumnae Professor at the Radcliffe Institute for Advanced Study.  Her numerous honors include a Frontiers of Science Award from the 2024 International Congress of Basic Science and the 2017 Ruth Lyle Satter Prize in Mathematics.  She is a fellow of the American Mathematical Society and a member of the National Academy of Sciences.