Emmy Noether Lectures

2023 Lecturer: Laura DeMarco

Rigidity and uniformity in algebraic dynamics

The periodic orbits and their structure are fundamental features of a dynamical system. In an algebraic setting, where the system is defined by polynomials, we can use tools from algebraic or arithmetic geometry to study these orbits. Important examples come from the study of abelian varieties, but already the setting of polynomials of one variable is a challenge. In this talk, I will describe some open questions and recent progress on families of complex and arithmetic dynamical systems.

Citation:

Laura DeMarco is a Professor of Mathematics at Harvard University and a Radcliffe Alumnae Professor at the Radcliffe Institute for Advanced Study. Before joining Harvard in 2020, she was on the faculty at Northwestern University 2014-2020, where she was appointed the Henry S. Noyes Professor of Mathematics in 2019; at the University of Illinois at Chicago 2007-2014, and University of Chicago 2002-2007. She received her Ph.D. in 2002 from Harvard University, where she studied with Curtis McMullen.

DeMarco has made fundamental and influential contributions to complex dynamics, arithmetic dynamics, and arithmetic geometry. In complex dynamics, she introduced the bifurcation current to study the stable locus in moduli spaces of rational maps and constructed a dynamically natural compactification of these spaces. Both groundbreaking ideas opened new directions of research in complex dynamics. She is a leading architect of the field of arithmetic dynamics. In her joint work with Matthew Baker, a far-reaching dynamical analog of the André-Oort conjecture in arithmetic geometry was formulated. Cases of the conjecture were proved using ingenious combinations of ideas from complex dynamics, logic, and number theory. In arithmetic geometry, her recent joint work with Krieger and Ye addressed a conjecture of Bogomolov, Fu and Tschinkel on uniform bounds on the number of common torsion points on two elliptic curves, and they obtained the first uniform result for a complex family of curves in the Manin-Mumford Conjecture. This paper, published in Annals of Math. in 2000, won the 2020 Alexanderson Award of the American Institute of Mathematics.

DeMarco is very active in serving the mathematics community in various capacities. She is also very active in fostering students and postdocs. Among the numerous conferences she organized are GROW, GROW II, GROW 2017 for undergraduate women in mathematics. DeMarco has won numerous accolades. She was in the inaugural class of Fellows of the American Mathematical Society in 2013; was awarded the Simons Foundation Fellowship in 2015, and the AMS Ruth Lyttle Satter Prize in Mathematics in 2017. She was an invited speaker at the 2018 International Congress of Mathematicians, and in 2020 she was elected a member of the National Academy of Sciences.