The Myhill Lecture Series 2019, *"Complex dynamics and arithmetic geometry"*, will be delivered by Dr. Laura DeMarco, Professor of Mathematics at Northwestern University. She earned her PhD in 2002 from Harvard. DeMarco's research is focused on the dynamics of polynomial or rational mappings on algebraic varieties, especially in dimension 1, with the primary goal of understanding notions of stability and bifurcation. Her recent work explores connections between dynamical properties of maps and the arithmetic geometry of the underlying varieties.

In a series of three talks, I will present connections between recent research in dynamical systems and the classical theory of elliptic curves and rational points. On the dynamical side — specifically in the study of iteration of rational functions (Julia sets, bifurcations, the Mandelbrot set), but originating in the mathematical study of planetary motion — the first connections to number theory were observed about 100 years ago. On the arithmetic side, it was probably the 1960s when dynamical ideas were first used as tools to understand the arithmetic geometry of elliptic curves and higher-dimensional varieties.

My goal is to provide examples of how these relationships developed and where they have brought us today. The first talk is designed for a general audience. The second and third talks are independent from the first but are connected to each other, and will be delivered in a colloquium style.

DeMarco was awarded the AMS Satter Prize in 2017 "for her fundamental contributions to complex dynamics, potential theory, and the emerging field of arithmetic dynamics."

*Excerpts from the AMS News Release:* The work of Laura DeMarco centers on dynamical systems, a branch of mathematics that studies systems that change and evolve, such as models of weather patterns that change over time. The systems can be built from very simple procedures, such as: start with a number, square it and add another number; use the result as the starting point for the next iteration; continue the procedure.

Even such simple dynamical systems can exhibit complex and beautiful patterns; an example is the well-known Mandelbrot set. Because these systems are based on arithmetical rules, connections with number theory arise. These connections have led to the emerging area of arithmetic dynamics, to which DeMarco has become one of the leading contributors.

DeMarco has brought groundbreaking new ideas into dynamical systems by using tools and ideas from other areas of mathematics. The prize citation notes in particular her joint work with M. Baker, which developed new methods through a remarkable confluence of ideas from dynamical systems theory and other areas such as logic, number theory, and analysis. Other work by DeMarco and K. Pilgrim led to striking new results, including a generalization of the well-known theorem that the Mandelbrot set is connected. In her most recent work, DeMarco pinpoints relations between dynamical systems and the arithmetic geometry of objects called elliptic curves.

Laura DeMarco earned her bachelor's degree from the University of Virginia and her master's degree from the University of California, Berkeley. In 2002, she received her PhD from Harvard University. At the University of Chicago, she was a National Science Foundation Postdoctoral Fellow, a Dickson Instructor, and an assistant professor before she joined the faculty at the University of Illinois at Chicago. While there, she received the NSF Career Award and a Sloan Fellowship. During the academic year 2013-14, DeMarco was the Kreeger-Wolf Distinguished Visiting Professor in the mathematics department at Northwestern University. In 2014 she assumed her present position as professor of mathematics at Northwestern. DeMarco was awarded a Simons Fellowship in 2015 and was named a Fellow of the AMS in 2012.

**September 18, 2019**

Wednesday, at 4:00 p.m.

Location: 14 Knox Hall

UB North Campus

**September 19, 2019**

Thursday at 4:00 p.m.

Location: 250 Mathematics Building

UB North Campus

**September 20, 2019**

Thursday at 4:00 p.m.

Location: 250 Mathematics Building

UB North Campus