UBC Professor's Work Helps Extend Groundbreaking Research by Fields Medalist Maryam Mirzakhani
University of British Columbia Computer Science Professor Joel Friedman's earlier work is now playing a key role in advancing the legacy of the late Maryam Mirzakhani—the first woman to receive the Fields Medal, the highest honour in mathematics.
As a graduate student, Mirzakhani pioneered innovative techniques to catalogue complex shapes that challenge our geometric intuition. She went on to make significant contributions to geometry, topology, and dynamical systems. Following her untimely death, mathematicians Nalini Anantharaman (Collège de France) and Laura Monk (University of Bristol) took up the challenge to build on her foundational work.
Hyperbolic surfaces can be characterized by their measure of connectedness. On a sphere, any path will eventually return to its starting point or reach any other point. In contrast, paths on complicated hyperbolic surfaces can continue much longer, due to their intricate structures. Mathematicians were stuck on how to prove a theoretical limit on how connected these complex shapes could be.
Anantharaman and Monk had the idea to approach the problem from a new angle using graph theory. Graphs, composed of points (called vertices) connected by lines (edges), are a fundamental concept in both mathematics and computer science. The researchers noticed that the relationships among paths and points on hyperbolic surfaces mirrored those in graphs.
To take their work further, they turned to Friedman's work from two decades ago. At that time, Professor Friedman proved that most graphs are as connected as possible. His proof, known for its difficulty and complexity, became a key reference point. Understanding and applying it required time and discussions with Professor Friedman.
By adapting relevant parts of Friedman's proof, Anantharaman and Monk succeeded in proving the theoretical limit they had been seeking. Their achievement sheds new light on the properties of hyperbolic surfaces that will contribute to other areas of mathematical research.
Read more about the story and the work here.
Professor Friedman is a professor in the computer science department at UBC. His research focuses on algorithms and graphs, and connecting aspects of graph theory to other fields. He earned his PhD at the University of California, Berkeley in 1987 then taught at Princeton University for 6 years before joining UBC in 1994.