Berkeley Mathematician Receives Three Million Dollar Prize for Advances in Topology
By Lina Sorg
On November 8th, Ian Agol from the University of California, Berkeley received the second annual Breakthrough Prize in Mathematics for his research in 3D geometric topology.
The field of topology assumes that all shapes are bendable and pliant, and involves the study of objects that appear flat but actually fold into several dimensions. Specifically, topologists analyze shape properties that remain the same even when a shape’s form is altered. Unlike geometry, which examines subtler properties, topology works with largescale properties of space. Although topologists have a solid understanding of twodimensional surfaces, 3D models prove more challenging because they cannot always be displayed visually.
Agol’s research is a continuation of the work of mathematician William Thurston. In 1982, Thurston published questions and preliminary answers regarding how to create and work with 3manifolds, or possible shapes for the universe; his questions have since driven topology research. Thurston’s geometrization conjecture (now the geometrization theorem) simplifies 3manifolds and permits mathematicians to use the twodimensional geometry of object surfaces to understand 3D properties. Hyperbolic geometry, however, does not fit this generalization.
Berkeley mathematician Ian Agol discusses the work that earned him the annual Breakthrough Prize in Mathematics.
Agol offers a simplified means of studying hyperbolic 3manifolds – negatively curved 3D spaces, similar to some proposed models of the universe – via their surfaces. He proves two significant conjectures: the Virtual Haken Conjecture (named for German mathematician Wolfgang Haken) and Thurston’s Virtual Fibering Conjecture. Agol demonstrates that some complex spaces can actually be understood through simpler methods. Essentially, 3manifolds can be defined in multiple ways – geometrically, dynamically, etc. – and Agol’s research unites these different portrayals. Read more about the two conjectures and virtual topology here.
Agol acknowledged the significance of collaboration in mathematics, given that his work builds off the research of other mathematicians. He also credited Thurston for inspiring his acute ability to work visually.
Despite the significance of his breakthrough, Agol says there is still work to be done in the field, and is interested in connecting hyperbolic geometry with other sectors of 3manifold topology. He plans to donate $100,000 of his winnings to help support mathematics grad students from developing countries.
Read more about the Breakthrough Prizes here.

Lina Sorg is the associate editor of SIAM News. 