On November 8th, Ian Agol from the University of California, Berkeley received the second annual Breakthrough Prize in Mathematics for his research in 3-D 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 large-scale properties of space. Although topologists have a solid understanding of two-dimensional surfaces, 3-D 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 3-manifolds, or possible shapes for the universe; his questions have since driven topology research. Thurston’s geometrization conjecture (now the geometrization theorem) simplifies 3-manifolds and permits mathematicians to use the two-dimensional geometry of object surfaces to understand 3-D properties. Hyperbolic geometry, however, does not fit this generalization.
Agol offers a simplified means of studying hyperbolic 3-manifolds – negatively curved 3-D 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, 3-manifolds 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 3-manifold 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.