Imperial mathematicians win Veblen Prize for counting theory in geometry

Dr Soheyla Feyzbakhsh and Professor Richard Thomas have won the prestigious Oswald Veblen Prize in Geometry from the American Mathematical Society.

The Veblen Prize is one of the most prestigious international prizes in geometry. Professor Thomas said, “I’m in awe at the list of recipients that have previously won it, who are all heroes of mine, including my former PhD supervisor.”

Their award celebrates groundbreaking work in Donaldson-Thomas (DT) theory, a framework for counting specific geometric objects that has become central in both pure mathematics and theoretical physics.

I’m in awe at the list of recipients that have previously won it, who are all heroes of mine, including my former PhD supervisor. Professor Richard Thomas Department of Mathematics

The prize recognises their remarkable achievement in showing that certain ‘higher-rank’ counts in DT theory can be fully described using simpler, previously known counts. This breakthrough not only simplifies complex calculations but also connects ideas across different mathematical areas, benefiting research into both geometry and string theory.

Simplifying counts in higher dimensions

DT theory is a tool mathematicians and physicists use to count objects on spaces called Calabi-Yau 3-folds, which appear in string theory as possible shapes for our universe's extra dimensions. It was developed 25 years ago by Sir Simon Donaldson and Professor Thomas, who was his PhD student.

Essentially, DT theory allows us to count geometric shapes, like curves and surfaces, within these spaces. While straightforward in small cases, these counts become challenging in higher ranks, which involve more intricate objects and relationships.

Dr Feyzbakhsh and Professor Thomas’s work takes this theory to a new level. They showed that the counts at higher ranks – where things usually get much more complicated – can actually be derived from simpler counts of lower ranks. This result is crucial because it simplifies a range of problems and makes DT theory much more manageable for researchers to use.

Connecting mathematics with string theory

Dr Feyzbakhsh and Professor Thomas’s work has applications in string theory. By simplifying counts at higher ranks, researchers can simplify problems involving the many space-time dimensions involved in string theory.

String theory conceptualises all particles in the universe as vibrating strings. DT theory allows us to count objects called ‘D-branes’ inside Calabi-Yau manifolds which these string attach to, determining the types of particles present in string theory.