Mirror partners

Dr Alan Thompson joined Loughborough in 2018. He is a mathematician whose research is in algebraic geometry, especially mirror symmetry for surfaces and Calabi-Yau threefolds.

Dr Alan Thompson works on mirror symmetry, which is a branch of algebraic geometry. At the heart of mirror symmetry lies the observation that many geometric objects seem to come in pairs, called "mirror partners". These mirror partners are very closely related and their geometric properties are intricately linked.

This idea was first proposed by theoretical physicists studying string theory, but it has since become a tremendously powerful mathematical tool. Often, a difficult mathematical question about a geometric object can be translated, through mirror symmetry, into a much simpler question about its mirror partner. This approach has been used to answer questions in geometry that were previously thought to be intractable.

However, there is a fundamental problem underlying all of this: given a geometric object, we usually have no idea what its mirror partner is! Over the last 30 years, attempts to solve this problem have led to a lot of ad-hoc constructions of mirror partners, each of which seems to work for some geometric objects and completely fail for others. This leads to the second fundamental problem of mirror symmetry: is there a bigger picture? A single overarching theory that combines all of the different mirror partner constructions into one consistent framework?

Dr Thompson's research aims to uncover more of this big picture, by showing that that two of the most widely-used mirror partner constructions, called "Calabi-Yau mirror symmetry" and "the Fano-LG correspondence", are actually two sides of the same story.