Scheme: Dorothy Hodgkin Fellowship
Organisation: Imperial College London
Dates: Jan 2015-Dec 2019
Summary: My current research concerns "enumerative invariants": counts of geometrical objects. For example: given 3 mutually tangent circles, how many circles are tangent to all 3 of them? The answer is always two, independent of exactly what the circles C1, C2, C3 are: hence the answer is "an invariant".
The enumerative invariants that I consider are more complex than this example: they are counts of two-dimensional surfaces inside a given space X. These surface-counts are powerful invariants of X that capture a great deal of its geometry. My field of study has been revolutionised recently by ideas coming from string theory, a candidate theory of quantum gravity capable of describing very small objects (quantum theory) and very large objects (gravity) at the same time. Thinking of surfaces embedded in a space X as being the paths of strings moving in the spacetime X predicts many surprising relationships between curve counts that we had not expected from pure mathematics alone. It also predicts that the curve counting problems are related to other, totally unexpected parts of mathematics such as the theory of differential equations. My research aims to verify these predictions where possible (with applications to geometry and other areas of mathematics) and to refine and correct them where necessary. This gives both a rigorous mathematical foundation and a powerful calculational tool for a class of physical models - the so-called "supersymmetric theories" - that are useful to approximate, and possibly extend, the Standard Model of Particle Physics.