Mirror Symmetry for Fano varieties

Project description

Fano varieties are one of the most fundamental spaces studied in algebraic geometry. They play an essential role in the Minimal Model Program, and their classification has been an open problem for centuries. Recent advances coming from birational geometry (for which Caucher Birkar - a former Nottingham PhD student - won the Fields Medal in 2018) and from Mirror Symmetry (an area of modern mathematics with its roots in theoretical physics) suggest that a general approach to classification may finally be possible.

Mirror Symmetry predicts a remarkable phenomenon: the numbers given by Gromov-Witten theory for a Fano variety - that is, the counts of the number of different paths a string can trace as it moves through space - can be reproduced by seemingly unrelated mathematical objects called Laurent polynomials. By understanding how to interpret the mathematics of Laurent polynomials in terms of geometry we are beginning to learn to see the structure of Fano varieties in new ways. In turn this is revealing previously unexplored commonalities between different areas of mathematics. The main aim of this project is develop the mathematics behind how Laurent polynomials can be used to classify three-dimensional terminal Fano varieties

This is an amazingly exciting area of modern algebraic geometry. The ideas are new and the mathematics involved is diverse and beautiful. One of the appealing aspects of this project is that you can get started straight away, producing results quickly and learning the material as you go. There is also the scope to adjust your approach depending on how your interests develop: this can be a very technical, abstract piece of work in deformation theory; it can focus on geometry or on combinatorics; and it can have large computational aspects.