Improving understanding of mirror symmetry for homogeneous spaces

The main objects of interest for this project are homogeneous spaces, which are geometric objects with so many symmetries that the neighbourhood of each point looks like that of any other. These symmetries are encoded by a set of matrices called a Lie group, and studying of this Lie group is the key understanding the geometry of the corresponding space. Mirror symmetry is a concept originally introduced in physics as a duality between two seemingly different physical theories. In mathematical terms, it is a conjectural relation between a given geometric object and its so-called mirror, which has proven extremely useful to solve difficult problems in various areas of mathematics. However, at the moment there is no general mirror construction for arbitrary spaces, and even in the special cases where a mirror has been discovered by physicists, mathematicians have generally not been able yet to prove each of its expected properties. Therefore it is important to develop the theory further and work out more examples in full mathematical detail; this is where homogeneous spaces come in. As they have so many symmetries, encoded by matrices, one may use a more sophisticated version of linear algebra called representation theory to study their mirrors, and thus extend the list of objects for which the mirror symmetry framework is proved to hold.

PhD student’s expected contribution: The goal of the project is to improve understanding of mirror symmetry for homogeneous spaces. It builds on a construction of Rietsch1, which has been further explicited by Marsh-Rietsch2 as well in joint work with my co-workers3. In this project the PhD student will:

1. Provide new mirror constructions using the methods mentioned in the aforementioned articles;
2. Design a computing package based on his/her results to encourage diffusion of these results to a wider mathematics and physics audience;
3. Study the combinatorial structure of the mirrors and investigate how it relates to the general mirror symmetry framework.

For more details, please do not hesitate to contact me.