Spherical Mirror Symmetry

Project description

Mirror symmetry, which originated in theoretical physics, is an exciting area of modern algebraic geometry toady. It has many applications within mathematics (e.g., classification of Fano varieties) and beyond (e.g., in theoretical physics). Mirror symmetry predicts a fascinating duality for certain geometric shapes (Calabi-Yau manifolds): Calabi-Yau manifolds seem to come in “mirror pairs” allowing mathematical notions and formulas on one side to be translated into corresponding counterparts on its mirror partner. It is expected that this deep duality will yield new insights into the geometry of algebraic varieties and produce new mathematics. However, the question of finding a rigorous explanation of this remarkable duality remains unsolved.

Finding explicit constructions to produce mirror pairs has turned out to be incredibly fruitful allowing to solve questions that previously seemed impossibly difficult. Batyrev and Borisov pioneered explicit constructions for mirror pairs by utilising toric geometry. A toric variety is a geometric shape (algebraic variety) with many symmetries. The symmetries of toric varieties prevent them from being Calabi-Yau varieties, however Calabi-Yau varieties can be embedded into toric varieties. The remarkable idea of Batyrev was to utilise a famous duality in combinatorics (polar duality) to produce to a given pair of toric ambient variety and Calabi-Yau subvariety a “toric mirror pair” with the predicted properties.

The aim of this project is to study extensions of the combinatorial Batyrev-Borisov mirror constructions for toric varieties in the context of a massively larger class of highly symmetric algebraic varieties: spherical varieties. Spherical varieties form a remarkable class of algebraic varieties containing those of toric varieties, flag varieties, and symmetric spaces. This project will allow you to get started from day 1, producing many examples of spherical mirror pairs and you learn the theory as you go. There is also the possibility to change your approach according to your interests and strengths: focus on developing new theories on the topology of spherical varieties; do a mix of theory and explicit computations exploiting the combinatorial nature of spherical varieties; or you can decide that your thesis contains a large computational component.