Mirror Symmetry and Classification of cluster varieties

Mirror Symmetry is an important tool for the task of classification of Fano varieties, which are important building blocks of algebraic varieties. The mirror dual to a Fano variety is conjecturally a cluster variety, so understanding and classifying cluster varieties allow us to find more examples of Fano varieties via this duality. The project explores cluster theory from the point of view of blow ups of Toric varieties to develop generalizations of Toric geometry. In particular, establishing a classification of an analogue of reflexive polytopes on tropicalisations of cluster varieties, and as a result a list of Log Calabi-Yau surfaces which arise as blowups of toric varieties.