Cluster Structures on Algebraic Varieties

At the border of Algebraic Geometry and Representation Theory, cluster algebras and their associated cluster varieties play a more and more important role in mirror symmetry and the construction of toric degenerations of projective varieties. However, their definition by Fomin-Zelevinsky in 2002 and Fock-Goncharov in 2009 still looks bizarre to Algebraic Geometers and makes it difficult to answer even basic algebraic-geometric questions on these structures:

1 It is widely assumed that cluster varieties are generalizations of toric varieties, but it is not known how to produce a cluster structure for a given toric varieties besides the algebraic torus itself. The first objective of the project is to devise such a construction.
2 The construction of a cluster variety can easily lead to non-Noetherian varieties, and there are no general criteria to decide when a cluster variety is Noetherian or of finite type. Another objective of the project is to provide such criteria.
3 Results in the directions of both objectives above will shed new light on the construction of a cluster variety. A third objective of the project is to use this new information to descibe more algebraic-geometric ways to construct cluster varieties.

A starting point for these objectives is given by Gross-Hacking-Keel's description of 2-dimensional cluster varieties as gluings of affine planes by elementary transformations. Elementary transformations also exist in higher dimension, and it is plausible to try to use them to generalize Groos-Hacking-Keel's construction. Also, using this construction to describe all 2-dimensional toric varieties as cluster varieties is an important first step towards the first objective.