Geometry of Cluster Algebras

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Cluster varieties are certain geometric spaces of parameters; certain functions of those parameters form so-called cluster algebras. These have found applications throughout mathematics and mathematical physics. This project proposes to study the geometry of cluster algebras in two senses. The first is to apply tools from algebraic geometry to the study of cluster varieties. This will hopefully enable us to understand the structure of integrals computed on cluster varieties, which are common in applications to particle physics. The second sense is to describe cluster varieties using the geometry of reflection groups. These are the symmetries formed by reflections over collections of mirrors. Such connections are already well known for finite reflection groups, but the typical cluster variety should be related to the properties of infinite reflection groups, and these are still very poorly understood.More precisely, the first project is to compute the mixed Hodge structure on the cohomology of cluster varieties. This is joint work with Thomas Lam. In previous work, the PI and his collaborator have discovered a "curious Lefschetz symmetry" in these mixed Hodge structures. They intend to build a spectral sequence to help compute these mixed Hodge structures, and they intend to study connections between these computations, rational Catalan theory and character varieties. The second project has as its ultimate goal describing cluster fans and cluster complexes in terms of infinite Coxeter groups. Nathan Reading and the PI previously did this for cluster varieties of finite type, which corresponded to finite Coxeter groups, and partially succeeded in doing so in general. The obstacle to further progress was that they needed a lattice into which they could embed an infinite Coxeter group; in the finite case, the Coxeter group itself is such a lattice. The current project, joint with Nathan Reading and Hugh Thomas, proposes to solve this problem by using ideas from the representation theory of quivers. They also intend to find combinatorial models of these lattices, and hope to eventually find applications to cluster varieties.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.