Mirror Symmetry for Cluster Varieties

The central objects of study in this proposal (called cluster varieties) strike a nice balance of richness and simplicity. They are a class of spaces defined over the complex numbers that come with a natural notion of volume. Moreover, they are made up of rather simple building blocks-- they are built out of tori glued together in a way that ensures the notion of volume in the cluster variety agrees with the notion of volume in the tori that make it up. Furthermore, cluster varieties come in pairs. Every torus has a dual torus. The same holds for cluster varieties, where duals are in fact built out of dual tori. We can learn a lot about a cluster variety by answering questions that seem entirely different for the dual cluster variety. This duality is an instance of a far more general phenomenon known as mirror symmetry that has been a vigorous research topic in mathematics and physics for the past 30 years. Cluster varieties and cluster duality carve out some territory within the broad setting of mirror symmetry where we can get our hands dirty with explicit computations and prove theorems that remain out of reach in full generality.

So, at least from a geometric point of view, we can learn a lot by studying cluster varieties. But another amazing aspect of the theory of cluster varieties is how widely they appear in mathematics. In fact, the mirror symmetry connection to cluster varieties is a recent development. Cluster algebras were originally invented by Fomin and Zelevinsky to study canonical bases for quantum groups, and they have close connections to representation theory of algebraic groups, representation theory of quivers, and hyperbolic and Poisson geometry. The cluster varieties we tend to study are interesting from many different perspectives, with each of these perspectives providing insight into the others. Our proposal deals with mirror symmetry for cluster varieties that appear naturally in the setting of representation theory of algebraic groups. In this context, the cluster variety is embedded in a larger space-- the space we are actually trying to study-- in a precise way. We propose to construct and study a dual embedding of the mirror cluster variety. Our principal question is how representation theory of the original space is related to geometry of this dual space. We hope knowledge will flow in both directions in this duality, and that a complete picture including relations between the two sides will be more beautiful than either side standing alone.

Dissemination. The research supported by this grant will impact a diverse group of mathematicians as outlined in the Academic Beneficiaries section. We will give talks at various seminars and conferences during the grant period. Specific travel funds are included for this purpose in the Justification of Resources section. The proposal includes an international group of collaborators with different areas of expertise, which will present the possibility of connections to more areas of mathematics, and help to facilitate the dissemination of our results and ideas to the broader mathematical community. We also propose to organise a three day workshop in London which will be another avenue of dissemination. Our results will be written up and posted on the preprint arxiv and submitted to high quality mathematical journals.

Impact to the local mathematical community. The proposal will have positive impact on the local mathematical community, including the vibrant community of PhD students London. The researcher co-I will be giving lectures for PhD students through the London Taught Course Centre (LTCC) as well as getting involved with supervision of mini-projects and PhD projects of LSGNT students. We have proposed collaborations with four mathematicians -- Lauren Williams and Man-Wai Cheung at Harvard and Alfredo Nájera Chávez and Lara Bossinger at UNAM Oaxaca. Lauren Williams and Lara Bossinger bring our team expertise in algebraic combinatorics, while Man-Wai Cheung and Alfredo Nájera Chávez offer expertise from the algebraic/categorical side of cluster theory. Visits to London by our mathematical collaborators will bring additional impact through seminar talks and interactions with local students and academics.

Public Engagement. The proposal includes a significant and innovative public engagement project which aims to illustrate the mathematical structure of a cluster variety in musical terms in collaboration with a composer. At the conclusion of the grant there will be a concert which will bring the output of this collaboration to a general audience, combined with a public lecture that will describe the connection between the music and the mathematics behind it in layperson's terms.