EPSRC Reference: 
EP/V002546/1 
Title: 
Mirror Symmetry for Cluster Varieties 
Principal Investigator: 
Rietsch, Professor K 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
Kings College London 
Scheme: 
Standard Research 
Starts: 
01 April 2021 
Ends: 
31 March 2024 
Value (£): 
465,294

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


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Summary on Grant Application Form 
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.

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