Masterclass: Cluster Algebras and Representation Theory

University of Copenhagen
November 14th-18th 2022

The goal of this master class is to provide an introduction to cluster algebras through the context of representation theory. Some topics introduced will be cluster categories, Fock and Goncharov cluster varieties, and applications to physics.

 

 

  • Karin Baur (University of Leeds) -- Cluster structures have been established on numerous algebraic varieties. These lectures focus on the Grassmannian variety and explain cluster structures on it. The tools include dimer models on surfaces, associated algebras and module categories, root systems, generalised Postnikov diagrams. 
  • Sira Gratz (Aarhus University) -- We will discuss a class of categories that exhibit infinite type A cluster combinatorics. They will serve as our playground to observe phenomena, previously hidden in the classical finite rank setting, that arise in the infinite rank case, both from a cluster theoretic as well as a representation theoretic perspective. We present different approaches to construct these categories, prove that they are equivalent, and finally discuss ways to exploit combinatorial models to improve our understanding of the structures they carry.
  • Gus Schrader (Northwestern University) -- I will give an introduction to the moduli spaces of local systems on decorated surfaces introduced by Fock and Goncharov in their work on higher Teichmuller theory. These moduli spaces are cluster Poisson varieties, and the cluster structure leads naturally to their quantization, which is closely related to the theory of quantum groups. As we will see, this relation sheds light on problems both in representation theory as well as in the 'quantum geometry' of the moduli spaces of local systems.

 

 

 

 

 

 

 

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The deadline for funding applications is 31.10.2022. 
If you are not applying for funding, please register by 7.11.2022

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