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 Grassmannian cluster algebras cluster categories, Fock and Goncharov cluster varieties, quantum groups, and quantised cluster algebras.

Graduate students and other early-career researchers can apply for financial support to partly cover travel to and from Copenhegen and accommodation expenses. The support should be roughly DKK 500 per day, at most DKK 2500 in total.

All of the lectures will be held in Auditorium 5 in the Department of Mathematical Sciences.

Lecturers/Invited speakers

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.

Programme

Monday 14th	Tuesday 15th	Wednesday 16th	Thursday 17th	Friday 18th
Registration (9:00-9:30)	Lecture 5 (Baur) (9:00-10:15)	Lecture 8 (Baur) (9:00-10:15)	Lecture 11 (Schrader)(9:00-10:15)	Lecture 13 (Baur) (9:00-10:15)
Lecture 1 (Baur) (9:30-10:45)	Coffee Break	Coffee Break	Coffee Break	Coffee Break
Lecture 2 (Gratz) (10:45-12:00)	Lecture 6 (Gratz) (10:45-12:00)	Lecture 9 (Gratz) (10:45-12:00)	Lecture 12 (Baur) (10:45-12:00)	Lecture 14 (Gratz) (10:45-12:00)
Lunch	Lunch	Lunch	Lunch	Lunch
Lecture 3 (Schrader) (14:00-15:15)	Lecture 7 (Schrader) (14:00-15:15)	Lecture 10 (Schrader) (14:00-15:15)	Free Afternoon	Lecture 15 (Schrader) (13:00-14:15)
Coffee Break	Coffee Break	Coffee Break		Closing
Lecture 4 (Gratz) (15:45-17:00)	Problem session 1 (15:45-17:00)	Problem session 2 (15:45-17:00)
Reception		Dinner Restaurant Maharaja Østerbro Strandboulevarden 67 2100 København

Venue and transportation

The conference/masterclass will take place at the Department of Mathematical Sciences, University of Copenhagen. See detailed instructions on how to reach Copenhagen and the conference venue.

Tickets and passes for public transportation can be bought at the Copenhagen Airport and every train or metro station. You can find the DSB ticket office on your right-hand side as soon as you come out of the arrival area of the airport. DSB has an agreement with 7-Eleven, so many of their shops double as selling points for public transportation.

A journey planner in English is available.

More information on the "find us" webpage.

Accommodation

We kindly ask the participants to arrange their own accommodation.

We recommend Hotel 9 Små Hjem, which is pleasant and inexpensive and offers rooms with a kitchen. Other inexpensive alternatives are CabInn, which has several locations in Copenhagen: the Hotel City (close to Tivoli), Hotel Scandivania (Frederiksberg, close to the lakes), and Hotel Express (Frederiksberg) are the most convenient locations; the latter two are 2.5-3 km from the math department. Somewhat more expensive – and still recommended – options are Hotel Nora and Ibsen's Hotel.

An additional option is to combine a stay at the CabInn Metro Hotel with a pass for Copenhagen public transportation (efficient and reliable). See information about tickets & prices.