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Masterclass June 2010

Cluster Algebras by Fomin and Di Francesco

2012.11.08 | Christine Dilling

Date Mon 14 Jun Fri 18 Jun
Time 11:30    16:00
Location QGM, Room (TBA)


  • Sergey Fomin, Robert M. Thrall, Department of Mathematics, University of Michigan

  • Phillippe Di Francesco, Institut de Physique Theorique, CEA Saclay, France


Cluster algebras arise in various algebraic and geometric contexts, with combinatorics providing a unifying framework. These lectures will introduce the fundamental concepts and results of the theory of cluster algebras, emphasizing its combinatorial aspects, and its connections with total positivity and Teichmüeller theory. The presentation will be guided by two families of examples: cluster algebras arising in representation theory and in the study of classical algebraic varieties, and cluster algebras associated with bordered oriented surfaces with marked points. No prior knowledge of cluster theory will be assumed.

Poster PDFAdditionally

One of the most exciting applications of cluster algebras in recent years has been to the theory of discrete integrable systems, such as the T-, Y-, and Q-systems arising in the theory of thermodynamic Bethe Ansatz (the work of Kuniba-Nakanishi et al., Di Francesco-Kedem, etc.). Therefore the last lecture on Tuesday-Friday will be held by

Phillippe Di Francesco under the title:

'Discrete integrable systems, paths and positivity'


Abstract: In these lectures, I will introduce discrete integrable recursive systems called Q and T systems arising from the physics of integrable quantum spin chains, and with links to various combinatorial objects such as dimers, alternating sign matrices, etc. These are all part of cluster algebra structures, and possess the positive Laurent property that any solution is expressible as a Laurent polynomial with non-negative integer coefficients in terms of any fixed initial data. To show this, we will use the existence of discrete integrals of motion for the systems to construct combinatorial models of hard particles, heaps and finally paths on target graphs, allowing to express explicitly the solutions in terms of the initial data. We will also present non-commutative generalizations of all these constructions with a particular emphasis on the case of rank two.

Master Class, Public/media, Staff