MAT 536: Algebra III

Fall 2011

Department of Mathematics
SUNY at Stony Brook

Cluster algebras and related topics

Course Description:

This is a graduate level course on cluster algebras. Cluster algebras are a class of combinatorially defined rings that provide a unifying structure for phenomena in a variety of algebraic and geometric contexts. A partial list of related areas includes quiver representations, statistical physics, and Teichmuller theory. This course will focus on the algebraic, geometric and combinatorial aspects of cluster algebras, thereby providing a concrete introduction to this rapidly-growing field. Besides providing background on the fundamentals of cluster theory, we will discuss complementary topics such as total positivity, quiver representations, the polyhedral geometry of cluster complexes, cluster algebras from surfaces, and connections to statistical physics.

Prerequisites: No prior knowledge of cluster algebras or representation theory will be assumed; although familiarity with groups, rings, and modules will be helpful.

Recommended (but not required) Texts:
Cluster Algebras and Poisson Geometry by Michael Gekhtman, Michael Shapiro, and Alek Vainshtein (2010, AMS Monograph). On reserve in the math library

Elements of the Representation Theory of Associative Algebras, Vol. 1, by Ibarahim Assem, Daniel Simson, and Andrzej Skowronski. (2006, Cambridge University Press) On reserve in the math library

Recommended Articles:

Surveys for Cluster Algebras:

• Total Positivity and cluster algebras (by Sergey Fomin, ICM 2010)

• Root systems and generalized associahedra (by Sergey Fomin and Nathan Reading, IAS/Park City 2004)

• Total Positivity: Tests and Parametrizations (by Sergey Fomin and Andrei Zelevinsky 1999)
  
  
  

Relevant Research articles

• Cluster Algebra I: Foundations (by Sergey Fomin and Andrei Zelevinsky 2002)

• Cluster Algebra II: Finite Type Classification (by Sergey Fomin and Andrei Zelevinsky 2003)

• The Laurent Phenomenon (by Sergey Fomin and Andrei Zelevinsky 2002)

• Cluster Algebras and triangulated surfaces (by Sergey Fomin, Michael Shapiro, and Dylan Thurston 2008)

• Positivity for Cluster Algebras from Surfaces (by Gregg Musiker, Ralf Schiffler, and Lauren Williams 2009)

• Laurent expansions in cluster algebras via quiver representations (by Philippe Caldero and Andrei Zelevinsky 2006)

• More articles , links to courses and relevant software are available at the Cluster Algebras Portal.

For Quiver Representations and later in the course:

• Lecture Notes on Quiver Representations (Lectures 1-5 Relevant for us)

• Introduction to representation theory (Section 5 covers Quiver Reps) by Pavel Etingof

• Cluster algebras, quiver representations and triangulated categories (by Bernhard Keller 2008)

• Representations of Quivers by M. Barot

• Cluster ensembles, quantization and the dilogarithm by V.V. Fock, A. B. Goncharov

• The pentagon relation for the quantum dilogarithm and quantized M_{0,5} by A. B. Goncharov

• Liouville-Arnold integrability of the pentagram map on closed polygons V. Ovsienko, R. E. Schwartz, S. Tabachnikov

• Quantum Dilogarithm by L.D. Faddeev, R.M. Kashaev
  More articles to be listed later
  
  SAGE Software for Cluster Algebras

Grading:

There will be no exams, but registered students are expected to present (orally) solutions of assigned problems (90% of each section) during office hours.The list of problems will be expanded during the course of the semester. Any form of collaboration on homework between students is welcomed.