Fall 2011

Department of Mathematics

SUNY at Stony Brook

**Lectures:** MWF 11:45am-12:40pm in Physics P130.
**Instructor:** Mikhail Movshev(mmovshev "at"
math.sunysb.edu)**Office Hours:** MWF TBA Math Tower
4-109. Also, by appointment, or feel free to knock.

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:

Relevant Research articles

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

For
Quiver Representations and later in the course:

# 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

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.