## Math 8680, Topics in Combinatorics: Cluster Algebras and Quiver Representations (Spring 2011)

**Lectures:**MW 4:00-5:15 in Vincent Hall 1.

**Instructor:**Gregg Musiker (musiker "at" math.umn.edu)

**Office Hours:**MWF 2:30-3:20 Vicent Hall 251. Also, by appointment, or feel free to knock. I usually keep my door open if I'm in.

## Course Description:

This is a graduate level course in algebraic combinatorics. The topic for this semester is cluster algebras and quiver representations. 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 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, as in Math 8202, 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

For Quiver Representations and later in the course:

*More articles to be listed later*

SAGE Software for Cluster Algebras (with Christian Stump)

Sage-Combinat Server (Experimental)

## Grading:

There will be no exams, but registered students are expected to attend, and should hand in the homework assignments. There will be homework every three weeks or so, tentatively three assignments over the semester. I encourage collaboration on the homework, as long as each person understands the solutions, writes them up in their own words, and indicates on the homework page their collaborators. The first homework will be collected on Wednesday, February 23th (Please note the change of date).## Tentative Lecture Schedule

Assignment | Due date | |
---|---|---|

Homework 1 | Wednesday 2/23 (Please note the change of date) | |

Homework 2 | Wednesday 3/30 (NOTE: 3/28: Some typos have been fixed and notes added in the attached pdf. ) | |

Homework 3 | Monday 5/2 |