## Math 8680, Topics in Combinatorics: Cluster Algebras and Quiver Representations (Fall 2016)

**Lectures:**MWF 11:15-12:05 in Vincent Hall 213.

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

**Office Hours:**MWF 2:30-3:20 in Vincent 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 topics 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 combinatorial and representation theoretic aspects of cluster algebras, thereby providing a concrete introduction to this rapidly-growing field.We begin with an introduction to quiver representations, which provide a class of representations (of finite dimensional associative algebras) that share some similar properties with representations of finite groups but also illuminating contrasts. We will follow a hands on approach with lots of examples when introducing these objects. Secondly, we turn our attention to our other main topic, cluster algebras, defining them algebraically and illustrating how they show up in a variety of combinatorial and geometric contexts. Finally, we will explore how cluster algebra theory can be studied via the language of tilting theory, cluster categories, and Auslander Reiten Translations coming full circle with the quiver representation theoretic approach presented earlier on.

While there is no required textbook, there are three recommended textbooks. We will cover Chapters 1-3 of "Quiver Representations" by Ralf Schiffler (and later chapters as time allows) and the majority of "Lecture Notes on Cluster Algebras" by Robert Marsh. Additionally, the first three chapters of Sergey Fomin, Lauren Williams, and Andrei Zelevinsky's text "Introduction to Cluster Algebras" was recently posted on the arXiv and this text will be used as a third recommended textbook. We will also supplement these three books with lecture notes, as well as survey and research articles that may be found online.

**Parts**of this course will cover material similar to my Spring 2011 Course on Cluster Algebras and Quiver Representations.

**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:**

*Quiver Representations*by Ralf Schiffler (2014, Springer CMS Books in Mathematics).

*Lecture Notes on Cluster Algebras*by Robert Marsh (2013, EMS Zurich Lectures in Advanced Mathematics).

On reserve in the math library

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

**Recommended Survey Articles:**

**(Just posted on the arxiv in August 2016! Highly Recommended)**

**Helpful Lecture Notes:**

**Relevant Research Articles:**

## Computer Software:

We will occasionally demo Sage Software during this course for cluster algebraic and quiver representation theoretic computations.You may easily access these via the Sage Math Cloud

Cloud worksheets will be listed here later in the course. For now see the (outdated) Compendium on the Cluster Algebra and Quiver Package in SAGE (with Christian Stump) for a tutorial.

Also, check out Keller's Quiver Applet in Java

.

## Grading:

There will be no exams, but registered students are expected to attend, and should hand in the homework assignments. There will be homework once a month with 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.## Tentative Lecture Schedule

Slides from 2012

Slides from 2010

Lecture 23 from 2011

MSRI Lecture Day 3 from 2011

Matrix Formuale and Skein Relations (with Lauren Williams)

Total Positivity by S. Fomin

## Homework assignments

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

Homework 1 | Friday 9/30 |

Homework 2 | Monday 11/21 (postponed) |

Homework 3 | Monday 12/12 |