## Math 8680, Topics in Combinatorics: Cluster Algebras and their Variations (Fall 2018)

**Lectures:**MWF 11:15-12:05 in Vincent Hall 311.

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

**Office Hours**(in Vincent Hall 251)

Monday 10:00-11:00

Wednesday 12:20 - 1:00

Friday 2:30 - 3:20

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 providing an introduction to cluster algebras as well as how these combinatorial structures connect to many different areas of mathematics and physics. Additionally, we will discuss extensions of the original definitions, including quantum and noncommutative cluster algebras, generalized cluster algebras, and higher cluster categories. Along the way, we will introduce and use techniques from tropical geometry, Teichmuller theory, and the theory of Grassmannians as we focus on certain classes of cluster algebras, such as those from surfaces or positroids. This course will also highlight recent developments such as maximal green sequences, greedy bases, and scattering diagrams.While there is no required textbook, I recommend "Cluster Algebras and Poisson Geometry'', which we will draw from throughout the course. These will be supplemented with survey articles, including excerpts of the forthcoming book by Fomin, Williams, and Zelevinsky, and a variety of research articles.

**Prerequisites:**No prior knowledge of cluster algebras 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).

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

On reserve in the math library

**Recommended Survey Articles:**

**Recommended Research Articles:**

**Suggestions for Twenty-five Minute Presentations:**

**Helpful Resources:**

Sage Math via CoCalc

Keller's Quiver Applet in Java

My previous graduate topics courses on cluster algebras

MSRI Graduate Summer School on Cluster Algebras (from Summer 2011)

**Grading:**There will be no exams, but registered students are expected to attend, and should hand in the homework assignments. There will be two to three assignments over the course of 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. Depending on interest, there may be an in-class presentation on a research paper instead of one of the homework assignments.

## Tentative Lecture Schedule

(See Mobius Functions Revealed by our very own Doug Arnold and Jon Rogness)

## Homework assignments

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

HW 1 | Monday October 1 |

HW 2 | Monday November 19 |

HW 3 | 25 Minute Presentations Dec 3rd, 5th, 7th, 10th, and 12th |