## Math 8680, Topics in Combinatorics: Cluster Algebras, Tilings, and Physics (Spring 2015)

**Lectures:**MW 4:00-5:30 in Vincent Hall 311.

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

**Office Hours:**TBA. 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, tilings, and physics. 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, Teichmuller theory, and string theory. This course will focus on combinatorial aspects of cluster algebras, as well as their connections to dimer models and gauge theories in physics. Besides providing background on the fundamentals of cluster theory, we will discuss Kastelyn matrices, Pfaffian orientations, Kuo condensation, Dodgson condensation, and other techniques for enumerating domino tilings (i.e. dimers). While there is no required textbook, we will begin with "Lecture Notes on Cluster Algebras" by Robert Marsh, followed by selections from "Lectures on Dimers" by Richard Kenyon and "Dimer Models and Calabi-Yau Algebras" by Nathan Broomhead. These will be supplemented with articles from both mathematics and the physics literature.**Part 1**of this course will closely follow my Spring 2011 Course on Cluster Algebras and Quiver Representations.

**Parts 2 and 3**will include cutting edge connections to dimer models and string theory.

**Prerequisites:**No prior knowledge of cluster algebras or physics will be assumed; although familiarity with groups, rings, and modules, as in Math 8202, will be helpful.

**Recommended (but not required) Texts:**

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

*Dimer Models and Calabi-Yau Algebras*by Nathan Broomhead (2012, Memoir of the AMS).

*Cluster Algebras and Poisson Geometry*by Michael Gekhtman, Michael Shapiro, and Alek Vainshtein (2010, AMS Monograph).

**Recommended Survey Articles:**

**Research Articles (Some Suggestions for Presentations):**

**Older Research Articles relevant for lectures:**

**Other Helpful Resources:**

A Compendium on the Cluster Algebra and Quiver Package in SAGE (with Christian Stump)

Sage-Combinat Server (for Cluster Algebra and other calculations)

Also Available via the Sage Math Cloud

Keller's Quiver Applet in Java

My Spring 2011 Course on Cluster Algebras and Quiver Representations.

MSRI Graduate Summer School on Cluster Algebras (from 2012)

## 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. Depending on interest, there may be student presentations instead of homework.In lieau of the third assignment, an in-class presentation on a research paper is encouraged.

## Tentative Lecture Schedule

### Part 1: Introduction to Cluster Algebras

### Part 2: Towards Dimer Models

### From March 30 - April 8, class will be 4:00-6:00

### We resume our regular 4:00-5:15 Schedule

### Part 3: From Dimer Models back to Cluster Algebras and Combinatorial Formulas

## Homework assignments

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

Homework 1 | Wednesday 2/25 |

Homework 2 | Wednesday 4/8 |