**Math 5020 Topics in Algebra:**

**Cluster Algebras**(Fall 2018)

**Instructor:**Ralf Schiffler

**Office:**Mont 336

**Email:** schiffler at math dot uconn dot edu

**Lectures: ** Mont 113, TT 12:30 – 1:45

**Description:** Cluster algebras are commutative algebras with a special combinatorial structure which are related to many different areas including Combinatorics, Representation Theory, Hyperbolic Geometry, Number Theory and Knot Theory. The subject is relatively young. The first paper on cluster algebras is published in 2002.

Topics covered in the course include the following. Definition and examples, Laurent phenomenon and positivity, classifications, coefficients, relation to representation theory, interpretation in terms of triangulated surfaces, snake graphs and perfect matchings, continued fractions.

** Homework:** There will be no graded assignments, but students are encouraged to work on exercises given in class. These exercises are crucial to the understanding of the material.

** Evaluation: ** Based on participation.

** Some references and surveys: **

Introduction to Cluster Algebras (Chapters 1-3 Preliminary Version) (by Sergey Fomin, Lauren Williams and Andrei Zelevinsky)

Cluster algebras, quiver representations and triangulated categories (by Bernhard Keller)

Cluster algebras from surfaces (by Ralf Schiffler) in CRM Short Courses, *Homological Methods, Representation Theory, and Cluster Algebras *(the same book has short courses by Pierre-Guy Plamondon on cluster characters and by Ibrahim Assem on cluster-tilted algebras)

Cluster algebras and cluster categories (by Ralf Schiffler)

Lecture notes on cluster algebras (by Robert Marsh), Zurich Lecture Notes in Advanced Mathematics, 2013.

Gekhtman, Shapiro, Vainstain: Cluster algebras and Poisson geometry, AMS Mathematical Surveys and Monographs, Volume 167, 2010.

**Tools: **

Quiver mutation app (by Bernhard Keller)

**Videos:**

Master class at Arhus University, 23 lectures (by Sergey Fomin and Philippe DiFrancesco)

Cluster Algebras and Cluster Combinatorics, 4 Short courses at MSRI, 20 lectures (by Lauren Williams, Nathan Reading, Gregg Musiker and Ralf Schiffler)

Cluster algebras and categorification, Lectures one, two, three at CIRM (by Claire Amiot)