Name: Calin Chindris
Address: School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A.
Office: Vincent Hall 331
Phone: (612)625-0172
Email: chindris@math.umn.edu
Webpage: http://www.math.umn.edu/~chindris
Meeting times: MWF 2:30-3:20pm in VinH 20
Office hours: MWF 3:20--4:20pm

Course description: A quiver is just a directed graph while a quiver representation assigns a vector space to each vertex and a linear map to each arrow. Quivers and their representations occur naturally in representation theory of algebras but they also have interesting connections with other areas such as algebraic combinatorics (Littlewood-Richardson coefficients, cluster combinatorics), algebraic geometry (quotient varieties) and physics (string theory).

The first part of the course covers classical aspects of the theory including reflection functors and Gabriel's classification of quivers of finite representation type. In the second part of the course we will introduce various ideas and techniques from quiver invariant theory. We will study (semi-)invariants of quivers and (rather special) rational convex polyhedral cones associated with quivers. Finally, we will use these techniques from quiver theory to solve conjectures from algebraic combinatorics/representation theory such as the Saturation Conjecture and Okounkov's Log-Concavity Conjecture for Littlewood-Richardson coefficients.

Prerequisites: Standard course in algebra, some basic knowledge of algebraic geometry (especially affine algebraic varieties). I will try to make the course essentially self-contained.

Grading: Based on several problem sets

Required Textbook: None. However, the following online lecture notes will be useful for the course:

• William Crawley-Boevey: Lectures on representations of quivers
• William Crawley-Boevey: Geometry of representations of algebras
• Harm Derksen: Quiver representations

• Assem, Ibrahim and Simson, Daniel, and Skowronski, Andrzej Elements of the representation theory of associative algebras. Vol. 1. Techniques of representation theory. London Mathematical Society Student Texts, 65. Cambridge University Press, Cambridge, 2006. x+458 pp.
• Auslander, Maurice and Reiten, Idun, and Smalo, Sverre O. Representation theory of Artin algebras. Cambridge Studies in Advanced Mathematics, 36. Cambridge University Press, Cambridge, 1995. xiv+423 pp.
• Ringel, Claus Michael Tame algebras and integral quadratic forms. Lecture Notes in Mathematics, 1099. Springer-Verlag, Berlin, 1984. xiii+376 pp.

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