Lecture 1, Quiver mutations.
Abstract: Quiver mutations are transformations of quivers (i.e., oriented graphs) defined by simple combinatorial rules. They play important roles in a number of mathematical contexts, some of which will be reviewed in the lecture. We will then recall a few general results about quiver mutations which are easy to state but surprisingly hard to prove. (We will not discuss the proofs.) This will be a colloquium-style lecture aimed at a general mathematical audience.
Lecture 2, Tensor diagrams and cluster algebras.
Abstract: The rings of SL(V) invariants of configurations of vectors and linear forms in a finite-dimensional complex vector space V were explicitly described by Hermann Weyl in the 1930s. It turns out that each of these rings carries a natural cluster algebra structure (conjecturally, many of them) whose cluster variables include Weyl's generators. When V is 3-dimensional, these cluster structures can be explicitly described using the combinatorial machinery of tensor diagrams. A key role is played by the web bases introduced by G.Kuperberg. This is joint work with Pasha Pylyavskyy (arXiv:1210.1888).
Abstract: Quiver mutations are transformations of quivers (i.e., oriented graphs) defined by simple combinatorial rules. They play important roles in a number of mathematical contexts, some of which will be reviewed in the lecture. We will then recall a few general results about quiver mutations which are easy to state but surprisingly hard to prove. (We will not discuss the proofs.) This will be a colloquium-style lecture aimed at a general mathematical audience.
Lecture 2, Tensor diagrams and cluster algebras.
Abstract: The rings of SL(V) invariants of configurations of vectors and linear forms in a finite-dimensional complex vector space V were explicitly described by Hermann Weyl in the 1930s. It turns out that each of these rings carries a natural cluster algebra structure (conjecturally, many of them) whose cluster variables include Weyl's generators. When V is 3-dimensional, these cluster structures can be explicitly described using the combinatorial machinery of tensor diagrams. A key role is played by the web bases introduced by G.Kuperberg. This is joint work with Pasha Pylyavskyy (arXiv:1210.1888).