Given a small (topological or discrete) category C, the functors from C to an additive category A may be thought of as A-valued representations or modules over C. The category of such functors (all or of a certain type) has often interesting homological algebra that reflects the combinatorial or topological structure of C. In this series of lectures, I will review several classical examples, starting with (co)homology theory of small categories (Quillen, Baues-Wirsching), cyclic and Hochschild homology (Connes), homology of crossed simplicial groups (Loday-Fiedorowicz). I will then discuss more recent topological examples, including higher order Hochschild homology (Pirashvili) and factorization homology of topological manifolds (Lurie, Francis). Finally, I will introduce a new example: representation homology of spaces that is a natural topological generalization of representation homology of algebras and groups. I will describe a small DG algebra model for computing representation homology for finite CW complexes and consider a few explicit examples, including surfaces, aspherical manifolds (knot complements) and lens spaces. Time permitting, I will also discuss a version of representation homology in the smooth category and the problem of quantization.
In these lectures we will identify the commutative algebra of cohomology of the partial glag varieties studied in algebraic geometry and topology, and the Bether algebra, the commutative algebra of linear operators associated to a particular lattice models in statistical physics. This describes a new instance of the so called Bethe/gauge correspondence discovered by Nekrasov and Shatashvilli and studied mathematically by Maulik, Okounkov.
We will define thoroughly both sides and explicitly identify them. We will emphasize the geometric interpretation of the key ingredients of the statistical Physics model: the R matrix, Bethe ansatz, Bethe states, monodromy matrix, etc. If time allows we will discuss the applications of this correspondence to the both subjects.
Moonshine is the surprising connection between finite sporadic groups and modular forms. The most famous example is monstrous moonshine, which connects the Klein j-invariant to the largest sporadic group, the Monster. This connection was explained using Vertex Operator Algebras. More recently, a similar connection was observed between the Mathieu group M24 and weak Jacobi forms. The weak Jacobi form in question is in fact the elliptic genus of K3, the unique Calabi-Yau manifold in two complex dimensions. So far this 'Mathieu moonshine' has not been explained. I will give an introduction to some of the necessary background material, and then give an overview over the current status of moonshine.
We will discuss a number of problems in representation theory, knot invariants, combinatorics and conformal field theory, whose underlying structure is a remarkable algebra called U_q(gl_1^^). Far from being trivial, this algebra will allow us to better understand rational Cherednik and DAHA representations, calculate refined Chern-Simons invariants of torus knots, formulate generalizations of the Pieri rules and the shuffle conjecture on symmetric functions, and obtain new vertex operator formulas for symmetric Dunkl operators.