Abstract: In this talk I will define spin structure and spin manifolds and give some examples. I will also quickly review some notions of Riemannian geometry like connections, curvature; in particular, the Levi-Civita connection. Then I will define the spin connection and using that I will introduce the Dirac operator.

Abstract: We will outline a certain program for Nakajima quiver varieties, in the cyclic quiver example. The picture includes two algebras that act on the K-theory of these varieties: one is the original picture by Nakajima, rephrased in terms of shuffle algebras, and the other one is the Maulik-Okounkov quantum toroidal algebra. The connection between the two is provided by the action of certain operators in the so-called "stable basis", and we will present formulas for this action. These formulas can be perceived as a generalization of Lascoux-Leclerc-Thibon ribbon tableau Pieri rules.