The geometric Satake equivalence identifies the Satake category of a reductive group $G$ – that is, the category of equivariant perverse sheaves on the affine Grassmannian $G_{rG}$ – with the representation category of its Langlands dual group $G^∨$. While the Satake category is topological in nature, it has a poorly understood algebro-geometric cousin: the category of perverse coherent sheaves on $G_{rG}$. This category is not semi-simple and its monoidal product is not symmetric. We show however that it is rigid and admits renormalized r-matrices similar to those appearing in the theory of quantum loop or KLR algebras. Applying the framework developed by Kang-Kashiwara-Kim-Oh in their proof of the dual canonical basis conjecture, we use these results to show that the coherent Satake category of $GL_n$ is a monoidal cluster categorification in the sense of Hernandez-Leclerc. This clarifies the physical meaning of the coherent Satake category: simple perverse coherent sheaves correspond to Wilson-'t Hooft operators in $\mathcal{N} = 2$ gauge theory, just as simple perverse sheaves correspond to 't Hooft operators in $\mathcal{N} = 4$ gauge theory following the work of Kapustin-Witten. Our results also explain the appearance of identical quivers in the work of Kedem-Di Francesco on $Q$-systems and in the context of BPS quivers. More generally, our construction of renormalized r-matrices works in any chiral $E_1$-category, providing a new way of understanding the ubiquity of cluster algebras in $\mathcal{N} = 2$ field theory: the existence of renormalized r-matrices, hence of iterated cluster mutation, is a formal feature of such theories after passing to their holomorphic-topological twists. This is joint work with Sabin Cautis (arXiv:1801.08111).[-]

The theory of cluster algebras has proved useful in proving theorems about the characters of graded tensor products or Demazure modules, via the $Q$-system. Upon quantization, the algebra associated with this system is shown to be related to a quantum affine algebra. Graded characters are related to a polynomial representation of the quantum cluster variables. This immediately suggests a further deformation to the spherical DAHA, quantum toroidal algebras and elliptic Hall algebras.[-]

The goal of this short course is to explain the concept of “triality”, which is an isomorphism between a large class of of (generalized) tame frieze patterns, certain spaces of linear difference equations, and the moduli space of configurations of points in the projective space. This approach will be used in several directions, in particular:
• to define “good” coordinates on moduli spaces related to cluster algebras and symplectic geometry
• to find simple proofs of some properties of friezes, such as periodicity
• to connect the subject to dynamical systems
• to create new types of friezes
• to count friezes of certain types.
The presentation is based on several joint papers with Sophie Morier-Genoud, Sergei Tabachnikov, and also Charles Conley, and Richard Schwartz. Coxeter friezes and geometry of the projective line. I will start with the classical Coxeter's frieze patterns and connect them to configurations of point in the 1-dimensional projective space P1. As a consequence, a (pre)symplectic structure on the space of Coxeter's friezes will be described. The basic notions of projective geometry, such as the cross-ratio and Schwarzian derivative will be recalled/explained and used.[-]

The goal of this talk is to explore the connections between various frieze patterns and representation theory of associative algebras. We begin with the classical Conway- Coxeter friezes over positive integers and their correspondence with Jacobian algebras of type A, where entries in the frieze count the number of submodules of indecompos- able representations. This can also be reinterpreted in terms of applying the Caldero- Chapoton map, providing a close connection to Fomin-Zelevinsky's cluster algebras. Extending these ideas beyond the classical case, we will also discuss higher dimen- sional friezes, called (tame) SLk friezes, as well as their relation to cluster algebras on coordinate rings of Grassmannians Gr(k,n) and their categorification. Furthermore, SLk friezes are a special type of SLk tilings, integer tilings of the plane satisfying the condition that every k x k square has determinant 1. We will present a characterization of SLk tilings in terms of pairs of bi-infinite sequences in Zk and discuss applications to duality and positivity.[-]