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.[-]

Skein algebras are certain diagrammatically defined algebras spanned by tangles drawn on the cylinder of a surface, with multiplication given by stacking diagrams. Quantum cluster algebras are certain systems of mutually birational quantum tori whose defining relations are encoded in a quiver drawn on the surface. The category of quantum character sheaves is a $q$-deformation of the category of ad-equivariant $D$-modules on the group $G$, expressed through an algebra $D_q (G)$ of “q-difference” operators on $G$.
In this I talk I will explain that these are in fact three sides of the same coin – namely they each arise as different flavors of factorization homology, and hence fit in the framework of four-dimensional topological field theory.[-]

Cluster algebras without coefficients admit (additive) categorifications given by certain 2-Calabi-Yau triangulated categories. Relative Calabi-Yau structures in the sense of Toën and Brav-Dyckerhoff appear when one tries to extend the theory to cluster algebras with coefficients. In the context of cluster algebras associated with marked surfaces, Merlin Christ has constructed the corresponding categorifications. We will report on a slightly different approach starting from an ice quiver with potential. The talk is based on work by Yilin Wu in his thesis and more recent joint work with him.[-]

I will give an introduction to the amplituhedron, a geometric object generalizing the positive Grassmannian, which was introduced by Arkani-Hamed and Trnka in the context of scattering amplitudes in N=4 super Yang Mills theory. I will focus in particular on its connections to cluster algebras, including the cluster adjacency conjecture. (Based on joint works with multiple coauthors, especially Evan-Zohar, Lakrec, Parisi, Sherman-Bennett, and Tessler.)[-]