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.[-]

In joint work with Konstanze Rietsch (arXiv:1712.00447), we use the $\mathcal{X}$-cluster structure on the Grassmannian and the combinatorics of plabic graphs to associate a Newton-Okounkov body to each $\mathcal{X}$-cluster. This gives, for each $\mathcal{X}$-cluster, a toric degeneration of the Grassmannian. We also describe the Newton-Okounkov bodies quite explicitly: we show that their facets can be read off from $\mathcal{A}$-cluster expansions of the superpotential. And we give a combinatorial formula for the lattice points of the Newton-Okounkov bodies, which has a surprising interpretation in terms of quantum Schubert calculus.[-]

Exact categories were introduced by Quillen in 1970s as part of his seminal work on algebraic K-theory. Exact categories provide a suitable enlargement of the class of abelian categories (for example, an extension-closed subcategory of an abelian category inherits the structure of an exact category) in which one "can do homological algebra". Recently, motivated also by questions in algebraic K-theory, Barwick introduced the class of exact infinity-categories, relying on the newly-developed theory of infinity-categories developed by Joyal, Lurie and others. This new class of mathematical objects includes not only the exact categories in the sense of Quillen but also the stable inftinty-categories in the sense of Lurie (the latter are to be regarded as refinements of triangulated categories in the sense of Verdier). The purpose of this lecture series is to motivate the theory of exact infinity-categories and sketch some of its applications. Familiarity with the theory of infinity-categories is not expected.[-]

Khovanov and Seidel introduced in the early 2000's an action of the braid group by autoequi-valences on the homotopy category of projective modules over the zig-zag algebra. This categorical action descends to the Burau representation, one of the most famous braid representations, but unlike the classical story, the lifting is faithful. It is interesting to notice that simultaneously, the Burau representation was also extended into a faithful finite-dimensional linear representation by Lawrence, Krammer and Bigelow, proving the linearity of the braid group.
I will review the basic constructions, both at the level of vector representations and at the ca-tegorical level. We will discuss possible extensions of these from classical braids (type A) to larger Artin-Tits groups, spherical or not, and try to relate Khovanov-Seidel's construction to Soergel bimodules and categorified quantum groups. I will also try to emphasize several metric aspects that appear in an elegant way from the categorical setting, with an emphasis on Bridgeland's stability conditions. Along the way, I would like to list several open questions and problems that I care about, hoping that someone in the audience will come up with a good idea.[-]

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.)[-]