A classical result usually attributed to Cayley and Bacharach asserts that if two plane curves of degrees c and d meet in cd points, then any curve of degree (c + d - 3) passing through all but one of these points must also pass through the remaining one. In the late 1970s, Griffiths and Harris showed that this is a special case of a general result about zero-loci of sections of a vector bundle. Inspired by a recent paper of Mu-Lin Li, I will describe a generalization allowing for excess vanishing. Multiplier ideals enter the picture in a natural way. Time permitting, I will also explain how a result due to Tan and Viehweg leads to statements of Cayley-Bacharach type for determinantal loci. This is joint work with Lawrence Ein.[-]

There is a very general story, due to Joyce and Kontsevich-Soibelman, which associates to a CY3 (three-dimensional Calabi-Yau) triangulated category equipped with a stability condition some rational numbers called Donaldson-Thomas (DT) invariants. The point I want to emphasise is that the wall-crossing formula, which describes how these numbers change as the stability condition is varied, takes the form of an iso-Stokes condition for a family of connections on the punctured disc, where the structure group is the infinite-dimensional group of symplectic automorphisms of an algebraic torus. I will not assume any knowledge of stability conditions, DT invariants etc.[-]

Homological mirror symmetry asserts that the connection, discovered by physicists, between a count of rational curves in a Calabi-Yau manifold and period integrals of its mirror should follow from an equivalence between the derived Fukaya category of the first manifold and the derived category of coherent sheaves on the second one. Physicists' observation can be reformulated as, or rather upgraded to, a statement about an isomorphism of certain Hodge-like data attached to both manifolds, and a natural first step towards proving the above assertion would be to try to attach similar Hodge-like data to abstract derived categories. The aim of the talk is to report on some recent progress in this direction and illustrate the approach in the context of what physicists call Landau-Ginzburg B-models.[-]

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).[-]