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.[-]

This talk has a dual aim: to provide a mathematical overview of Stone duality theory, and to invite collaboration on its Lean formalization.
Stone duality is an algebraic way of looking at profinite topologies. A profinite set is a compact, T2, totally disconnected space, or, equivalently, a topological space which can be obtained as the projective limit of finite discrete spaces. Stone proved in the 1930s that the category of profinite sets is dually equivalent to that of Boolean algebras, and, more generally, that the category of spectral spaces is dually equivalent to that of bounded distributive lattices. I will explain how spectral spaces can be advantageously understood as profinite posets, also known as Priestley spaces. I will also point to more modern research that takes Stone duality further, and may touch upon some mathematical contexts where it pops up, notably topos theory and condensed mathematics.
Elements of Stone duality theory have been formalized in Lean over the past few years, and I will report on some of the most recent progress. I will also propose a number of concrete formalization goals at various levels of estimated difficulty, to provide the audience with some potential project ideas for this week.[-]

This talk begins with examples of rigid and non-rigid geometric structures, followed by an in-depth discussion of the Fundamental Theorem of Riemannian Geometry, on existence and uniqueness of a torsion-free connection compatible with a Riemannian metric. This result is interpreted as giving a framing on the orthonormal frame bundle uniquely determined by the metric. It is seen to be a consequence of the vanishing of the first prolongation of the orthogonal Lie algebra.[-]

For a class of fundamental groups of closed oriented hyperbolic 3-manifolds acting on their Gromov boundary, we compute the K-theory of the associated crossed products in terms of the first homology group of the manifold. Using classification results of purely infinite C*-algebras, we conclude that there exist infinitely many pairwise nonisomorphic torsion-free hyperbolic groups acting on their boundary, for which all crossed products are isomorphic. As in all these cases the boundary is homeomorphic to the 2-sphere, we find infinitely many pairwise non-conjugate Cartan subalgebras with spectrum $S^2$ in such crossed products. This is joint work with Johannes Ebert and Julian Kranz.[-]

I will talk about certain higher algebraic structure, governed by Kontsevich's Lie graph complex, that can be associated to an oriented fibration with Poincaré duality fiber. To obtain it, we prove a parametrized version of the classical result, due to Kadeishvili and Stasheff, that the cohomology of a Poincaré duality space carries a cyclic C-infinity algebra structure. I will also discuss how this higher structure can be used to relate seemingly disparate problems in commutative algebra and differential topology: on one hand, the problem of putting multiplicative structures on minimal free resolutions and, on the other hand, the question of whether a given Poincaré duality fibration can be promoted to a smooth manifold bundle.[-]

Functors on the category gr of finitely generated free groups and group homomorphisms appear naturally in different contexts of topology. For example, Hochschild-Pirashvili homology for a wedge of circles or Jacobi diagrams in handlebodies give rise to interesting functors on gr. Some of these natural examples satisfy further properties: they are analytic and/or exponential. Pirashvili proves that the category of exponential contravariant functors from gr to the category k-Mod of k-modules is equivalent to the category of cocommutative Hopf algebras over k. Powell proves an equivalence between the category of analytic contravariant functors from gr to k-Mod, and the category of linear functors on the linear PROP associated to the Lie operad when k is a field of characteristic 0. In this talk, after explaining these two equivalences of categories, I will explain how they interact with each other. (This is a joint work with Minkyu Kim).[-]

Cellular A1-homology is a new homology theory for smooth algebraic varieties over a perfect field, which is often entirely computable and is expected to give the correct motivic analogue of Poincaré duality for smooth manifolds in classical topology. I will introduce cellular A1-homology, describe the precise conjectures about cellular A1-homology of smooth projective varieties and discuss how they can be verified for smooth projective rational surfaces. The talk is based on joint work with Fabien Morel.[-]