Following Grothendieck's vision that a motive of an algebraic variety should capture many of its cohomological invariants, Voevodsky introduced a triangulated category of motives which partially realises this idea. After describing some of the properties of this category, I explain how to define the motive of certain algebraic stacks. I will then focus on defining and studying the motive of the moduli stack of vector bundles on a smooth projective curve and show that this motive can be described in terms of the motive of this curve and its symmetric powers. If there is time, I will give a conjectural formula for this motive, and explain how this follows from a conjecture on the intersection theory of certain Quot schemes. This is joint work with Simon Pepin Lehalleur.[-]

Given a nilpotent Lie algebra over a characteristic zero field, one can construct a group in a universal way via the Baker-Campbell-Hausdorff formula. This integration procedure admits generalizations to dg Lie or L∞-algebras, giving in general ∞-groupoid of deformations that it encodes, as by the Lurie-Pridham correspondence, infinitesimal deformation problems are equivalent to dg Lie algebras. The recent work of Brantner-Mathew establishes a correspondence between infinitesimal deformation problems and partition Lie algebras over a positive characteristic field. In this talk, I will explain how to construct an analogue of the integration functor for certain point-set models of (spectral) partition Lie algebras, and how this integration functor can recover the associated deformation problem under some assumptions. Furthermore, I will discuss some applications of these constructions to unstable p-adic homotopy theory.[-]

I will discuss applications of geometric representation theory to topological and quantum invariants of character stacks. In particular, I will explain how generalized Springer correspondence for class $D$-modules and Koszul duality for Hecke categories encode surprising structure underlying the homology of character stacks of surfaces (joint work with David Ben-Zvi and David Nadler). I will then report on some work in progress with David Jordan and Pavel Safronov concerning a q-analogue of these ideas. The applications include an approach towards Witten's conjecture on the fi dimensionality of skein modules, and methods for computing these dimensions in certain cases.[-]