Given a cyclic subgroup G of GL(2,C) acting on C^2, it was first noticed by Wunram in the 80s that there is a correspondence between certain special representations of G and the exceptional curves appearing in the minimal resolution Y of the surface singularity C^2/G. In modern terms, this was reformulated by Ishii and Ueda as the existence of a fully faithful functor from the derived category of sheaves of Y to the G-equivariant derived category of C^2. In this talk, I will describe a mirror symmetric interpretation of this which exhibits the fully faithful inclusion in algebraic geometry as a sequence of positive Lefschetz stabilizations in symplectic geometry.

Skein lasanga modules are a smooth 4-manifold invariant that was introduced by Morrison, Walker and Wedrich using Khovanov homology. This invariant was recently used by Ren and Willis to give the first analysis free proof of the existence of exotic 4-manifolds. However, even for simple handlebodies it remains difficult to compute. A generalisation was introduced by Chen using Knot Floer homology, which in principle should be easier to compute due to cabling formulas for knot Floer homology. I will give a general introduction to lasagna modules assuming no knowledge of Khovanov or knot Floer homology, and then explain some methods, from upcoming work, for computing Floer Lasagna modules.

Gauge/gravity duality is more than AdS/CFT.  In this talk I will discuss how the holographic dictionary generalises to non-conformal settings, focusing on maximally supersymmetric Yang-Mills theories in diverse dimensions and their Dp-brane supergravity duals. Scaling covariance replaces conformal invariance as the unifying principle on both sides of the duality. On the gravity side, I will show how to systematically organise effective actions and Witten diagram rules for arbitrary correlators of scalar and spin-1 Kaluza-Klein modes. On the field theory side, scale covariance fixes the kinematic structure of 2- and 3-point functions at strong coupling, with the latter admitting closed-form expressions in terms of Appell functions. I will illustrate these results with explicit examples, focussing on 3d MSYM.

Theta operators are weight-shifting differential operators on  automorphic forms. They play an important role in studying congruences between Hecke eigenforms and their p-adic variation. For instance, the classical theta operator, which acts on q-expansions of modular forms as q·(d/dq), is used crucially in Edixhoven’s proof of the weight part of Serre’s conjecture, Katz’s construction of p-adic L-functions over CM fields, and Coleman’s classicality theorem.

Recent years have witnessed extensive works on understanding theta operators over general Shimura varieties, from both geometric and representation-theoretic perspectives. In this talk, I will hint at some aspects of this fascinating area of research. If time permits, I will discuss my ongoing work on overconvergent theta operators over Siegel Shimura varieties.