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I give an overview of work with Aasen and Mong on topologically invariant defects in two-dimensional classical lattice models, quantum spin chains and tensor networks. We show how to find defects that satisfy commutation relations guaranteeing the partition function depends only on their topological properties. These relations and their solutions can be extended to allow defect lines to branch and fuse, again with properties depending only on topology. These lattice topological defects have a variety of useful applications. In the Ising model, the fusion of duality defects allows Kramers-Wannier duality to be enacted on the torus and higher genus surfaces easily, implementing modular invariance directly on the lattice. These results can be extended to a very wide class of models, giving generalised dualities previously unknown in the statistical-mechanical literature. A consequence is an explicit definition of twisted boundary conditions that yield the precise shift in momentum quantization and thus the spin of the associated conformal field. Other universal quantities we compute exactly on the lattice are the ratios of g-factors for conformal boundary conditions. 

Black-hole recoils are arguably the strong-gravity phenomena with the most striking astrophysical consequences. In the late inspiral and final coalescence of black-hole binaries, anisotropic emission of gravitational waves causes significant linear momentum loss. The remnant black hole, therefore, recoils in the opposite direction. These final kicks can reach magnitudes up to 5000 km/s (“superkicks”), larger than the escape speed of even the most massive galaxies, thus opening the possibility of black hole ejections. In this talk, I summarize recent advances in modeling black-hole recoils and their astrophysical environments. I will address the relevance of disk-assisted spin alignment, new approaches to model black-hole recoils with waveform approximants, and prospects to directly detect superkicks with gravitational-wave detectors.

Large deviation theory gives a general framework for studying nonequilibrium systems which in many ways parallels equilibrium thermodynamics. In transport, according to the large deviation principle, the distribution of rare fluctuations of the total transfer (of charge, energy, etc.) between two baths take a special form encoded by the large deviation function, which plays the role of a free energy. Its Legendre transform is the scaled cumulant generating function (SCGF). For instance, in mesoscopic physics, full counting statistics for charge transport through quantum impurities are SCGFs. In this talk I propose a formalism giving access to SCGFs for ballistic transport in homogeneous, stationary states. The formalism is conjectured to hold for classical and quantum systems alike, and give exact results in terms of the theory of linear fluctuating hydrodynamics. I will explain how it applies to nonequilibrium steady states of integrable models such as the classical hard rod gas, the quantum Lieb-Liniger model and the XXZ chain, using generalised hydrodynamics. This largely extends earlier results for free fermions (Levitov-Lesovik) and for 1+1-dimensional conformal field theory (Bernard-Doyon). The formalism also applies to transport in nonequilibrium critical systems of arbitrary dimension. It is, in a sense, the ballistic counterpart to macroscopic fluctuation theory.

In the context of the AdS(4)/BCFT(3) correspondence, we study the holographic entanglement entropy for spatial regions having arbitrary shape. An analytic expression for the subleading term with respect to the area law is discussed. When the bulk spacetime is a part of AdS(4),
this formula becomes the Willmore functional with a proper boundary term evaluated on the minimal surface viewed as a submanifold of the three dimensional flat Euclidean space with a boundary. 
Numerical checks of this formula are performed through a code which allows to construct minimal area surfaces anchored to generic curves. For some simple regions like infinite strips and disks, analytic results are obtained and they confirm the general expression for the subleading term. In particular, when the spatial region contains corners adjacent to the boundary, a logarithmic divergence occurs whose coefficient is determined by a so-called corner function which depends on the boundary conditions. An analytic expression for the holographic corner function and its checks are also discussed.

The localization theorem, which has played a central role in representation theory since its discovery in the 1980s, identifies a regular block of Category O for a semisimple Lie algebra with certain D-modules on its flag variety. In this talk we will explain work in progress which produces a similar picture for the Virasoro algebra and more generally for affine W-algebras. Some new purely algebraic input is (i) a version of the Feigin-Fuchs duality between Verma modules for Vir at central charges c and 26 - c, which applies to all smooth representations and other affine W-algebras, and (ii) a linkage principle for representations in category O of a W-algebra. As geometric input, we will explain how to (i) adapt the Beilinson-Drinfeld construction of vertex algebras via factorization spaces to also produce representations and in particular (ii) develop a factorizable version of affine Borel--Weil--Bott.