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The original spinfoam amplitude, Ponzano-Regge, has two properties in seeming contradiction: (1.) It can be written as an integral of a product of Dirac delta functions imposing that holonomies be exactly flat, and (2.) In its original sum-over-spins form, its leading order large spin asymptotics consist in Regge calculus, modified to include an additional local discrete orientation variable for each tetrahedron, which, when fixed inhomogeneously, leads to critical point equations for the edge lengths which do not necessarily imply flatness, but allow spikes. Of course, this apparent contradiction between flatness and spikes appears only for triangulations with bubbles, for which both of these formulations of the model are divergent and ill-defined anyway, and this may be the resolution of the paradox. However, we explore the possibility of another resolution of this paradox which may also have relevance for the semiclassical regime of 4D spinfoams, in which a similar sum over local orientations appears.

The recent detection of a low frequency gravitational wave background by pulsar timing arrays provides solid evidence that the observable universe contains a population of in-spiralling supermassive black hole binaries. Such binaries likely form as a result of collisions between galaxies and can offer clues as to how black holes and galaxies grow over time. Moreover, since galactic centers are often densely populated by gas and stars, these systems are much more likely to be actively accreting and radiating compared to stellar mass-sized black hole binaries. This makes them promising candidates for multi-messenger detection (combining electromagnetic and gravitational wave information), particularly when LISA comes online or as pulsar timing arrays improve their sensitivity. In order to facilitate this goal, it is important that, in addition to predictions for the possible gravitational wave signals from numerical relativity, we also develop predictions for electromagnetic signals from simulations of black hole binary accretion. In this talk I will present some of our recent results from 3D general relativistic magnetohydrodynamic simulations that utilize a strong-field approximation to the in-spiralling spacetime metric. Specifically, I will showcase several possible distinctive electromagnetic signals from black hole binaries, including quasi-periodic emission, jet precession, and two processes analogous to flaring activity frequently observed in the outer layer of the Sun.

Numerical methods are powerful tools for advancing our understanding of quantum gravity. In this talk, I will introduce two complementary numerical approaches. The first focuses on solving nonlinear partial differential equations that arise in Loop Quantum Gravity (LQG)-inspired effective models. This framework enables us to investigate the formation and evolution of shock waves in spherically symmetric gravitational collapse. The second approach involves the use of complex critical points, Lefschetz thimble techniques, and the Metropolis Monte Carlo algorithm to study the Lorentzian path integral in Spinfoam models and Quantum Regge Calculus. These methods offer new insights into quantum cosmology and black-to-white hole transitions.

Einstein's theory admits interesting limits with different causal structures obtained by letting the speed of light go to infinity (Galilean or "non-relativistic" limit) or to zero (Carrollian, sometimes called "ultrarelativistic", limit). The Carroll limit turns out to be relevant near spacelike (cosmological) singularities, and particularly so when p-form gauge fields are coupled to gravity as in the context of extended supergravities. The resulting differential equations possess a remarkable interpretation in terms of infinite-dimensional Kac-Moody algebras. The talk will review various aspects of Carroll invariant theories, of the Belinski-Khalatnikov-Lifshitz analysis of spacelike singularities and of the related symmetries.