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In this talk we will construct a family complete bases for the non perturbative Hilbert space of quantum gravity, and use this to address two puzzles regarding the Hilbert space of quantum gravity. Firstly, Gibbons and Hawking proposed that the Euclidean gravity path integral with periodic boundary conditions in time computes the thermal partition sum of gravity. As a corollary, the derivative of the associated free energy with respect to the Euclidean time period results in the celebrated black hole entropy formula  S=A/4G. Why is this interpretation is correct? That is, why does this path integral compute a trace over the gravity Hilbert space? We show that the quantity computed by the Gibbons-Hawking path integral is equal to an  a priori different object -- an explicit thermal trace over the Hilbert space spanned by states produced by the Euclidean gravity path integral. Secondly, Holography suggests that the Hilbert space of quantum gravity with two asymptotic boundaries should factorise into two copies of the single-boundary gravity Hilbert space.  However, the existence of spacetimes with Einstein-Rosen bridges connecting the two boundaries seems to contradict this. We will show that the non-perturbative, two-boundary  Hilbert space factorises nonetheless. One implication of our results is that universes containing a horizon can sometimes be understood as superpositions of horizonless geometries entangled with a closed universe.

In this talk, we will review the asymptotic structure of three-dimensional Carroll gravity with a negative cosmological constant. We formulate a consistent set of boundary conditions preserved by an infinite-dimensional extension of the AdS3 Carroll algebra, which turns out to be isomorphic to a precise generalised BMS3 algebra. We also introduce a new solution endowed with a Carroll extremal surface that fulfils this set of asymptotic conditions. By taking advantage of the Chern-Simons formulation of the theory, Carroll's thermal properties, obtained from regularity conditions, and the entropy of the configuration are also addressed. Some comments about the 3D Galilean gravity case are addressed at the end of the talk. 

Holography provides a powerful framework for probing the intricate structure of black hole interiors. Remarkably, the seminal BKL analysis from the early '70s has recently found a natural realization within AdS/CFT, offering a way to obtain and study Kasner-like dynamics inside AdS black holes. In this talk, I'll discuss how these behaviors manifest in the dual CFT as exotic RG flows, and how CFT observables can sometimes capture — or fail to capture — the physics near the singularity. I will highlight various probes, including the thermal c-function, two-sided correlators, and holographic complexity. Time permitting, I will also touch on the role of higher-derivative corrections and their impact on the structure of deep interiors.