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I discuss some analogies between the Haag-Kastler approach to QFT and quantum statistical mechanics of lattice systems. As an illustrative example, I consider the interpretation of the Hall conductance of gapped 2d lattice systems as an obstruction to gauging a global symmetry of a gapped state. I argue that in order to define a proper analog of the net of algebras of observables one needs to study a category of subsets of the lattice equipped with a natural Grothendieck topology.

Ultra-cold Fermi gases exhibit a rich array of quantum mechanical properties, including the transition from a fermionic superfluid Bardeen-Cooper-Schrieffer (BCS) state to a bosonic superfluid Bose-Einstein condensate (BEC), which can be precisely probed experimentally. However, accurately describing these properties poses significant theoretical challenges due to strong pairing correlations and non-perturbative interactions. In this talk, I will discuss our recent development—a Pfaffian-Jastrow neural-network quantum state equipped with a message-passing architecture, designed to efficiently capture pairing and backflow correlations. We benchmark our approach against existing Slater-Jastrow frameworks and state-of-the-art diffusion Monte Carlo methods. Analysis of pair distribution functions and pairing gaps reveals the emergence of strong pairing correlations around unitarity. We demonstrate that transfer learning stabilizes the training process in the presence of strong, short-ranged interactions, allowing for an effective exploration of the BCS-BEC crossover region. Our findings highlight the potential of neural-network quantum states as a promising strategy for investigating ultra-cold Fermi gases.

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Extended BMS symmetry is believed to be a fundamental symmetry of any classical gravitational scattering process, as well as of the quantum gravity S-matrix. We explore this property using the BRST formulation of BMS symmetry, which allows to construct a non trivial solution of the Wess-Zumino consistency condition at the null boundaries of spacetime. We interpret this solution as an anomaly for asymptotic BRST Ward identities for superrotations. By relating them to the leading and subleading soft graviton theorem, we recover the well known results that the leading soft theorem is exact at all loop in perturbation theory without ever referring to Feynman diagrams, while the anomaly for superrotations is at the origin of the 1-loop correction to the subleading soft factor. This construction provides a rigorous, fully quantum origin for the invariance of the S-matrix under asymptotic symmetries.

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Recent horizon-scale images of Messier 87* and Sagittarius A* have been used to demonstrate that their spacetimes are well-described by the Kerr metric. The latter is a solution to the vacuum Einstein equations of general relativity, and is used to describe spinning black holes. While of fundamental importance, it has undesirable features such as a spacetime singularity or a Cauchy horizon. To find phenomenological resolutions of such features, using observations, studies of astrophysical processes in non-Kerr spacetimes have recently gained prominence. We will begin by briefly reviewing the current status of observational constraints on such alternatives. We will then demonstrate how future observations of the "photon ring" can grant access to new observables that will refine our physical understanding of strong-gravity. We will end by sketching how, using state-of-the-art numerical simulations, the energetics of relativistic outflows (jets) is universally described by a simple electromagnetic Penrose process (the Blandford-Znajek mechanism).

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The prototypical example of a symmetric tensor category is Rep(G) for G a compact group. The other main source of symmetric tensor categories are Deligne's interpolation categories (S_t, GL_t, and O_t) which extend a family of group representation categories defined at positive integers to all complex numbers. Harman and Snowden have developed a theory of symmetric tensor categories coming from Oligomorphic permutation groups together with a well-behaved measure on G-sets. This includes Deligne's S_t which comes from the infinite symmetric group together with a measure where the usual permutation G-set has volume t. The simplest new example of their theory is the group of order-preserving bijections of the real line with the measure given by Euler characteristic. In joint work with Harman and Snowden (arXiv:2211.15392) we give a detailed description of this new symmetric tensor category, which has a number of novel properties. We call this category the Delannoy category because the dimensions of Hom spaces are given by Delannoy numbers. In this talk I'll outline our main results, including the classification of simple objects, the tensor product rules, and a combinatorial model for the category using Delannoy paths.

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