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Recent progress in AdS/CFT has provided a good understanding of how the bulk spacetime is encoded in the entanglement structure of the boundary CFT. However, little is known about how spacetime emerges directly from the bulk quantum theory. We address this question in AdS3 pure gravity, which we formulate as a topological quantum field theory. We explain how gravitational entropy can be viewed as bulk entanglement entropy of gravitational edge modes.  These edge modes transform under a quantum group symmetry. This suggests an effective description of bulk microstates in terms of collective, anyonic degrees of freedom whose entanglement leads to the emergence of the bulk spacetime.  Time permitting we will discuss a proposal for how our bulk TQFT arises from an ensemble of approximate CFT’s, generalizing the relation between JT gravity and random matrix ensemble.

I will talk about supersymmetric quantum field theories with 8 supercharges in dimensions 3-6. After a brief introduction I will mostly speak about the moduli space of vacua of such theories, and in particular their Higgs branches, which are so called symplectic singularities (or mild generalisations thereof). Powerful theorems from mathematics say that a singular Higgs branch is stratified into a disjoint union of smooth open subsets, so called symplectic leaves. This stratification matches exactly the pattern of partial Higgsings of the theory in question. After introducing the stratification and explaining its physical interpretation, I will show how brane systems and so called magnetic quivers can be used to compute it.

We review the construction of non-invertible duality defects in various dimensions. We explain how they can be preserved along RG flows and how their realization on gapped phases contains their 't Hooft anomalies. We finally give a presentation of the anomalies from the Symmetry TFT. Time permitting I will discuss some possible future applications.

The Post Correspondence Problem (PCP) is a classical problem in computer science that can be stated as: is it decidable whether given two morphisms g and h between two free semigroups $A$ and $B$, there is any nontrivial $x$ in $A$ such that $g(x)=h(x)$? This question can be phrased in terms of equalisers, asked in the context of free groups, and expanded: if the `equaliser' of $g$ and $h$ is defined to be the subgroup consisting of all $x$ where $g(x)=h(x)$, it is natural to wonder not only whether the equaliser is trivial, but what its rank or basis might be. 

While the PCP for semigroups is famously insoluble and acts as a source of undecidability in many areas of computer science, the PCP for free groups is open, as are the related questions about rank, basis, or further generalisations. In this talk I will give an overview of what is known about the PCP in hyperbolic groups, nilpotent groups and beyond (joint work with Alex Levine and Alan Logan).

One of the more common ways to study a residually finite group (or its profinite completion) is via breaking it down into a graph of groups in some way. The descriptions of this theory generally found in the literature are highly algebraic and difficult to digest. I will present alternative, more geometric, definitions and perspectives on these theories based on properties of virtually free groups and their profinite completions.

The celebrated Kirchberg-Phillips classification theorem classifies so-called Kirchberg algebras by K-theory. Many examples of Kirchberg algebras can be constructed via the crossed product construction starting from a group action on a compact space. One might ask: When exactly does the crossed product construction produce a Kirchberg algebra? In joint work with Gardella, Geffen, and Naryshkin, we obtained a dynamical answer to this question for a large class of nonamenable groups which we call "groups with paradoxical towers". Our class includes many non-positively curved groups such as acylindrically hyperbolic groups and lattices in Lie groups. I will try to advertise our notion of paradoxical towers, outline how we use it, and pose some open questions.

In a recent work with Sam Mellick and Amanda Wilkens, we proved that higher rank semisimple Lie groups satisfy a generalization of Gaboriau fixed price property (originally defined for countable groups) to the setting of locally compact second countable groups. As one of the corollaries, under mild conditions, we can prove that the rank (minimal number of generators) or the first mod-p Betti number of a higher rank lattice grow sublinearly in the covolume.  The proof relies on surprising geometric properties of Poisson-Voronoi tessellations in higher-rank symmetric spaces, which could be of independent interest. 

We shall consider a crossing equation of the Euclidean conformal group in terms of conformal partial waves and in particular, a position independent representation of this equation. We shall briefly discuss the relevance of this equation to the problem of cosmological bootstrap. Thereafter, we shall sketch the derivation of the Biedenharn-Eliiot identity (a pentagon identity) for the 6j symbols of the conformal group and show how this provides us with an exact solution to said crossing equation. For the conformal group (which is non-compact), this involves some careful bookkeeping of the spinning representations. Finally, we shall discuss some consistency checks on the result obtained, and some open questions. 

Recently, a relation was introduced connecting codes of various types with the space of abelian (Narain) 2d CFTs. We extend this relation to provide holographic description of code CFTs in terms of abelian Chern-Simons theory in the bulk. For codes over the alphabet Z_p corresponding bulk theory is, schematically, U(1)_p times U(1)_{-p} where p stands for the level. Furthermore, CFT partition function averaged over all code theories for the codes of a given type is holographically given by the Chern-Simons partition function summed over all possible 3d geometries. This provides an explicit and controllable example of holographic correspondence where a finite ensemble of CFTs is dual to "topological/CS gravity" in the bulk. The parameter p controls the size of the ensemble and "how topological" the bulk theory is. Say, for p=1 any given Narain CFT is described holographically in terms of U(1)_1^n times U(1)_{-1}^n Chern-Simons, which does not distinguish between different 3d geometries (and hence can be evaluated on any of them). When p approaches infinity, the ensemble of code theories covers the whole Narain moduli space with the bulk theory becoming "U(1)-gravity" proposed by Maloney-Witten and Afkhami-Jeddi et al.