L6

 While extremal black hole microstates are reproduced by index calculations, the study of near-BPS black holes requires special care to account for quantum fluctuations. A semiclassical analysis indicates that the spectrum of such black holes has a large extremal degeneracy followed by a mass gap up to a continuum of non-BPS states. The inclusion of a theta angle term alters the properties of the spectrum (Witten effect shifting the mass gap and mixed 't Hooft anomaly). This journal club will study two papers by Toldo and Heydeman, [2412.03695] and [2412.03697] where they study 4d near-BPS black holes. As we shall see, a key point of their derivation is the reduction to 2d JT gravity. The dual CFTs are ABJM and some class R (non lagrangian) theories. Since these theories are strongly coupled, the gravity analysis offers a powerful tool to describe their specturm at finite temperature.

Given an FP-infinity subgroup G of SL(2,C), we are interested in the asymptotic behavior of the cohomology of G with coefficients in an irreducible complex representation V of SL(2,C). We prove that, as the dimension of V grows, the dimensions of these cohomology groups approximate the L2-Betti numbers of G. We make no further assumptions on G, extending a previous result of W. Fu. This yields a new method to compute those Betti numbers for finitely generated hyperbolic 3-manifold groups. We will give a brief idea of the proof, whose main tool is a completion of the universal enveloping algebra of the p-adic Lie algebra sl(2, Zp).

Virtually planar groups (that is, those groups with a finite-index subgroup admitting a planar Cayley graph) exhibit many fairly unique coarse geometric properties. Often, we find that any one of these properties completely characterises this class of groups. 

Abelian gauge theories in 2+1 dimensions are very interesting QFTs: they are strongly coupled and exhibit non-trivial dynamics. However, they are somewhat more tractable than non-Abelian theories in 3+1 dimensions. In this talk, I will first review the known properties of fermions in 2+1 dimensions and some conjectures about QED_3 with a single Dirac fermion. I will then present the recent proposal from [arXiv:2409.17913] regarding the phase diagram of QED_3 with two fermions. The findings reveal surprising (yet compelling) features: while semiclassical analysis would suggest two trivially gapped phases and a single phase transition, the actual dynamics indicate the presence of two distinct phase transitions separated by a "quantum phase." This intermediate phase exists over a finite range of parameters in the strong coupling regime and is not visible semiclassically. Moreover, these phase transitions are second-order and exhibit symmetry enhancement. The proposal is supported by several non-trivial checks and is consistent with results from numerical bootstrap, lattice simulations, and extrapolations from the large-Nf expansion.

I will present some contributions to the quasiisometry classification of solvable Lie groups of exponential growth that we obtain using sublinear bilipschitz equivalences, which are generalized quasiisometries. This is joint work with Ido Grayevsky.

Fundamental to the study of hyperbolic groups is their Gromov boundaries. The classical Cannon--Thurston map for a closed fibered hyperbolic 3-manifolds relates two such boundaries: it gives a continuous surjection from the boundary of the surface group (a circle) to the boundary of the 3-manifold group (a 2-sphere). Mj (Mitra) generalized this to all hyperbolic groups with hyperbolic normal subgroups. A generalization of the Gromov boundary to all finitely generated groups is called the Morse boundary. It collects all the "hyperbolic-like" rays in a group. In this talk we will discuss Cannon--Thurston maps for Morse boundaries. This is joint work with Ruth Charney, Antoine Goldsborough, Alessandro Sisto and Stefanie Zbinden.

(EFC-DGAs) lead to an algebraic approach to derived analytic geometry, pioneered for more general Fermat theories by Carchedi and Roytenberg.
 
They are well-suited to modelling finite-dimensional analytic spaces, and classical theorems in analysis ensure they give a largely equivalent theory to Lurie's more involved approach via pregeometries. DG dagger affinoid spaces provide a well-behaved class of geometric building blocks whose homotopy theory is governed by the underlying EFC-DGAs. 

Time permitting, I might also say a little about non-commutative generalisations.
 

The BNSR Invariant is a classical geometric invariant that encodes the finite generation of all coabelian subgroups of a given finitely generated group. The aim of this talk is to present a conjecture about the structure of the BNSR invariant of an Artin group and to present a new family in which the conjecture is true in terms of graph colorings.

We introduce ultrasolid modules, a variant of complete topological vector spaces. In this setting, we will prove some results in commutative algebra and apply them to the deformation of algebraic varieties in the language of derived algebraic geometry.

We consider two approximation schemes for the construction of a class of non-Gaussian multiplicative chaos, and show that they give rise to the same limit in the entire subcritical regime. Our approach uses a modified second moment method with the help of a new coupling argument, and does not rely on any Gaussian approximation or thick point analysis. As an application, we extend the martingale central limit theorem for partial sums of random multiplicative functions to L^1 twists. This is a joint work with Ofir Gorodetsky.