L2

One of the major insights gained from holographic duality is the relation between the physics of black holes and quantum chaotic systems. This relation is made precise in the duality between two dimensional JT gravity and random matrix theory.  In this work, we generalize this to a duality between AdS3 gravity and a random ensemble of approximate CFT's.  The latter is described by a combined tensor and matrix model, describing the OPE coefficients and spectrum of a theory that approximately satisfies the bootstrap constraints.   We show that the Feynman diagrams of the random ensemble produce a sum over 3 manifolds that agrees with the partition function of 3d gravity.  A crucial element of this dictionary is the Virasoro TQFT, which defines the bulk gravitational path integral via the cutting and sewing relations of topological field theory.  Time permitting, we will explain why this TQFT has gravitational edge modes degrees of freedom whose entanglement gives rise to gravitational entropy.

I shall review some aspects of the relationship between scale and conformal invariance in 2-dimensional sigma models.  Then, I shall explain how such an investigation is related to the Perelman's ideas of proving the Poincare' conjecture.  Using this, I shall demonstrate that scale invariant sigma models  with B-field coupling and  compact target space  are conformally invariant. Several examples will also be presented that elucidate the results.  The talk is based on the arXiv paper 2404.19526.

TBC

TBC

Khintchine's Theorem is one of the cornerstones in metric Diophantine approximation. The question of removing the monotonicity condition on the approximation function in Khintchine's Theorem led to the recently proved Duffin-Schaeffer conjecture. Gallagher showed an analogue of Khintchine's Theorem for multiplicative Diophantine approximation, again assuming monotonicity. In this talk, I will discuss my joint work with L. Frühwirth about a Duffin-Schaeffer version for Gallagher's Theorem. Furthermore, I will give a broader overview on various questions in metric Diophantine approximation and demonstrate the deep connection to both analytic and combinatorial number theory that is hidden inside the proof of these statements.

Functoriality is a key feature in Langlands’ conjectured relationship between automorphic representations and Galois representations; it predicts that certain Galois representations are automorphic, i.e. should come from automorphic representations. We discuss the idea of $p$-adic propagation of automorphy, which seeks to establish the automorphy of everything in a “neighborhood” given the automorphy of something in that neighborhood. The “neighborhoods” that we consider will be the irreducible components of a $p$-adic analytic space called the eigenvariety, which parameterizes $p$-adic automorphic representations. This technique was introduced by Newton and Thorne in their proof of symmetric power functoriality, and can be adapted to investigate similar problems.

In this talk I will describe the systematic construction of strongly interacting RG fixed points with a finite disorder strength.  Such random-field disorder is quite common in condensed matter experiment, necessitating an understanding of the effects of this disorder on the properties of such fixed points. In the past, such disordered fixed points were accessed using e.g. epsilon expansions in perturbative quantum field theory, using the replica method to treat disorder.  I will show that holography gives an alternative picture for RG flows towards disordered fixed points.  In holography, spatially inhomogeneous disorder corresponds to inhomogeneous boundary conditions for an asymptotically-AdS spacetime, and the RG flow of the disorder strength is captured by the solution to the Einstein-matter equations. Using this construction, we have found analytically-controlled RG fixed points with a finite disorder strength.  Our construction accounts for, and explains, subtle non-perturbative geometric effects that had previously been missed.  Our predictions are consistent with conformal perturbation theory when studying disordered holographic CFTs, but the method generalizes and gives new models of disordered metallic quantum criticality.