I will explain how the model-theoretic algebraic closure in Zilber’s pseudo-exponential field can be described in terms of the self-sufficient closure. I will sketch a proof and show how the Mordell-Lang conjecture for algebraic tori comes into play. If time permits, I’ll also talk about the characterisation of strongly minimal sets and their geometries. This is joint work (still in progress) with Jonathan Kirby.

Modular forms are holomorphic functions on the upper half plane satisfying a transformation property under the action of Mobius transformations. While they are a priori complex-analytic objects, they have applications to number theory thanks to their connection with Galois representations. Weight one modular forms are special because their Galois representations factor through a finite quotient. In this talk, we will explain a different degeneracy: they contribute to the cohomology of a line bundle over the modular curve in degrees 0 and 1. We propose an arithmetic explanation for this: an action of a unit group associated to the Galois representation of the modular form. This extends the conjectures of Venkatesh, Prasanna, and Harris. Time permitting, we will discuss a generalization to Hilbert modular forms.

In recent progress on the black hole information paradox, Page curves consistent with unitarity have been obtained in 2d models and in bottom-up braneworld models using the notion of double holography. In this talk we discuss top-down models realizing 4d black holes coupled to a bath in Type IIB string theory and obtain Page curves. We make the ideas behind double holography precise in these models and address causality puzzles which have arisen in the bottom-up models, leading to a refinement of their interpretation.
 

We study supersymmetric gauge theories with 8 supercharges in d=3,4,5,6. For these theories, one can perform Higgsings by turning on VEVs of scalar fields. However, this process can often be difficult when dealing with superconformal field theories (SCFTs) where the Lagrangian is often not known. Using techniques of magnetic quivers and a new algorithm we call "Inverted Quiver Subtraction", we show how one can easily obtain the SCFT(s) after Higgsing. This technique can be equally well applied to SCFTs in d=3,4,5,6. 

Argyres—Douglas (AD) theories are 4d N=2 SCFTs which have some unusual features, and until recently, explicit holographic duals of these theories were unknown. We will consider a concrete class of these theories obtained by wrapping the 6d N=(2,0) ADE theories on a (twice) punctured sphere: one irregular and one regular puncture, and construct their holographic duals. The novel aspects of these solutions require a relaxation of the regularity conditions of the usual Gaiotto—Maldacena framework and to allow for brane singularities. We show how to construct the dictionary between the AdS(5) solutions and the field theory and match observables between the two. If time allows, I will comment on some on-going work about further compactifying the AD theories on spindles, or the 6d theories on four-dimensional orbifolds.