Erdos and Renyi introduced a model for studying random graphs of a given "density" and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected.  Motivated by ideas in geometric group theory we will explain some new threshold theorems we have discovered for random graphs.  We will then, explain applications of these results to the geometry of Coxeter groups.  Some of this talk will be on joint work with Hagen and Sisto; other parts are joint work with Hagen, Susse, and Falgas-Ravry.

Form factors form a bridge between the purely on-shell amplitudes and the purely off-shell correlation functions. In this talk, we study the form factors of general gauge-invariant local composite operators in N=4 SYM theory via on-shell methods. At tree-level and for a minimalnumber of external fields, the form factor exactly realises the spin-chain picture of N=4 SYM theory in the language of scattering amplitudes. Via generalised unitarity, we obtain the cut-constructible part of the one-loop correction to the minimal form factor of a generic operator. Its UV divergence yields the complete one-loop dilatation operator of the theory. At two-loop order, we employ unitarity to calculate the minimal form factors and thereby the dilatation operator for the Konishi primary operator and all operators in the SU(2) sector. For the former operator as well as other non-protected operators, important subtleties arise which require an extension of the method of unitarity.

Recently, as part of a project to find CY manifolds for which both the Hodge numbers (h^{11}, h^{21}) are small, manifolds have been found with Hodge numbers (4,1) and (1,1). The one-dimensional special geometries of their complex structures are more complicated than those previously studied. I will review these, emphasising the role of the fundamental period and Picard-Fuchs equation. Two arithmetic aspects arise: the first is the role of \zeta(3) in the monodromy matrices and the second is the fact, perhaps natural to a number theorist, that through a study of the CY manifolds over finite fields, modular functions can be associated to the singular manifolds of the family. This is a report on joint work with Volker Braun, Xenia de la Ossa and Duco van Straten.

Graphons and permutons are analytic objects associated with convergent sequences of graphs and permutations, respectively. Problems from extremal combinatorics and theoretical computer science led to a study of graphons and permutons determined by finitely many substructure densities, which are referred to as finitely forcible. The talk will contain several results on finite forcibility, focusing on the relation between finite forcibility of graphons and permutons. We also disprove a conjecture of Lovasz and Szegedy about the dimension of the space of typical vertices of finitely forcible graphons. The talk is based on joint work with Roman Glebov, Andrzej Grzesik and Dan Kral.

The study of infrared singularities, due to the emission of “soft” (low momentum) gauge bosons, remains a highly active research area in a variety of quantum field theories. After motivating both phenomenological and formal reasons as to why we should care about IR singularities, this talk will review their structure in QED, QCD and quantum gravity, examining the similarities and differences between these three contexts. The role of Wilson lines will be examined, which provide a useful unifying language. Finally, I will examine recent work on moving beyond the soft approximation, and why this might be useful.