Oxford University

Detectors in collider experiments are modeled by light-ray operators in Quantum Field Theory. For example, energy detectors are certain null integrals of the stress-energy tensor, localized at an angle on the celestial sphere, where they collect quanta that escape in their direction. In this talk, I will discuss a series of work developing a nonperturbative, convergent operator product expansion (OPE) for light-ray operators in Conformal Field Theories (CFTs). Objects appearing in the expansion are more general light-ray operators, whose matrix elements can be computed by the generalized Lorentzian inversion formula. An important application is to event shapes in collider physics, which correspond to correlation functions of light-ray operators within the state created by the incoming particles. I will discuss some applications of the light-ray OPE in CFT, and mention some extensions to QCD which make contact with measurements at the LHC. Talk based primarily on [1905.01311] and [2010.04726].

We consider close-packed tiling models of geometric objects -- a mixture of hardcore dimers and plaquettes -- as a generalisation of the familiar dimer models. Specifically, on an anisotropic cubic lattice, we demand that each site be covered by either a dimer on a z-link or a plaquettein the x-y plane. The space of such fully packed tilings has an extensive degeneracy. This maps onto a fracton-type `higher-rank electrostatics', which can exhibit a plaquette-dimer liquid and an ordered phase. We analyse this theory in detail, using height representations and T-duality to demonstrate that the concomitant phase transition occurs due to the proliferation of dipoles formed by defect pairs. The resultant critical theory can be considered as a fracton version of the Kosterlitz-Thouless transition. A significant new element is its UV-IR mixing, where the low energy behavior of the liquid phase and the transition out of it is dominated by local (short-wavelength) fluctuations, rendering the critical phenomenon beyond the renormalization group paradigm.

The integral of mean curvature squared is a conformal invariant of surfaces reintroduced by Willmore in 1965 whose study exercised a tremendous influence on geometric analysis and most notably on minimal surfaces in the last years.

On the other hand, the Loewner energy is a conformal invariant of planar curves introduced by Yilin Wang in 2015 which is notably linked to SLE processes and the Weil-Petersson class of (universal) Teichmüller theory.

In this presentation, after a brief historical introduction, we will discuss some recent developments linking the Willmore energy to the Loewner energy and mention several open problems.

Joint work with Yilin Wang (MIT/MSRI)

I will discuss a string theory way to study G_2 instanton moduli space and explain how to compute the instanton partition function for a certain G_2 manifold. An important insight comes from the twisted M-theory on the G_2 manifold. Building on the example, I will explain a possibility to extend the story to a large set of conjectural G_2 manifolds and a possible connection to 4d N=1 SCFT via geometric engineering. This talk is based on https://arxiv.org/abs/2109.01110 and a series of works in progress with Michele Del Zotto and Yehao Zhou.

I will discuss the regularity of solutions to quasilinear systems satisfying a Legendre-Hadamard ellipticity condition. For such systems it is known that weak solutions may which fail to be C^1 in any neighbourhood, so we cannot expect a general regularity theory. However if we assume an a-priori regularity condition of the solutions we can rule out such counterexamples. Focusing on solutions to Euler-Lagrange systems, I will present an improved regularity results for solutions whose gradient satisfies a suitable BMO / VMO condition. Ideas behind the proof will be presented in the interior case, and global consequences will also be discussed.

Nonlinear wave equations are ubiquitous in physics, and in three spatial dimensions they can exhibit a wide range of interesting behaviour even in the small data regime, ranging from dispersion and scattering on the one hand, through to finite-time blowup on the other. The type of behaviour exhibited depends on the kinds of nonlinearities present in the equations. In this talk I will explore the boundary between "good" nonlinearities (leading to dispersion similar to the linear waves) and "bad" nonlinearities (leading to finite-time blowup). In particular, I will give an overview of a proof of global existence (for small initial data) for a wide class of nonlinear wave equations, including some which almost fail to exist globally, but in which the singularity in some sense takes an infinite time to form. I will also show how to construct other examples of nonlinear wave equations whose solutions exhibit very unusual asymptotic behaviour, while still admitting global small data solutions.

Aimed: An introduction to the nonlinear theory of hyperbolic PDEs, as well as its close connections with the other areas of mathematics and wide range of applications in the sciences.