Week 02.05.2022 – 08.05.2022

We study the deformation theory of nearly G_2 manifolds. These are seven dimensional manifolds admitting real Killing spinors. We show that the infinitesimal deformations of nearly G_2 structures are obstructed in general. Explicitly, we prove that the infinitesimal deformations of the homogeneous nearly G_2 structure on the Aloff--Wallach space are all obstructed to second order. We also completely describe the cohomology of nearly G_2 manifolds. The talk is based on a joint work with Shubham Dwivedi (HU, Berlin).

General Relativity in asymptotically flat spacetimes gives rise to an infinite number of symmetries which form the celebrated BMS group comprising superrotations and supertranslations. These symmetries are closely related to soft theorems of gravitational scattering amplitudes. Recently it was shown that supertranslations and superrotations are only the lowest levels of a whole tower of symmetries of tree level gravitational scattering amplitudes that form a w_{1+\infty} algebra. The fate of this symmetry once loop effects are taken into account is currently unknown.
In this talk I will review the emergence of this symmetry algebra based on the celestial CFT program and argue that the w_{1+\infty} algebra persists quantum corrections in self-dual gravity.
This talk is based on 2111.10392 with A.Ball, S. Narayanan, and A. Strominger.

Abstract: In this talk we will discuss Montgomery's pair correlation conjecture for the zeros of the Riemann zeta function. This is a deep spectral conjecture, closely related to several arithmetic conjectures on the distribution of primes. For example, even assuming a strong form of the twin prime conjecture, one would only resolve Montgomery's conjecture in a limited range. Yet, building on work of Goldston and Gonek from the late 1990s, we will present a recent conditional lower bound on the Fourier transform of Montgomery's pair correlation function, valid under milder hypotheses. The new technical ingredient is a correlation estimate for the Selberg sieve weights, for which the level of support of the weights lies beyond the classical square-root barrier.