Dept Math & Stat UiT, Forskningsparken B459 & Zoom
  L1: 12 Oct 2021 Tue 10:00-11:00
  L2: 18 Oct 2021 Mon 13:00-14:00
  L3: 25 Oct 2021 Mon 13:00-14:00
  L4: 29 Oct 2021 Fri 14:00-15:00

Arman Taghavi-Chabert (POLS, Warsaw)

"Lectures on almost Robinson geometry"

An almost Robinson structure on an even-dimensional Lorentzian (conformal) manifold is a totally null complex distribution of maximal rank. Such a structure singles out a real null line distribution together with a bundle Hermitian structure on its screen bundle. When the complex distribution is involutive, it is referred to as a Robinson structure, a notion introduced by Nurowski and Trautman in 2002. In this case the congruence of curves tangent to the null line distribution is geodesic, and its leaf space acquires a Cauchy-Riemann structure.

The history of this topic in fact goes back to Elie Cartan's seminal work on pure spinors. Later, during the Golden Age of general relativity, Robinson structures, under the guise of non-shearing congruences of null geodesics, proved fundamental in the study of exact solutions to the Einstein field equations in dimension four. They also provide an elegant geometric articulation of important results of mathematical relativity such as the Robinson, Goldberg-Sachs and Kerr theorems, which were influential in the early formulation of Sir Roger Penrose's twistor theory.

In this series of lectures, I will give an introduction to almost Robinson geometry starting from elementary considerations in linear algebra, before moving on to a geometric description of almost Robinson structures as G-structures, emphasising the relation with conformal and CR geometries. Applications to mathematical relativity will illustrate the exposition. I will follow the approach of my recent joint work with Anna Fino and Thomas Leistner, but will also draw on the traditions of the Oxford and Warsaw schools.

(To join Zoom: please contact Boris Kruglikov.)

Here are the lecture notes from the course.