The first geometry seminar of 2026 will be given on 19th January by Kirill Krasnov (Nottingham UK). The seminar this term remains in the regular time and place: 2pm on Mondays in the Salle des Profs. Kirill’s title is “G-structures, elliptic complexes and 4D Einstein equations” and his abstract is below. Hope to see you there!
I will begin with a brief overview of the theory of G-structures and intrinsic torsion. I will then explain how a natural elliptic complex of differential operators arises when G is metric, i.e. when G\subset O(n), and how this framework allows one to reinterpret the Einstein equations in the special case G=O(n).
More generally, I will show how the elliptic complex associated with a G-structure leads to natural second-order partial differential equations governing the geometry. I will then describe a construction that is specific to four dimensions. In this case one has the exceptional isomorphism Spin(4)=SU+ (2)×SU− (2). Taking G=SU− (2), the complement of G in SO(4) is itself a Lie group, namely SU+ (2). I will show that imposing SU+ (2)-invariance on the equations for an SU− (2)-structure leads precisely to the four-dimensional Einstein equations. I will conclude with some speculations on possible generalisations of this picture to higher dimensions.