Symmetric trilinear forms and Einstein-like equations: from affine spheres to Griess algebras p. III
Channel: Centrum Fizyki Teoretycznej PAN
Date: 2022-12-05

Prof. Daniel J F Fox (Universidad Politécnica de Madrid).
Symmetric trilinear forms and Einstein-like equations: from affine spheres to Griess algebras - Part III.
Abstract: First there will be explained what is meant by Einstein-like equations in the context of equations coupling a metric to a tensor with prescribed symmetries. In this generality there will be described a hierarchy of equations generalizing the usual hierarchy constant sectional curvature, Einstein, constant scalar curvature. Next there will be reviewed the affine differential geometry of nondegenerate hypersurface and there will be introduced a class of geometric structures generalizing statistical structures and Weyl structures and for this class there will be described Einstein equations generalizing the usual Einstein-Weyl equations and such that the conjugate statistical structures induced on a nondegenerate hypersurface are Einstein if and only if the hypersurface is an affine sphere. Finally, further examples not locally equivalent to these will be obtained via purely algebraic constructions. This will entail a discussion of metrized commutative algebras of the flavor of simple Jordan algebras and Griess algebras of VOAs. The element uniting these a priori apparently little related contexts is a symmetric trilinear form.

This is a bi-weekly seminar for the Grieg collaboration project SCREAM: Symmetry, Curvature Reduction, and EquivAlence Methods, funded by the Norwegian Financial Mechanism 2014-2021. The registration number of the project is 2019/34/H/ST1/00636. 

The focus of the project is on a broad class of differential geometric structures known as Cartan and parabolic geometries, which include conformal, projective, CR, and ODE geometry, (2, 3, 5)-distributions, parabolic contact structures, and many more besides. Moreover, we explore the interaction of these structures with a variety of subjects such as mechanical systems and geometric robots, the theory of integrable systems, and Penrose's Conformal Cyclic Cosmology programme. 

More information can be found in www.cft.edu.pl/grieg.