Symmetric Trilinear Forms and Einstein-like Equations: From Affine Spheres to Griess Algebras

Explore a comprehensive lecture on symmetric trilinear forms and Einstein-like equations, spanning from affine spheres to Griess algebras. Delve into the concept of Einstein-like equations in the context of coupling metrics to tensors with prescribed symmetries. Examine a hierarchy of equations generalizing constant sectional curvature, Einstein, and constant scalar curvature. Review affine differential geometry of nondegenerate hypersurfaces and discover a class of geometric structures that extend statistical and Weyl structures. Investigate Einstein equations that generalize Einstein-Weyl equations and their relationship to affine spheres. Uncover further examples through algebraic constructions, including discussions on metrized commutative algebras akin to simple Jordan algebras and Griess algebras of VOAs. Gain insights into the unifying element of these diverse contexts: symmetric trilinear forms. This lecture is part of the SCREAM project, focusing on Cartan and parabolic geometries and their interactions with mechanical systems, integrable systems, and Penrose's Conformal Cyclic Cosmology programme.