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

Explore a comprehensive lecture on symmetric trilinear forms and Einstein-like equations, covering topics 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 generalize statistical and Weyl structures. Investigate Einstein equations that extend Einstein-Weyl equations and their relationship to affine spheres. Explore additional examples through algebraic constructions, including discussions on metrized commutative algebras similar to simple Jordan algebras and Griess algebras of VOAs. Understand the unifying element of these diverse contexts: a symmetric trilinear form. 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.