Algebras of geometries

I will give a couple of constructions of rings that are generated by classes of algebraic or symplectic varieties, such as the Grothendieck ring of varieties or the ring of bordisms. These rings are endowed with algebraic operations such as symmetric powers or Landweber-Novikov operations. One can prove interesting formulas in these rings, for example there is a formula that relates a cubic hypersurface with its variety of lines. Also I will discuss possible applications of this operations and identities for studying ratinality questions (arXiv:1405.5154) and for a proof of gamma-conjectures (1404.6407).