Explicit composition identities for higher composition laws

In 2001, Manjul Bhargava gave a new proof of Gauss composition of binary quadratic forms by using 2x2x2 integer cubes. Moreover, he showed that there are five higher composition laws which are related to quadratic rings similar to the case of binary quadratic forms. The proof of these higher composition laws relies on bijections between certain orbits of the spaces on which the composition is defined under some natural group action and certain suitable (tuples of) ideal classes of quadratic rings. In my PhD thesis, we formulated the higher composition laws in a manner similar to Gauss' formulation of composition of binary quadratic forms. More precisely, we provided explicit composition identities for the higher composition laws in the quadratic case. In this talk, I will briefly outline Bhargava's work on composition laws in the quadratic case, (and the cubic case if time permits), and describe our results on the composition identities.