Binary quadratic forms, higher composition laws and explicit composition identities

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 (tuples of) ideal classes of quadratic rings. In my PhD thesis, under the supervision of Gautam Chinta, 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 give an accessible introduction to the theory of binary quadratic forms, describe briefly Bhargava's approach, and then state the explicit identities that we formulated in our work.