Mathematical Conversations

Cubic forms are homogeneous polynomials of degree 3 in several 
variables. Number theory is interested in their zeros over the rational 
numbers. Algebraic geometry studies the cubic hypersurfaces defined by 
them (e.g., the 27 lines on smooth cubic surfaces over the complex 
numbers). How is the arithmetic of cubic hypersurfaces (expected to be) 
governed to their geometry? I will discuss old and new conjectures and results.