Clemens-Griffiths discovered in 1971 a surprising connection
between two questions (all terms will be defined in the talk): 

1. When is a 3-dimensional hypersurface rational (i.e. birational to
projective 3-space)?

2. When is a flat torus isometric to the Jacobian of an algebraic curve?

I will explain this connection and how I used it to give a simple proof of the (already known) irrationality of the general quartic 3-fold.  More importantly, we will see some fundamental classical ideas in action: resolution of singularities, the theory of periods, automorphism groups of varieties, and a beautiful example of Felix Klein.  If time permits I will also explain a differential-geometric answer to question 2 that I found with Curt McMullen (inspired by an old result of Gromov).