GAAG seminar - A triangular path from number theory to algebraic geometry

A basic problem of Number theory is to understand integral or rational solutions of Diophantine Equations. One makes use of the algebraic geometry underlying the complex (or real) solutions. A primary example is furnished by the Congruent numbers, which are positive integers arising as areas of rational right triangles; it has been understood quite well due to some striking recent results (due to others). We will discuss a generalization to the setting of right tetrahedra with rational edges, which leads to some intriguing geometry of elliptic surfaces and cubic twists; this ongoing work is joint with Christopher Lyons.