A problem on right tetrahedra, and rational points

Title: A problem on right tetrahedra, and rational points

Abstract:  A basic problem of Number theory is to understand  integral or rational solutions of polynomial equations with integer coefficients. One makes use of the geometry underlying the complex (or real) solutions. A prime 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 certain complex surfaces; this ongoing work is joint with Christopher Lyons.