Polytopes are higher-dimensional generalizations of polygons in the plane. Not only are they intrinsically beautiful objects, they lie at the crossroads of diverse areas of mathematics. In this talk, I will discuss a quantity associated with polytopes, called the mixed volume, the fundamental Alexandrov—Fenchel Inequality for mixed volumes, and the many connections that these quantities have with other parts of mathematics.

A gentle yet rapid introduction to class field theory, its conjectured generalization – the Langlands program, and the role of L-functions therein. Our starting point is a density result due to Frobenius, which implies that the Galois (hence all) extensions of a number field are uniquely determined by the set of primes that split completely. The rest of the talk traces historically the quest of understanding this set culminating in its close connection with Artin L-functions, certain meromorphic functions on the complex plane reminiscent of the Riemann zeta function.