Jordan Ellenberg: Smyth’s conjecture and a non-deterministic Hasse principle

Smyth asked in the 1980s which linear relations with integral coefficients a_1 x_1 + ... + a_r x_r could hold when x_1, ... x_r are Galois conjugates.  He found a necessary condition, which he conjectured was sufficient.  Surprisingly, this problem, which appears to be about algebraic number theory, ends up touching on many different areas.  I’ll explain how to express this problem in terms of eigenvalues of linear combinations of permutation matrices, and finally how to solve it by means of a “non-deterministic Hasse principle,” in which we solve Diophantine equations but take our variables to be rational-valued random variables rather than deterministic rational numbers.  There will be almost no advanced math beyond the definition of the p-adic numbers in this talk, but we will at one point use Brianchon’s theorem on ellipses inscribed in hexagons.