BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Jordan Ellenberg (University of Wisconsin) DTSTART:20240723T220000Z DTEND:20240723T230000Z DTSTAMP:20260406T162115Z UID:agstanford/148 DESCRIPTION:Title: Smyth’s conjecture and a non-deterministic Hasse principle\nby Jordan Ellenberg (University of Wisconsin) as part of Stanford algebraic g eometry seminar\n\nLecture held in 383-N.\n\nAbstract\nSmyth 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 fou nd a necessary condition\, which he conjectured was sufficient. Surprisin gly\, this problem\, which appears to be about algebraic number theory\, e nds up touching on many different areas. I’ll explain how to express th is problem in terms of eigenvalues of linear combinations of permutation m atrices\, and finally how to solve it by means of a “non-deterministic H asse principle\,” in which we solve Diophantine equations but take our v ariables to be rational-valued random variables rather than deterministic rational numbers. There will be almost no advanced math beyond the defini tion of the p-adic numbers in this talk\, but we will at one point use Bri anchon’s theorem on ellipses inscribed in hexagons.\n LOCATION:https://researchseminars.org/talk/agstanford/148/ END:VEVENT END:VCALENDAR