BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Jordan Ellenberg (University of Wisconsin) DTSTART:20240206T210000Z DTEND:20240206T220000Z DTSTAMP:20260423T125532Z UID:MITNT/85 DESCRIPTION:Title: A round Smyth’s conjecture\nby Jordan Ellenberg (University of Wiscons in) as part of MIT number theory seminar\n\nLecture held in Room 2-449 in the Simons Building (building 2).\n\nAbstract\nAre there algebraic numbers $x\,y\,z$ which are Galois conjugate to each other over $\\mathbb{Q}$ and which satisfy the equation $5x + 6y + 7z = 0$? In 1986\, Chris Smyth prop osed an appealingly simple conjecture about linear relations between Galoi s conjugates\, which would provide answers to the above questions and all questions of the same form\, and which has remained unsolved. My experien ce is that most people\, upon seeing Smyth’s conjecture\, immediately th ink it must be false (I certainly did!)\, but I have come to think it’s true\, and I’ll talk about a provisional solution\, joint with Will Hard t\, in the case of three conjugates. I’ll explain why (as Smyth observe d) this is really a conjecture about linear combinations of permutation ma trices (related question\, solved by Speyer: which algebraic numbers can be eigenvalues of the sum of two permutation matrices?)\, and why our appr oach can be thought of as proving a “Hasse principle for probability dis tributions” in a particular case\, plus a bit of additive number theory. Much of this talk\, maybe all\, will be suitable for undergraduates.\n LOCATION:https://researchseminars.org/talk/MITNT/85/ END:VEVENT END:VCALENDAR