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SUMMARY:Smyth’s conjecture and a probabilistic local-to-global principle
  - Jordan Ellenberg (University of Wisconsin-Madison)
DTSTART:20240410T143000Z
DTEND:20240410T153000Z
UID:TALK214018@talks.cam.ac.uk
DESCRIPTION:Let a\,b\,c be integers.&nbsp\; Are there algebraic numbers x\
 ,y\,z which are Galois conjugate to each other over Q and which satisfy th
 e equation ax + bx + cy = 0? In 1986\, Chris Smyth proposed an appealingly
  simple conjecture that a set of plainly necessary local conditions on a\,
 b\,c guarantees the existence of such an x\,y\,z.&nbsp\; I will present a 
 provisional proof of this conjecture\, joint with Will Hardt.&nbsp\; As Sm
 yth observed\, this problem\, which appears on its face to be a question a
 bout algebraic number theory\, is really a question in combinatorics (rela
 ted to a problem solved by David Speyer: what can the eigenvalues of the s
 um of two permutation matrices be?)&nbsp\; It turns out\, though\, that th
 is combinatorics problem can be interpreted in number-theoretic terms\, as
  a question about whether certain kinds of equations in probability distri
 butions (I&rsquo\;ll explain what&rsquo\;s meant by this!) &nbsp\;have sol
 utions over Q whenever they have solutions over every completion of Q. &nb
 sp\;Solving this number theoretic problem\, in the end\, comes down to a b
 it of additive combinatorics.\n&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
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