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SUMMARY:Candida Bowtell (University of Oxford)
DTSTART:20211213T140000Z
DTEND:20211213T150000Z
DTSTAMP:20260423T035406Z
UID:EPC/82
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EPC/82/">The
  n-queens problem</a>\nby Candida Bowtell (University of Oxford) as part o
 f Extremal and probabilistic combinatorics webinar\n\n\nAbstract\nHow many
  ways are there to place n queens on an n by n chessboard so that no two c
 an attack one another? What if the chessboard is embedded on the torus? Le
 t Q(n) be the number of ways on the standard chessboard and T(n) the numbe
 r on the toroidal board. The toroidal problem was first studied in 1918 by
  Pólya who showed that T(n)>0 if and only if n is not divisible by 2 or 3
 . Much more recently Luria showed that T(n) is at most $\\left((1+o(1))ne^
 {-3}\\right)^n$ and conjectured equality when n is not divisible by 2 or 3
 . We prove this conjecture\, prior to which no non-trivial lower bounds we
 re known to hold for all (sufficiently large) n not divisible by 2 or 3. W
 e also show that Q(n) is at least $\\left((1+o(1))ne^{-3}\\right)^n$ for a
 ll natural numbers n which was independently proved by Luria and Simkin an
 d\, combined with our toroidal result\, completely settles a conjecture of
  Rivin\, Vardi and Zimmerman regarding both Q(n) and T(n).\n\nIn this talk
  we'll discuss our methods used to prove these results. A crucial element 
 of this is translating the problem to one of counting matchings in a 4-par
 tite 4-uniform hypergraph. Our strategy combines a random greedy algorithm
  to count `almost' configurations with a complex absorbing strategy that u
 ses ideas from the methods of randomised algebraic construction and iterat
 ive absorption.\n\nThis is joint work with Peter Keevash.\n
LOCATION:https://researchseminars.org/talk/EPC/82/
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