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DTSTART;TZID=America/New_York:20211026T090000
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DTSTAMP:20240304T101126Z
CREATED:20240213T113529Z
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UID:10002507-1635238800-1635242400@cmsa.fas.harvard.edu
SUMMARY:The n-queens problem
DESCRIPTION:Abstract: The n-queens problem asks how many ways there are to place n queens on an n x n chessboard so that no two queens can attack one another\, and the toroidal n-queens problem asks the same question where the board is considered on the surface of a torus. Let Q(n) denote the number of n-queens configurations on the classical board and T(n) the number of toroidal n-queens configurations. 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 ((1+o(1))ne^{-3})^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 were known to hold for all (sufficiently large) n not divisible by 2 or 3. We also show that Q(n) is at least ((1+o(1))ne^{-3})^n for all natural numbers n which was independently proved by Luria and Simkin and\, combined with our toroidal result\, completely settles a conjecture of Rivin\, Vardi and Zimmerman regarding both Q(n) and T(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-partite 4-uniform hypergraph. Our strategy combines a random greedy algorithm to count `almost’ configurations with a complex absorbing strategy that uses ideas from the methods of randomised algebraic construction and iterative absorption. \nThis is joint work with Peter Keevash.
URL:https://cmsa.fas.harvard.edu/event/10-26-2021-combinatorics-physics-and-probability-seminar/
CATEGORIES:Combinatorics Physics and Probability
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