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SUMMARY:Ofir Gorodetsky (Tel Aviv University)
DTSTART:20200316T123000Z
DTEND:20200316T133000Z
DTSTAMP:20260423T005834Z
UID:HSPETDS/9
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/HSPETDS/9/">
 The anatomy of integers and Ewens permutations</a>\nby Ofir Gorodetsky (Te
 l Aviv University) as part of Horowitz seminar on probability\, ergodic th
 eory and dynamical systems\n\n\nAbstract\nWe will discuss an analogy betwe
 en integers and permutations\, an analogy which goes back to works of Erd
 ős and Kac and of Billingsley which we shall survey. Certain statistics o
 f the prime factors of a uniformly drawn integer (between $1$ and $x$) agr
 ee\, in the limit\, with similar statistics of the cycles of a uniformly d
 rawn permutation from the symmetric group on $n$ elements. This analogy is
  beneficial to both number theory and probability theory\, as one can ofte
 n prove new number-theoretical results by employing probabilistic ideas\, 
 and vice versa.\nThe Ewens measure with parameter Θ\, first discovered in
  the context of population genetics\, is a non-uniform measure on permutat
 ions. We will present an analogue of this measure on the integers\, and sh
 ow how natural questions on the integers have answers which agree with ana
 logous problems for the Ewens measure. For example\, the size of the prime
  factors of integers which are sums of two squares\, and the cycle lengths
  of permutations drawn according to the Ewens measure with parameter 1/2\,
  both converge to the Poisson-Dirichlet process with parameter 1/2. We wil
 l convey some of the ideas behind the proofs.\nJoint work with Dor Elboim.
 \n
LOCATION:https://researchseminars.org/talk/HSPETDS/9/
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