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SUMMARY:Product-mixing in compact Lie groups - David Ellis (Bristol)
DTSTART:20240314T143000Z
DTEND:20240314T153000Z
UID:TALK212845@talks.cam.ac.uk
CONTACT:103978
DESCRIPTION:We say a subset S of a group G is product-free if there are no
 \nsolutions to the equation xy=z with x\,y and z all elements of S.\nWe ma
 ke progress on a question of Gowers concerning the maximal measure of\na m
 easurable product-free subset of SU(n)\, showing that this measure is\nat 
 most exp(-cn^{1/3}) for all n\, where c>0 is an absolute constant.\nIn fac
 t\, we prove similar 'stretched exponential' upper bounds for any\ncompact
 \, connected Lie group\, in terms of the minimal dimension of a\nnontrivia
 l irreducible continuous ordinary representation of the group. The\nbest-k
 nown lower bound (for SU(n)) remains exp(-cn) for an absolute\nconstant c\
 , which may well be the truth. Our bounds for SU(n) are\nbest-possible for
  a slightly stronger phenomenon\, known as product-mixing.\n\nOur main too
 ls are some new hypercontractive inequalities\, one of which\nsays that if
  f is a real-valued function on SU(n)\, then the q-norm of Tf\n(for some q
 uite large value of q > 2) is no larger than the 2-norm of\nf\, where T is
  a natural `noise' operator which\, roughly speaking\,\nreplaces the value
  of f at x with the average of f over a small\nsphere centred at x (for ea
 ch x). Our methods are easiest to explain if\nwe replace SU(n) with SO(n)\
 , where we obtain the desired\nhypercontractive inequality by a coupling w
 ith Gaussian space\, combined\nwith a variant of Weyl's unitary trick. In 
 this talk I will try to give a\ngood (high-level) picture of these methods
 .
LOCATION:MR12
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