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SUMMARY:Itay Glazer (University of Oxford)
DTSTART:20240516T153000Z
DTEND:20240516T162000Z
DTSTAMP:20260416T215547Z
UID:GiG2024/5
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GiG2024/5/">
 Fourier and Small ball estimates for word maps on unitary groups</a>\nby I
 tay Glazer (University of Oxford) as part of Groups in Galway 2024\n\nLect
 ure held in McMunn lecture theatre.\n\nAbstract\nLet $w(x\,y)$ be a word i
 n a free group. For any group $G$\, $w$ induces a word map $w:G^2 \\to G$.
  For example\, the commutator word $w=xyx^{-1}y^{-1}$ induces the commutat
 or map. If $G$ is finite\, one can ask what is the probability that $w(g\,
 h)$ is equal to the identity element $e$\, for a pair $(g\,h)$ of elements
  in $G$\, chosen independently at random. \nIn the setting of finite simpl
 e groups\, Larsen and Shalev showed there exists $\\epsilon(w)>0$ (dependi
 ng only on $w$)\, such that the probability that $w(g\,h)=e$ is smaller th
 an $|G|^{-\\epsilon(w)}$\, whenever $G$ is large enough (depending on $w$)
 . \nIn this talk\, I will discuss analogous questions for compact groups\,
  with a focus on the family of unitary groups\; For example\, given a word
  $w$\, and given two independent Haar-random $n$ by $n$ unitary matrices $
 A$ and $B$\, what is the probability that $w(A\,B)$ is contained in a smal
 l ball around the identity matrix?\n\nBased on a joint work with Nir Avni 
 and Michael Larsen.\n
LOCATION:https://researchseminars.org/talk/GiG2024/5/
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