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{\Large Department of Mathematics, BGU}

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{\Huge BGU Probability and Ergodic Theory  (PET) seminar}\\[0.2\baselineskip]

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\textbf{On} \emph{Thursday, November  6, 2025}
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\textbf{At} \emph{11:10 -- 12:00}
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\textbf{In} \emph{-101}

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{\large\scshape Itay Glazer 
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will talk about
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{\Large\bfseries Small ball estimates and mixing for word maps on unitary groups\par}
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\textsc{Abstract:}
Let w(x,y) be a word in a free group. For any group G, w induces a word map w:G\^{}2--\textgreater{}G. For example, the commutator word w=xyx\^{}(-1)y\^{}(-1) induces the commutator map. 
In the setting of finite simple groups, Larsen, Shalev and Tiep showed there exists epsilon(w)\textgreater{}0 (depending only on the word w), such that for all sufficiently large G, the probability that a random pair (g\_1,g\_2) in G\^{}2 satisfies w(g\_1,g\_2)=g is smaller than \textbar{}G\textbar{}\^{}(-epsilon(w)). They further obtained uniform upper bounds on the L\^{}1- and L\^{}infty-mixing times for the random walks induced by the corresponding word measures.  \newline
I will discuss analogous results for the family of unitary groups in all ranks.








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