BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Mikołaj Frączyk (University of Chicago) DTSTART:20210301T160000Z DTEND:20210301T170000Z DTSTAMP:20260423T023050Z UID:WarsawNT/22 DESCRIPTION:Title: Sarnak’s density hypothesis in horizontal families\nby Mikołaj Fr ączyk (University of Chicago) as part of Warsaw Number Theory Seminar\n\n Lecture held in currently online.\n\nAbstract\nLet $G$ be a real semi simp le Lie group with an irreducible unitary representation $\\pi$. The non-te mperedness of $\\pi$ is measured by the real parameter $p(\\pi)$ which is defined as the infimum of $p$ such that $\\pi$ has non-zero matrix coeffic ients in $L^p(G)$. Sarnak and Xue conjectured that for any arithmetic latt ice $\\Gamma\\subset G$ and principal congruence subgroup $\\Gamma(q)\\sub set \\Gamma$\, the multiplicity of $\\pi$ in $L^2(G/\\Gamma(q))$ is at mos t $O(V(q)^{2/p(\\pi) +\\varepsilon})$\, where $V(q)$ is the covolume of $\ \Gamma(q)$. Sarnak and Xue proved this conjecture for $G=SL(2\,\\mathbb R) \,SL(2\,\\mathbb C)$. I will talk about the joint work with Gergely Harcos \, Peter Maga and Djordje Milicevic where we prove bounds of the same qual ity that hold uniformly for families of pairwise non-commensurable lattice s in $G=SL(2\,\\mathbb R)^a\\times SL(2\,\\mathbb C)^b$. These families of lattices\, which we call horizontal\, are given as unit groups of maximal orders of quaternion algebras over number fields.\n LOCATION:https://researchseminars.org/talk/WarsawNT/22/ END:VEVENT END:VCALENDAR