BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Yuan Liu (University of Michigan) DTSTART:20200604T173000Z DTEND:20200604T183000Z DTSTAMP:20260423T004659Z UID:MAGIC/6 DESCRIPTION:Title: Pr esentations of Galois groups of maximal extensions with restricted ramific ations\nby Yuan Liu (University of Michigan) as part of MAGIC (Michiga n - Arithmetic Geometry Initiative - Columbia)\n\n\nAbstract\nIn previous work with Melanie Matchett Wood and David Zureick-Brown\, we conjecture th at an explicitly-defined random profinite group model can predict the dist ribution of the Galois groups of maximal unramified extension of global fi elds that range over $\\Gamma$-extensions of $\\mathbb{Q}$ or $\\mathbb{F} _q(t)$. In the function field case\, our conjecture is supported by the mo ment computation\, but very little is known in the number field case. Inte restingly\, our conjecture suggests that the random group should simulate the maximal unramified Galois groups\, and hence suggests some particular requirements on the structure of these Galois groups. In this talk\, we wi ll prove that the maximal unramified Galois groups are always achievable b y our random group model\, which verifies those interesting requirements. The proof is motivated by the work of Lubotzky on the profinite presentati ons and by the work of Koch on the $p$-class tower groups. We will also di scuss how the techniques used in the proof can be applied to the cases tha t are not covered by the Liu--Wood--Zureick-Brown conjecture\, which poten tially could help us obtain random group models for those cases.\n\n(Zoom password = order of the alternating group on six letters)\n LOCATION:https://researchseminars.org/talk/MAGIC/6/ END:VEVENT END:VCALENDAR