BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Rob Benedetto (Amherst College) DTSTART:20241021T200000Z DTEND:20241021T210000Z DTSTAMP:20260423T035958Z UID:NTBU/6 DESCRIPTION:Title: Arb oreal Galois groups with colliding critical points\nby Rob Benedetto ( Amherst College) as part of Boston University Number Theory Seminar\n\nLec ture held in CDS Room 548 in Boston University (*NOT the usual room for th e semester*).\n\nAbstract\nLet $f\\in K(z)$ be a rational function of degr ee $d\\geq 2$ defined over a field $K$ (usually $\\mathbb{Q}$)\, and let $ x_0\\in K$. The backward orbit of $x_0$\, which is the union of the iterat ed preimages $f^{-n}(x_0)$\, has the natural structure of a $d$-ary rooted tree. Thus\, the Galois groups of the fields generated by roots of the eq uations $f^n(z)=x_0$ are known as arboreal Galois groups. In 2013\, Pink o bserved that when $d=2$ and the two critical points $c_1\,c_2$ of $f$ coll ide\, meaning that $f^m(c_1)=f^m(c_2)$ for some $m\\geq 1$\, then the arbo real Galois groups are strictly smaller than the full automorphism group o f the tree. We study these arboreal Galois groups when $K$ is a number fie ld and $f$ is either a quadratic rational function (as in Pink's setting o ver function fields) or a cubic polynomial with colliding critical points. We describe the maximum possible Galois groups in these cases\, and we pr esent sufficient conditions for these maximum groups to be attained.\n\nJo int BU Number Theory and Dynamics seminar. Note the non-standard room.\n LOCATION:https://researchseminars.org/talk/NTBU/6/ END:VEVENT END:VCALENDAR