BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Terence Tao (UCLA) DTSTART:20210126T161500Z DTEND:20210126T181500Z DTSTAMP:20260404T095357Z UID:ICMS/1 DESCRIPTION:Title: Sendov’s conjecture for sufficiently high degree polynomials\nby Terence Tao (UCLA) as part of International Center for Mathematical Scien ces\n\n\nAbstract\nIn 1958\, Blagovest Sendov made the following conjectur e: if a polynomial $f$ of degree $n \\geq 2$ has all of its zeroes in the unit disk\, and $a$ is one of these zeroes\, then at least one of the crit ical points of $f$ lies within a unit distance of $a$. Despite a large amo unt of effort by many mathematicians and several partial results (such as the verification of the conjecture for degrees $n \\leq 8$)\, the full con jecture remains unresolved. In this talk\, we present a new result that es tablishes the conjecture whenever the degree $n$ is larger than some suffi ciently large absolute constant $n_0$. A result of this form was previousl y established in 2014 by Degot assuming that the distinguished zero $a$ st ayed away from the origin and the unit circle. To handle these latter case s we study the asymptotic limit as $n \\to \\infty$ using techniques from potential theory (and in particular the theory of balayage)\, which has co nnections to probability theory (and Brownian motion in particular). Apply ing unique continuation theorems in the asymptotic limit\, one can control the asymptotic behavior of both the zeroes and the critical points\, whic h allows us to resolve the case when $a$ is near the origin via the argume nt principle\, and when $a$ is near the unit circle by careful use of Tayl or expansions to gain fine asymptotic control on the polynomial $f$.\n\nTh is talk is jointly organized with the Institute of the Mathematical Scienc es of the Americas at the University of Miami (IMSA)and the Union of Bulga rian Mathematicians (UBM).\n LOCATION:https://stable.researchseminars.org/talk/ICMS/1/ END:VEVENT END:VCALENDAR