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:20260423T035536Z
UID:ICMS/1
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ICMS/1/">Sen
 dov’s conjecture for sufficiently high degree polynomials</a>\nby Terenc
 e Tao (UCLA) as part of International Center for Mathematical Sciences\n\n
 \nAbstract\nIn 1958\, Blagovest Sendov made the following conjecture: if a
  polynomial $f$ of degree $n \\geq 2$ has all of its zeroes in the unit di
 sk\, and $a$ is one of these zeroes\, then at least one of the critical po
 ints of $f$ lies within a unit distance of $a$. Despite a large amount of 
 effort by many mathematicians and several partial results (such as the ver
 ification of the conjecture for degrees $n \\leq 8$)\, the full conjecture
  remains unresolved. In this talk\, we present a new result that establish
 es the conjecture whenever the degree $n$ is larger than some sufficiently
  large absolute constant $n_0$. A result of this form was previously estab
 lished in 2014 by Degot assuming that the distinguished zero $a$ stayed aw
 ay from the origin and the unit circle. To handle these latter cases we st
 udy the asymptotic limit as $n \\to \\infty$ using techniques from potenti
 al theory (and in particular the theory of balayage)\, which has connectio
 ns to probability theory (and Brownian motion in particular). Applying uni
 que continuation theorems in the asymptotic limit\, one can control the as
 ymptotic behavior of both the zeroes and the critical points\, which allow
 s us to resolve the case when $a$ is near the origin via the argument prin
 ciple\, and when $a$ is near the unit circle by careful use of Taylor expa
 nsions to gain fine asymptotic control on the polynomial $f$.\n\nThis talk
  is jointly organized with the Institute of the Mathematical Sciences of t
 he Americas at the University of Miami (IMSA)and the Union of Bulgarian Ma
 thematicians (UBM).\n
LOCATION:https://researchseminars.org/talk/ICMS/1/
END:VEVENT
END:VCALENDAR