BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:Lower bounds for incidences and Heilbronn's triangle problem - Dmi
 trii Zakharov (Massachusetts Institute of Technology)
DTSTART:20250129T133000Z
DTEND:20250129T150000Z
UID:TALK225118@talks.cam.ac.uk
CONTACT:Julia Wolf
DESCRIPTION:Upper bounds on the number of incidences between points and li
 nes\, tubes\, and other geometric objects\, have many applications in comb
 inatorics and analysis. On the other hand\, much less is known about lower
  bounds. We prove a general lower bound for the number of incidences betwe
 en points and tubes in the plane under a natural spacing condition. In par
 ticular\, if you take n points in the unit square and draw a line through 
 each point\, then there is a non-trivial point-line pair with distance at 
 most n^-2/3+o(1)^. This quickly implies that any n points in the unit squa
 re define a triangle of area at most n^-7/6+o(1)^\, giving a new upper bou
 nd for the Heilbronn's triangle problem.\nJoint work with Alex Cohen and C
 osmin Pohoata.
LOCATION:MR4\, CMS
END:VEVENT
END:VCALENDAR