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SUMMARY:A Proto Inverse Szemerédi–Trotter Theorem - Olivine Silier (UC 
 Berkeley)
DTSTART:20241106T133000Z
DTEND:20241106T150000Z
UID:TALK223849@talks.cam.ac.uk
CONTACT:125830
DESCRIPTION:A point-line incidence is a point-line pair such that the poin
 t is on the line. The Szemerédi-Trotter theorem says the number of point-
 line incidences for n (distinct) points and lines in R^2^ is tightly upper
 bounded by O(n^4/3^). We advance the inverse problem: we geometrically cha
 racterize ‘sharp’ examples which saturate the bound by proving the exi
 stence of a nice cell decomposition we call the two bush cell decompositio
 n. The proof crucially relies on the crossing number inequality from graph
  theory and has a traditional analysis flavor.\n\nOur two bush cell decomp
 osition also holds in the analogous point-unit circle incidence problem. T
 his constitutes an important step towards obtaining an ε improvement in t
 he unit-distance problem.
LOCATION:MR4\, CMS
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