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SUMMARY:Mehdi Makhul (London School of Economics)
DTSTART:20250523T133000Z
DTEND:20250523T135500Z
DTSTAMP:20260423T010235Z
UID:CANT2025/8
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2025/8/"
 >Web geometry and the orchard problem</a>\nby Mehdi Makhul (London School 
 of Economics) as part of Combinatorial and additive number theory (CANT 20
 25)\n\nLecture held in CUNY Graduate Center - Science Center (4th floor).\
 n\nAbstract\nLet $P$ be a set of $n$ points in the plane\, not all lying o
 n a single line. \nThe orchard planting problem asks for the maximum numbe
 r of lines passing through exactly three points of $P$. Green and Tao show
 ed that the maximum possible number of such lines \nfor an $n$-element set
  is~$\\lfloor \\frac{n(n-3)}{6} \\rfloor+1$. Lin and Swanepoel also invest
 igated a generalization of the orchard problem in higher dimensions. \nSpe
 cifically\, if $P$ is a set of $n$ points \nin $d$-dimensional space\, the
 y established an upper bound for the maximum number of hyperplanes passing
  through exactly $d+1$ points of $P$. Our goal is to describe the structur
 al properties of configurations that achieve near-optimality in the asympt
 otic regime. \nLet $C \\subset \\mathbb{R}^d$ be an algebraic curve of deg
 ree~$r$\, and suppose that $P \\subset C$ is a set of $n$ points.  If $P$ 
 determines at least~$cn^d$ hyperplanes\, each passing through exactly $d+1
 $ points of $P$\, then the following must hold: The degree of $C$ must be 
 $d+1$\; and the curve $C$ is the complete intersection of ${d\\choose 2}-1
 $ quadric hypersurfaces. Our approach relies on the theory of web geometry
  and the Elekes-Szab\\'o Theorem-a cornerstone of incidence geometry-both 
 of which provide the structural basis for our analysis. \nJoint work with 
 Konrad Swanepoel.\n
LOCATION:https://researchseminars.org/talk/CANT2025/8/
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