Use this URL to cite or link to this record in EThOS: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.798351
Title: Structure theorems and extremal problems in incidence geometry
Author: Lin, Hiu Chung Aaron
ISNI:       0000 0004 8507 2408
Awarding Body: London School of Economics and Political Science (LSE)
Current Institution: London School of Economics and Political Science (University of London)
Date of Award: 2019
Availability of Full Text:
Access from EThOS:
Full text unavailable from EThOS. Please try the link below.
Access from Institution:
Abstract:
In this thesis, we prove variants and generalisations of the Sylvester-Gallai theorem, which states that a finite non-collinear point set in the plane spans an ordinary line. Green and Tao proved a structure theorem for sufficiently large sets spanning few ordinary lines, and used it to find exact extremal numbers for ordinary and 3-rich lines, solving the Dirac-Motzkin conjecture and the classical orchard problem respectively. We prove structure theorems for sufficiently large sets spanning few ordinary planes, hyperplanes, circles, and hyperspheres, showing that such sets lie mostly on algebraic curves (or on a hyperplane or hypersphere). We then use these structure theorems to solve the corresponding analogues of the Dirac-Motzkin conjecture and the orchard problem. For planes in 3-space and circles in the plane, we are able to find exact extremal numbers for ordinary and 4-rich planes and circles. We also show that there are irreducible rational space quartics such that any n-point subset spans only O(n8=3) coplanar quadruples, answering a question of Raz, Sharir, and De Zeeuw [51]. For hyperplanes in d-space, we are able to find tight asymptotic bounds on the extremal numbers for ordinary and (d + 1)-rich hyperplanes. This also gives a recursive method to compute exact extremal numbers for a fixed dimension d. For hyperspheres in d-space, we are able to find a tight asymptotic bound on the minimum number of ordinary hyperspheres, and an asymptotic bound on the maximum number of (d + 2)-rich hyperspheres that is tight in even dimensions. The recursive method in the hyperplanes case also applies here. Our methods rely on Green and Tao's results on ordinary lines, as well as results from classical algebraic geometry, in particular on projections, inversions, and algebraic curves.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.798351  DOI: Not available
Keywords: QA Mathematics
Share: