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Title page for ETD etd-08092010-093437


Type of Document Dissertation
Author Fitzpatrick, Justin Liam
Author's Email Address justin.l.fitzpatrick@vanderbilt.edu
URN etd-08092010-093437
Title The Geometry of Optimal and Near-Optimal Riesz Energy Configurations
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Douglas Hardin Committee Chair
Edward B. Saff Committee Member
Emmanuele DiBenedetto Committee Member
Gieri Simonett Committee Member
James Dickerson Committee Member
Keywords
  • Riesz energy
  • geometric inequalities
  • Voronoi diagrams
  • best-packing
Date of Defense 2010-08-09
Availability unrestricted
Abstract
This thesis discusses recent and classical results concerning the asymptotic properties (as N gets large) of ``ground state' configurations of N particles restricted to a compact set A of Hausdorff dimension d interacting through through an inverse power law 1/r^s for some s>0.

It has been observed that, as s becomes large, ground state configurations approach best-packing configurations on A. When d=2, it is generally believed that ground state configurations form a hexagonal lattice. This thesis aims to justify this belief in the case d=2 through the study of geometric inequalities for polygons. Specifically, it is shown that, when s is large, a normalized energy associated to interactions from particles that are ``nearest neighbors' to a fixed point in the configuration is minimized when the nearest neighbors form a regular polygon. This technique provides new lower bounds for the energy for 2-dimensional ground state configurations.

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