University of Illinois Urbana-Champaign

Some problems in polynomial interpolation and topological complexity

Fieldsteel, Nathan Mulvey

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FIELDSTEEL-DISSERTATION-2017.pdf

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https://hdl.handle.net/2142/97758

Description

Title
Some problems in polynomial interpolation and topological complexity
Author(s)
Fieldsteel, Nathan Mulvey
Issue Date
2017-04-21
Director of Research (if dissertation) or Advisor (if thesis)
Schenck, Hal
Doctoral Committee Chair(s)
Nevins, Tom
Committee Member(s)
  • Baryshnikov, Yuliy
  • Hirani, Anil
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2017-08-10T20:33:17Z
Keyword(s)
  • Approximation theory
  • Polynomial interpolation
  • Motion planning
Abstract
This thesis is comprised of two projects in applied computational mathematics. In Chapter 1, we discuss the geometry and combinatorics of geometrically characterized sets. These are finite sets of n+d choose n points in R^d which impose independent conditions on polynomials of degree n, and which have Lagrange polynomials of a special form. These sets were introduced by Chung and Yao in a 1977 paper in the SIAM Journal of Numerical Analysis in the context of polynomial interpolation. There are several conjectures on the nature and geometric structure of these sets. We investigate the geometry and combinatorics of GC sets for d at least 2, and prove they are closely related to simplicial complexes which are Cohen-Macaulay and have a Cohen-Macaulay dual. In Chapter 2, we will discuss the motion planning problem in complex hyperplane arrangement complements. The difficulty of constructing a minimally discontinuous motion planning algorithm for a topological space X is measured by an integer invariant of X called topological complexity or TC(X). Yuzvinsky developed a combinatorial criterion for hyperplane arrangement complements which guarantees that their topological complexity is as large as possible. Applying this criterion in the special case when the arrangement is graphic, we simplify the criterion to an inequality on the edge density of the graph which is closely related to the inequality in the arboricity theorem of Nash-Williams.
Graduation Semester
2017-05
Type of Resource
text
Permalink
http://hdl.handle.net/2142/97758
Copyright and License Information
Copyright 2017 Nathan Fieldsteel

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