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Title:  Some problems in polynomial interpolation and topological complexity 
Author(s):  Fieldsteel, Nathan Mulvey 
Director of Research:  Schenck, Hal 
Doctoral Committee Chair(s):  Nevins, Tom 
Doctoral Committee Member(s):  Baryshnikov, Yuliy; Hirani, Anil 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(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 CohenMacaulay and have a CohenMacaulay 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 NashWilliams. 
Issue Date:  20170421 
Type:  Text 
URI:  http://hdl.handle.net/2142/97758 
Rights Information:  Copyright 2017 Nathan Fieldsteel 
Date Available in IDEALS:  20170810 
Date Deposited:  201705 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois