Some problems in polynomial interpolation and topological complexity

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2017

Author dc:creator

• Fieldsteel, Nathan Mulvey

Contributors dc:contributor

• Schenck, Hal
• Nevins, Tom
• Baryshnikov, Yuliy
• Hirani, Anil

• Approximation theory
• Polynomial interpolation
• Motion planning

Statement dc:rights

• Copyright 2017 Nathan Fieldsteel

Language dc:language

en

Handle dc:identifier

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

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/97758