Some problems in polynomial interpolation and topological complexity

Fieldsteel, Nathan Mulvey

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

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

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

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

Copyright and License Information

Copyright 2017 Nathan Fieldsteel

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