Graduation Year

2020

Date of Submission

12-2019

Document Type

Campus Only Senior Thesis

Degree Name

Bachelor of Arts

Department

Mathematics

Reader 1

Lenny Fukshansky

Terms of Use & License Information

Terms of Use for work posted in Scholarship@Claremont.

Abstract

This thesis explores several problems in discrete geometry, focusing on covering problems. We first go over some well known results, explaining Keith Ball's solution to the symmetric Tarski plank problem, as well as results of Alon and F\"uredi on covering all but vertices of a cube with hyperplanes. The former extensively utilizes techniques from matrix analysis, and the latter applies polynomial method. We state and explore the related problem, asking for the number of parallel hyperplanes required to cover a given discrete set of points in $\mathbb{Z}^{d}$ whose entries are bounded, and prove that there exist sets which are ``difficult'' to cover in every dimension for entries whose absolute values are bounded by~1 using a similar polynomial-based approach.

This thesis is restricted to the Claremont Colleges current faculty, students, and staff.

Share

COinS