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A Primer for Undergraduate Research

Part of the Foundations for Undergraduate Research in Mathematics Book Series

Editors: Wootton, Aaron; Peterson, Valerie; Lee, Christopher

DOI: 10.1007/978-3-319-66065-3

ISBN: 978-3-319-66064-6

Publisher: Springer International Publishing - Springer Journals

Jan 6, 2018

1 - 10 of 11 Chapters

Available

Coxeter Groups and the Davis Complex

Schroeder, Timothy A.

Pages: 1 –33

[We take a constructive and active look at group theory, by focusing on the action of finitely presented groups on CW-complexes. In particular, we focus on the action of Coxeter groups on the so-called Davis complex. Students are invited to participate in several constructions and investigate the...
Available

A Tale of Two Symmetries: Embeddable and Non-embeddable Group Actions on Surfaces

Peterson, Valerie; Wootton, Aaron

Pages: 35 –59

[Plainly speaking, a compact Riemann surface, S, can be thought of as the layer of glaze on a g-holed doughnut. A group of symmetries of S is a group that acts on S while preserving some of its underlying structure. We provide an easily understood exposition of the modern techniques used to...
Available

Tile Invariants for Tackling Tiling Questions

Hitchman, Michael P.

Pages: 61 –84

[This article provides an introduction to techniques developed to address three types of tiling questions in the integer lattice: questions of tileability - which regions in a collectionℛ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}...
Available

Forbidden Minors: Finding the Finite Few

Mattman, Thomas W.

Pages: 85 –97

[The Graph Minor Theorem of Robertson and Seymour associates, to any graph property whatsoever, a finite, characteristic list of graphs. We view this as an impressive generalization of Kuratowski’s theorem, which defines planarity in terms of two forbidden subgraphs, K5 and K3,3. Robertson and...
Available

Introduction to Competitive Graph Coloring

Dunn, C.; Larsen, V.; Nordstrom, J. F.

Pages: 99 –126

[We consider a collection of coloring games played on finite graphs. These games all involve two players, Alice and Bob, alternating coloring the vertices (or edges or both) from a finite set of colors. At each step, the players must use legal colors, where the definition of legal depends on...
Available

Matroids

McNicholas, Erin; Neudauer, Nancy Ann; Starr, Colin

Pages: 127 –144

[Graph Theory is an accessible topic for undergraduate research. Matroids are a generalization of graphs that, while more abstract, provide projects of reasonable scope for undergraduates. In this chapter, we introduce graphs and matroids, then propose several research questions focused around a...
Available

Finite Frame Theory

Datta, Somantika; Oldroyd, Jesse

Pages: 145 –170

[Given a signal, whether it is a discrete vector or a continuous function, one desires to write it in terms of simpler components. Typically, these components or “building blocks” form what is called a basis. A basis is an optimal set, containing the minimal number of elements needed to uniquely...
Available

Mathematical Decision-Making with Linear and Convex Programming

Kotas, Jakob

Pages: 171 –191

[Operations research (OR) is a branch of applied mathematics dedicated to analytical decision-making, and has close ties with optimization, statistics, and mathematical modeling. As a discipline, it is often housed in schools of engineering or business, and undergraduate students of mathematics...
Available

Computing Weight Multiplicities

Harris, Pamela E.

Pages: 193 –222

[Central to the study of the representation theory of Lie algebras is the computation of weight multiplicities, which are the dimensions of vector subspaces called weight spaces. The multiplicity of a weight can be computed using a well-known formula of Kostant that consists of an alternating sum...
Available

Vaccination Strategies for Small Worlds

Just, Winfried; Highlander, Hannah Callender

Pages: 223 –264

[Infectious diseases are a serious threat to our health. Vaccination often can prevent their spread, but typically it is not feasible to vaccinate absolutely everyone. Sometimes it is necessary to carefully target the group of individuals to whom a limited supply of vaccine should be administered...

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