A Primer for Undergraduate Research: From Groups and Tiles to Frames and Vaccines (Foundations for Undergraduate Research in Mathematics)

"This highly readable book aims to ease the many challenges of starting undergraduate research. It accomplishes this by presenting a diverse series of self-contained, accessible articles which include specific open problems and prepare the reader to tackle them with ample background material and references. Each article also contains a carefully selected bibliography for further reading. The content spans the breadth of mathematics, including many topics that are not normally addressed by the undergraduate curriculum (such as matroid theory, mathematical biology, and operations research), yet have few enough prerequisites that the interested student can start exploring them under the guidance of a faculty member. Whether trying to start an undergraduate thesis, embarking on a summer REU, or preparing for graduate school, this book is appropriate for a variety of students and the faculty who guide them"--Publisher's website Contents......Page 6 1 Introduction......Page 7 2 Group Presentations and Graphs......Page 8 2.1.1 A Constructive Approach......Page 9 2.2 Some Basic Graph Theory......Page 10 2.3 Cayley Graphs for Finitely Presented Groups......Page 11 3 Coxeter Groups......Page 13 3.1 The Presentation of a Coxeter Group......Page 14 3.2 Coxeter Groups and Geometry......Page 16 3.2.3 Hyperbolic Geometry and Reflections......Page 17 3.2.4 The Poincaré Disk Model for Hyperbolic Space......Page 18 4 Group Actions on Complexes......Page 20 4.1 CW-Complexes......Page 21 4.2 Group Actions on CW-Complexes......Page 24 5 The Cellular Actions of Coxeter Groups: The Davis Complex......Page 29 5.1.2 The Strict Fundamental Domain......Page 30 5.3 The Mirror Cellulation of Σ......Page 32 5.4 The Coxeter Cellulation......Page 33 5.4.1 Euclidean Representations......Page 34 5.4.2 The Coxeter Cell of Type T......Page 35 6 Closing Remarks and Suggested Projects......Page 36 References......Page 39 1 Introduction......Page 40 2.1 Realizing A4 as a Group of Rotations......Page 42 2.2 Preliminary Examples......Page 44 2.3 Signatures......Page 46 2.4 Generating Vectors and Riemann's Existence Theorem......Page 49 3.1 Signatures for A4-Actions......Page 52 4 Embeddable A4-Actions......Page 55 4.1 Necessary and Sufficient Conditions for Embeddability of A4......Page 56 5 Suggested Projects......Page 60 References......Page 63 1 Prologue......Page 65 2 Tiling Basics......Page 67 3 Tile Invariants......Page 69 3.1 Coloring Invariants......Page 70 3.2 Boundary Word Invariants......Page 71 3.3 Invariants from Local Connectivity......Page 74 3.4 The Tile Counting Group......Page 76 4 Tile Invariants and Tileability......Page 78 5 Enumeration......Page 83 6 Concluding Remarks......Page 85 References......Page 87 1 Introduction......Page 89 2 Properties with Known Kuratowski Set......Page 91 3 Strongly Almost–Planar Graphs......Page 94 4 Additional Project Ideas......Page 98 References......Page 100 1 Introduction......Page 102 1.1 Trees and Forests......Page 103 1.3 Edge Coloring and Total Coloring......Page 105 2.1 Forests with Game Chromatic Number 2......Page 106 2.2 Smallest Tree with Game Chromatic Number 4......Page 107 3 Relaxed-Coloring Games......Page 111 4 The Clique-Relaxed Game......Page 114 5 Edge Coloring......Page 117 6 Total Coloring......Page 123 7 Conclusions and Problems to Consider......Page 125 References......Page 128 1 Introduction......Page 130 2 Graphs......Page 131 2.1 UPC Graphs......Page 132 3.1 Introduction to Matroids......Page 134 3.2 The Cycle and Vector Matroids......Page 135 3.3 Beyond the Basics......Page 137 4.1 UPC Matroids......Page 139 4.2 Binary Matroids......Page 141 4.3 Binary UPC Matroids......Page 143 5 Further Reading......Page 146 References......Page 147 1 Introduction......Page 148 2 Frames in Finite Dimensional Spaces......Page 154 3 Equiangular Tight Frames......Page 162 3.1 k-Angle Tight Frames......Page 163 3.2 Tight Frames and Graphs......Page 165 4.2 k-Angle Tight Frames and Regular Graphs......Page 167 4.3 Frame Design Issues......Page 170 4.5 Reconstruction in Presence of Random Noise......Page 171 References......Page 173 1 Introduction......Page 174 2.1 Diet Problem......Page 176 2.2 Standard Forms......Page 177 2.3 Solution of LPs......Page 179 2.3.1 The Simplex Method......Page 180 2.4 Transportation Problem......Page 181 2.5 Duality......Page 183 3 Convex Programming......Page 184 3.1 Convex Sets......Page 185 3.2 Convex Functions......Page 186 3.3 Classes of Convex Programs......Page 188 References......Page 193 1 Introduction......Page 195 2.1 History......Page 199 2.2 Technical Background and Definitions......Page 200 3 Learning Through Examples......Page 205 4 Open Problems......Page 221 References......Page 223 1 Introduction......Page 225 1.1 Modeling Infectious Diseases and Contact Networks......Page 226 1.1.2 The Basic Reproductive Number......Page 228 1.1.3 Models of Contact Networks: Graphs......Page 229 1.1.4 Network-Based Models......Page 231 2.1 Random Graphs......Page 232 2.2 Definition of Erdős-Rényi Random Graphs......Page 233 2.3 Properties That Hold Asymptotically Almost Surely......Page 234 2.4 Exploring the Connected Components of Erdős-Rényi Random Graphs......Page 235 3.1 A Motivating Example......Page 238 3.2 Definitions of Clustering Coefficients......Page 240 3.4 A General Theorem About Strong Clustering......Page 243 4.1 Milgram's Famous ``Six Degrees of Separation'' Experiment......Page 244 4.2 The IONTW Guide to the Small World (Property)......Page 245 5.1 The Small-World Property and Small-World Networks......Page 250 5.2 Small-World Models......Page 251 5.3.1 Small World Models in IONTW......Page 252 5.3.2 Mathematical Derivations of Some Properties of GSWdim(N, d, λ)......Page 253 5.4 Vaccination Strategies in Small-World Models......Page 255 6 Suggested Research Projects......Page 260 Appendix: Hints for Selected Exercises......Page 263 References......Page 265 1 Introduction......Page 267 2.1 Overview of Differential Equations......Page 268 2.2 Overview of Numerical Methods......Page 271 2.3 Overview of Software......Page 276 3 Error Analysis......Page 278 3.1 Verifying Accuracy......Page 279 3.2 Convergence......Page 282 3.3 Stability......Page 283 3.4 Oscillatory Behavior......Page 288 4.1 Steady-State Solutions......Page 291 4.2 Newton's Method......Page 292 4.3 Traveling Wave Solutions......Page 297 5 Further Investigations......Page 298 5.1 Proof of Method Accuracy......Page 299 5.2 Main Program for Graphing Numerical Solutions......Page 301 5.3 Support Program for Setting up Numerical Analysis of PDE......Page 302 5.5 Support Program for Verifying Method Accuracy......Page 303 References......Page 304 Index......Page 306