Difference Sets: Connecting Algebra, Combinatorics, and Geometry
About this Title
Emily H. Moore, Grinnell College, Grinnell, IA and Harriet S. Pollatsek, Mount Holyoke College, South Hadley, MA
Publication: The Student Mathematical Library
Publication Year 2013: Volume 67
ISBNs: 978-0-8218-9176-6 (print); 978-1-4704-0973-9 (online)
MathSciNet review: MR3077008
MSC: Primary 05-01; Secondary 05B10, 20C15, 94B99
Difference sets belong both to group theory and to combinatorics. Studying them requires tools from geometry, number theory, and representation theory. This book lays a foundation for these topics, including a primer on representations and characters of finite groups. It makes the research literature on difference sets accessible to students who have studied linear algebra and abstract algebra, and it prepares them to do their own research.
This text is suitable for an undergraduate capstone course, since it illuminates the many links among topics that the students have already studied. To this end, almost every chapter ends with a coda highlighting the main ideas and emphasizing mathematical connections. This book can also be used for self-study by anyone interested in these connections and concrete examples.
An abundance of exercises, varying from straightforward to challenging, invites the reader to solve puzzles, construct proofs, and investigate problems—by hand or on a computer. Hints and solutions are provided for selected exercises, and there is an extensive bibliography. The last chapter introduces a number of applications to real-world problems and offers suggestions for further reading.
Both authors are experienced teachers who have successfully supervised undergraduate research on difference sets.
Undergraduate students, graduate students, and research mathematicians interested in algebra and combinatorics.
Table of Contents
- Chapter 1. Introduction
- Chapter 2. Designs
- Chapter 3. Automorphisms of designs
- Chapter 4. Introducing difference sets
- Chapter 5. Bruck-Ryser-Chowla theorem
- Chapter 6. Multipliers
- Chapter 7. Necessary group conditions
- Chapter 8. Difference sets from geometry
- Chapter 9. Families from Hadamard matrices
- Chapter 10. Representation theory
- Chapter 11. Group characters
- Chapter 12. Using algebraic number theory
- Chapter 13. Applications
- Appendix A. Background
- Appendix B. Notation
- Appendix C. Hints and solutions to selected exercises