Inevitable Randomness in Discrete Mathematics
About this Title
József Beck, Rutgers, The State University of New Jersey, Piscataway, NJ
Publication: University Lecture Series
Publication Year 2009: Volume 49
ISBNs: 978-0-8218-4756-5 (print); 978-1-4704-1644-7 (online)
MathSciNet review: MR2543141
MSC: Primary 60C05; Secondary 05D10, 11K38, 60F05, 68Q87, 91A46
Mathematics has been called the science of order. The subject is remarkably good for generalizing specific cases to create abstract theories. However, mathematics has little to say when faced with highly complex systems, where disorder reigns. This disorder can be found in pure mathematical arenas, such as the distribution of primes, the $3n+1$ conjecture, and class field theory.
The purpose of this book is to provide examples—and rigorous proofs—of the complexity law:
(1) discrete systems are either simple or they exhibit advanced pseudorandomness;
(2) a priori probabilities often exist even when there is no intrinsic symmetry.
Part of the difficulty in achieving this purpose is in trying to clarify these vague statements. The examples turn out to be fascinating instances of deep or mysterious results in number theory and combinatorics.
This book considers randomness and complexity. The traditional approach to complexity—computational complexity theory—is to study very general complexity classes, such as P, NP and PSPACE. What Beck does is very different: he studies interesting concrete systems, which can give new insights into the mystery of complexity.
The book is divided into three parts. Part A is mostly an essay on the big picture. Part B is partly new results and partly a survey of real game theory. Part C contains new results about graph games, supporting the main conjecture. To make it accessible to a wide audience, the book is mostly self-contained.
Graduate students and research mathematicians interested in discrete mathematics, combinatorics, and number theory.
Table of Contents
Part A. Reading the shadows on the wall and formulating a vague conjecture
- Chapter 1. Complex systems
- Chapter 2. Collecting data: Apparent randomness of digit sequences
- Chapter 3. Collecting data: More randomness in number theory
- Chapter 4. Laplace and the principle of insufficient reason
- Chapter 5. Collecting proofs for the SLG conjecture
Part B. More evidence for the SLG conjecture: Exact solutions in real game theory
- Chapter 6. Ramsey theory and games
- Chapter 7. Practice session (I): More on Ramsey games and strategies
- Chapter 8. Practice session (II): Connectivity games and more strategies
- Chapter 9. What kind of games?
- Chapter 10. Exact solutions of games: Understanding via the equiprobability postulate
- Chapter 11. Equiprobability postulate with constraints (endgame policy)
- Chapter 12. Constraints and threshold clustering
- Chapter 13. Threshold clustering and a few bold conjectures
Part C. New evidence: Games and graphs, the surplus, and the square root law
- Chapter 14. Yet another simplification: Sparse hypergraphs and the surplus
- Chapter 15. Is surplus the right concept? (I)
- Chapter 16. Is surplus the right concept? (II)
- Chapter 17. Working with a game-theoretic partition function
- Chapter 18. An attempt to save the variance
- Chapter 19. Proof of theorem 1: Combining the variance with an exponential sum
- Chapter 20. Proof of theoem 2: The upper bound
- Chapter 21. Conclusion (I): More on theorem 1
- Chapter 22. Conclusion (II): Beyond the SLG conjecture