Proofs and Confirmations
Proofs and Confirmations
This introduction to recent developments in algebraic combinatorics illustrates how research in mathematics actually progresses. The author recounts the dramatic search for and discovery of a proof of a counting formula conjectured in the late 1970s: the number of n x n alternating sign matrices, objects that generalize permutation matrices. While it was apparent that the conjecture must be true, the proof was elusive. As a result, researchers became drawn to this problem and made connections to aspects of the invariant theory of Jacobi, Sylvester, Cayley, MacMahon, Schur, and Young; to partitions and plane partitions; to symmetric functions; to hypergeometric and basic hypergeometric series; and, finally, to the six-vertex model of statistical mechanics. This volume is accessible to anyone with a knowledge of linear algebra, and it includes extensive exercises and Mathematica programs to help facilitate personal exploration. Students will learn what mathematicians actually do in an interesting and new area of mathematics, and even researchers in combinatorics will find something unique within Proofs and Confirmations.
- Requires only linear algebra as preparation but leads the reader to current research at the boundary of algebraic combinatorics and statistical mechanics
- Uses contemporary mathematics to illustrate the nature of mathematical research to those beyond the research community
- Includes extensive exercises and Mathematica programs to facilitate personal exploration
Reviews & endorsements
"Bressoud has created a beautiful new genre of mathematical exposition. It is neither popular mathematics, nor textbook, nor research monograph, nor problem book. It is all these and much more: a historical novel, a detective story and, implicitly, a philosophical manifesto. Yet the mathematics is deep, and all the proofs are complete...Proofs and Confirmations is destined to be a classic." American Mathematical Monthly
"Bressoud's book provides an opportunity to learn about all the mainstays of combinatorics like partitions, lattice paths, plane partitions, and hypergeometric functions by tracing a narrative that reads like a taut detective novel." Choice
"Bressoud's book provides an opportunity to learn about all the mainstays of combinatorics like partitions, lattice paths, plane partitions, and hypergeometric functions by tracing a narrative that reads like a taut detective novel." Choice
"Bressoud has done a very nice job of presenting us with a readable book which delivers a self-contained look at some current mathematics. And he's done a wonderful job at exposing the flavor of research mathematics. Take a look." MAA Online
"the book will appeal to anyone who likes algebra and combinatorics, and is curious as to what is currently going on at intersection of these two disciplines." William Gasarch
Product details
August 1999Paperback
9780521666466
292 pages
229 × 152 × 17 mm
0.43kg
49 b/w illus. 307 exercises
Available
Table of Contents
- 1. The conjecture
- 2. Fundamental structures
- 3. Lattice paths and plane partitions
- 4. Symmetric functions
- 5. Hypergeometric series
- 6. Explorations
- 7. Square ice.
