Polynomial Methods in Combinatorics
About this Title
Larry Guth, Massachusetts Institute of Technology, Cambridge, MA
Publication: University Lecture Series
Publication Year: 2016; Volume 64
ISBNs: 978-1-4704-2890-7 (print); 978-1-4704-3214-0 (online)
MathSciNet review: MR3495952
MSC: Primary 05-01; Secondary 11B75, 13F20, 14N10, 52C10, 94B25
This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. One of the first results in this story is a short elegant solution of the Kakeya problem for finite fields, which was considered a deep and difficult problem in combinatorial geometry. The author also discusses in detail various problems in incidence geometry associated to Paul Erdős's famous distinct distances problem in the plane from the 1940s. The proof techniques are also connected to error-correcting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and far-reaching, it should be accessible to first- and second-year graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book.
Some of the greatest advances in geometric combinatorics and harmonic analysis in recent years have been accomplished using the polynomial method. Larry Guth gives a readable and timely exposition of this important topic, which is destined to influence a variety of critical developments in combinatorics, harmonic analysis and other areas for many years to come.
—Alex Iosevich, University of Rochester, author of “The Erdős Distance Problem” and “A View from the Top”
It is extremely challenging to present a current (and still very active) research area in a manner that a good mathematics undergraduate would be able to grasp after a reasonable effort, but the author is quite successful in this task, and this would be a book of value to both undergraduates and graduates.
—Terence Tao, University of California, Los Angeles, author of “An Epsilon of Room I, II” and “Hilbert's Fifth Problem and Related Topics”
Graduate students and research mathematicians interested in combinatorial incidence geometry, algebraic geometry, and harmonic analysis.
Table of Contents
- Chapter 1. Introduction
- Chapter 2. Fundamental examples of the polynomial method
- Chapter 3. Why polynomials?
- Chapter 4. The polynomial method in error-correcting codes
- Chapter 5. On polynomials and linear algebra in combinatorics
- Chapter 6. The Bezout theorem
- Chapter 7. Incidence geometry
- Chapter 8. Incidence geometry in three dimensions
- Chapter 9. Partial symmetries
- Chapter 10. Polynomial partitioning
- Chapter 11. Combinatorial structure, algebraic structure, and geometric structure
- Chapter 12. An incidence bound for lines in three dimensions
- Chapter 13. Ruled surfaces and projection theory
- Chapter 14. The polynomial method in differential geometry
- Chapter 15. Harmonic analysis and the Kakeya problem
- Chapter 16. The polynomial method in number theory