Softcover ISBN:  9781470428907 
Product Code:  ULECT/64 
List Price:  $48.00 
MAA Member Price:  $43.20 
AMS Member Price:  $38.40 
Electronic ISBN:  9781470432140 
Product Code:  ULECT/64.E 
List Price:  $48.00 
MAA Member Price:  $43.20 
AMS Member Price:  $38.40 

Book DetailsUniversity Lecture SeriesVolume: 64; 2016; 273 ppMSC: Primary 05;
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 errorcorrecting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and farreaching, it should be accessible to first and secondyear graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book.
The subject is a beautiful one that has seen contributions by many leading mathematicians, including the author. The applications of the polynomial method covered in the book are quite wideranging, and run from combinatorial geometry to diophantine equations.
One particular feature of the book—and of the author's writing more generally—is the very inviting and discursive style, emphasising ideas where possible. This is *not* a dry monograph.—Ben Joseph Green, Oxford University
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”
ReadershipGraduate students and research mathematicians interested in combinatorial incidence geometry, algebraic geometry, and harmonic analysis.

Table of Contents

Chapters

Chapter 1. Introduction

Chapter 2. Fundamental examples of the polynomial method

Chapter 3. Why polynomials?

Chapter 4. The polynomial method in errorcorrecting 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


Additional Material

Reviews

One of the strengths that combinatorial problems have is that they are understandable to nonexperts in the field...One of the strengths that polynomials have is that they are well understood by mathematicians in general. Larry Guth manages to exploit both of those strengths in this book and provide an accessible and enlightening drive through a selection of combinatorial problems for which polynomials have been used to great effect.
Simeon Ball, Jahresbericht der Deutschen MathematikerVereinigung 
In the 273 page long book, a huge number of concepts are presented, and many results concerning them are formulated and proved. The book is a perfect presentation of the theme.
Béla Uhrin, Mathematical Reviews


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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 errorcorrecting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and farreaching, it should be accessible to first and secondyear graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book.
The subject is a beautiful one that has seen contributions by many leading mathematicians, including the author. The applications of the polynomial method covered in the book are quite wideranging, and run from combinatorial geometry to diophantine equations.
One particular feature of the book—and of the author's writing more generally—is the very inviting and discursive style, emphasising ideas where possible. This is *not* a dry monograph.
—Ben Joseph Green, Oxford University
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.

Chapters

Chapter 1. Introduction

Chapter 2. Fundamental examples of the polynomial method

Chapter 3. Why polynomials?

Chapter 4. The polynomial method in errorcorrecting 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

One of the strengths that combinatorial problems have is that they are understandable to nonexperts in the field...One of the strengths that polynomials have is that they are well understood by mathematicians in general. Larry Guth manages to exploit both of those strengths in this book and provide an accessible and enlightening drive through a selection of combinatorial problems for which polynomials have been used to great effect.
Simeon Ball, Jahresbericht der Deutschen MathematikerVereinigung 
In the 273 page long book, a huge number of concepts are presented, and many results concerning them are formulated and proved. The book is a perfect presentation of the theme.
Béla Uhrin, Mathematical Reviews