Polynomial Methods in Combinatorics

Häftad, Engelska, 2016

815 kr

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Beskrivning

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 Erdos'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.

Produktinformation

Utgivningsdatum:2016-05-30
Mått:178 x 254 x undefined mm
Vikt:505 g
Format:Häftad
Språk:Engelska
Serie:University Lecture Series
Antal sidor:273
Förlag:American Mathematical Society
ISBN:9781470428907

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Mer om författaren

Larry Guth, Massachusetts Institute of Technology, Cambridge, MA, USA.

Recensioner i media

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 Erdos 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"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 "One of the strengths that combinatorial problems have is that they are understandable to non-experts 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 Mathematiker-Vereinigung

Innehållsförteckning

IntroductionFundamental examples of the polynomial methodWhy polynomials?The polynomial method in error-correcting codesOn polynomials and linear algebra in combinatoricsThe Bezout theoremIncidence geometryIncidence geometry in three dimensionsPartial symmetriesPolynomial partitioningCombinatorial structure, algebraic structure, and geometric structureAn incidence bound for lines in three dimensionsRuled surfaces and projection theoryThe polynomial method in differential geometryHarmonic analysis and the Kakeya problemThe polynomial method in number theoryBibliography