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OverviewThe past decade has seen numerous major mathematical breakthroughs for topics such as the finite field Kakeya conjecture, the cap set conjecture, Erdős's distinct distances problem, the joints problem, as well as others, thanks to the introduction of new polynomial methods. There has also been significant progress on a variety of problems from additive combinatorics, discrete geometry, and more. This book gives a detailed yet accessible introduction to these new polynomial methods and their applications, with a focus on incidence theory. Based on the author's own teaching experience, the text requires a minimal background, allowing graduate and advanced undergraduate students to get to grips with an active and exciting research front. The techniques are presented gradually and in detail, with many examples, warm-up proofs, and exercises included. An appendix provides a quick reminder of basic results and ideas. Full Product DetailsAuthor: Adam Sheffer (Bernard M. Baruch College, City University of New York)Publisher: Cambridge University Press Imprint: Cambridge University Press Edition: New edition Dimensions: Width: 15.70cm , Height: 2.20cm , Length: 23.50cm Weight: 0.540kg ISBN: 9781108832496ISBN 10: 1108832490 Pages: 260 Publication Date: 24 March 2022 Audience: College/higher education , Tertiary & Higher Education Format: Hardback Publisher's Status: Active Availability: Manufactured on demand  We will order this item for you from a manufactured on demand supplier. Table of ContentsIntroduction; 1. Incidences and classical discrete geometry; 2. Basic real algebraic geometry in R^2; 3. Polynomial partitioning; 4. Basic real algebraic geometry in R^d; 5. The joints problem and degree reduction; 6. Polynomial methods in finite fields; 7. The Elekes–Sharir–Guth–Katz framework; 8. Constant-degree polynomial partitioning and incidences in C^2; 9. Lines in R^3; 10. Distinct distances variants; 11. Incidences in R^d; 12. Incidence applications in R^d; 13. Incidences in spaces over finite fields; 14. Algebraic families, dimension counting, and ruled surfaces; Appendix. Preliminaries; References; Index.Reviews'This book gives a very nice introduction to the areas of incidence geometry and the polynomial method ... Since this area of mathematics is still rather young, the book contains many open problems - this helps to bring the reader to the front of research. Furthermore, each chapter is followed by a generous amount of exercises.' Audie Warren, zbMATH Open Author InformationAdam Sheffer is Mathematics Professor at CUNY's Baruch College and the CUNY Graduate Center. Previously, he was a postdoctoral researcher at the California Institute of Technology. Sheffer's research work is focused on polynomial methods, discrete geometry, and additive combinatorics. Tab Content 6Author Website:Countries AvailableAll regions |
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