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This is a detailed introduction to the new polynomial methods responsible for numerous major mathematical breakthroughs in the past decade. It requires a minimal background and includes many examples, warm-up proofs, and exercises, allowing graduate and advanced undergraduate students to get to grips with an active and exciting research front.
List of contents
Introduction; 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.
About the author
Adam 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.
Summary
This is a detailed introduction to the new polynomial methods responsible for numerous major mathematical breakthroughs in the past decade. It requires a minimal background and includes many examples, warm-up proofs, and exercises, allowing graduate and advanced undergraduate students to get to grips with an active and exciting research front.
