Polynomial Methods in Combinatorics 1st editon by Larry Guth ISBN 1470428903 978-1470428907

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ISBN 10: 1470428903
ISBN 13: 978-1470428907
Author: Larry Guth

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.

Polynomial Methods in Combinatorics 1st Table of contents:

Chapter 1. Introduction 
1.1. Incidence geometry 
1.2. Connections with other areas 
1.3. Outline of the book 
1.4. Other connections between polynomials and combinatorics 
1.5. Notation 
Chapter 2. Fundamental examples of the polynomial method 
2.1. Parameter counting arguments 
2.2. The vanishing lemma 
2.3. The finite-field Nikodym problem 
2.4. The finite field Kakeya problem 
2.5. The joints problem 
2.6. Comments on the method 
2.7. Exercises 
Chapter 3. Why polynomials? 
3.1. Finite field Kakeya without polynomials 
3.2. The Hermitian variety 
3.3. Joints without polynomials 
3.4. What is special about polynomials? 
3.5. An example involving polynomials 
3.6. Combinatorial structure and algebraic structure 
Chapter 4. The polynomial method in error-correcting codes 
4.1. The Berlekamp-Welch algorithm 
4.2. Correcting polynomials from overwhelmingly corrupted data 
4.3. Locally decodable codes
4.4. Error-correcting codes and finite-field Nikodym 
4.5. Conclusion and exercises 
Chapter 5. On polynomials and linear algebra in combinatorics 
Chapter 6. The Bezout theorem 
6.1. Proof of the Bezout theorem 
6.2. A Bezout theorem about surfaces and lines 
6.3. Hilbert polynomials 
Chapter 7. Incidence geometry 
7.1. The Szemer´edi-Trotter theorem 
7.2. Crossing numbers and the Szemer´edi-Trotter theorem 
7.3. The language of incidences 
7.4. Distance problems in incidence geometry 
7.5. Open questions 
7.6. Crossing numbers and distance problems 
Chapter 8. Incidence geometry in three dimensions 
8.1. Main results about lines in R3 
8.2. Higher dimensions 
8.3. The Zarankiewicz problem 
8.4. Reguli 
Chapter 9. Partial symmetries 
9.1. Partial symmetries of sets in the plane 
9.2. Distinct distances and partial symmetries 
9.3. Incidence geometry of curves in the group of rigid motions 
9.4. Straightening coordinates on G 
9.5. Applying incidence geometry of lines to partial symmetries 
9.6. The lines of L(P) don’t cluster in a low degree surface 
9.7. Examples of partial symmetries related to planes and reguli 
9.8. Other exercises 
Chapter 10. Polynomial partitioning 
10.1. The cutting method 
10.2. Polynomial partitioning 
10.3. Proof of polynomial partitioning 
10.4. Using polynomial partitioning 
10.5. Exercises 
10.6. First estimates for lines in R3 
10.7. An estimate for r-rich points 
10.8. The main theorem 
Chapter 11. Combinatorial structure, algebraic structure,
and geometric structure 
11.1. Structure for configurations of lines with many 3-rich points 
11.2. Algebraic structure and degree reduction 
11.3. The contagious vanishing argument 
11.4. Planar clustering 
11.5. Outline of the proof of planar clustering 
11.6. Flat points 
11.7. The proof of the planar clustering theorem 
11.8. Exercises 
Chapter 12. An incidence bound for lines in three dimensions 
12.1. Warmup: The Szemer´edi-Trotter theorem revisited 
12.2. Three-dimensional incidence estimates 
Chapter 13. Ruled surfaces and projection theory 
13.1. Projection theory 
13.2. Flecnodes and double flecnodes 
13.3. A definition of almost everywhere 
13.4. Constructible conditions are contagious 
13.5. From local to global 
13.6. The proof of the main theorem 
13.7. Remarks on other fields 
13.8. Remarks on the bound L3/2 
13.9. Exercises related to projection theory 
13.10. Exercises related to differential geometry 
Chapter 14. The polynomial method in differential geometry 
14.1. The efficiency of complex polynomials 
14.2. The efficiency of real polynomials 
14.3. The Crofton formula in integral geometry 
14.4. Finding functions with large zero sets 
14.5. An application of the polynomial method in geometry 
Chapter 15. Harmonic analysis and the Kakeya problem 
15.1. Geometry of projections and the Sobolev inequality 
15.2. Lp estimates for linear operators 
15.3. Intersection patterns of balls in Euclidean space 
15.4. Intersection patterns of tubes in Euclidean space 
15.5. Oscillatory integrals and the Kakeya problem 
15.6. Quantitative bounds for the Kakeya problem 
15.7. The polynomial method and the Kakeya problem 
15.8. A joints theorem for tubes 
15.9. Hermitian varieties 
Chapter 16. The polynomial method in number theory 
16.1. Naive guesses about diophantine equations 
16.2. Parabolas, hyperbolas, and high degree curves 
16.3. Diophantine approximation 
16.4. Outline of Thue’s proof 
16.5. Step 1: Parameter counting 
16.6. Step 2: Taylor approximation 
16.7. Step 3: Gauss’s lemma 
16.8. Conclusio

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