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E-raamat: Polynomial Methods in Combinatorics

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  • Formaat: 273 pages
  • Sari: University Lecture Series
  • Ilmumisaeg: 06-Sep-2016
  • Kirjastus: American Mathematical Society
  • ISBN-13: 9781470432140
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Polynomial Methods in CombinatoricsSuurem pilt
  • Formaat: 273 pages
  • Sari: University Lecture Series
  • Ilmumisaeg: 06-Sep-2016
  • Kirjastus: American Mathematical Society
  • ISBN-13: 9781470432140
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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 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.

Arvustused

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

Preface ix
Chapter 1 Introduction
1(8)
1.1 Incidence geometry
2(2)
1.2 Connections with other areas
4(2)
1.3 Outline of the book
6(1)
1.4 Other connections between polynomials and combinatorics
7(1)
1.5 Notation
7(2)
Chapter 2 Fundamental examples of the polynomial method
9(10)
2.1 Parameter counting arguments
9(1)
2.2 The vanishing lemma
10(1)
2.3 The finite-field Nikodym problem
11(1)
2.4 The finite field Kakeya problem
12(1)
2.5 The joints problem
13(2)
2.6 Comments on the method
15(2)
2.7 Exercises
17(2)
Chapter 3 Why polynomials?
19(18)
3.1 Finite field Kakeya without polynomials
19(3)
3.2 The Hermitian variety
22(5)
3.3 Joints without polynomials
27(5)
3.4 What is special about polynomials?
32(1)
3.5 An example involving polynomials
33(1)
3.6 Combinatorial structure and algebraic structure
34(3)
Chapter 4 The polynomial method in error-correcting codes
37(14)
4.1 The Berlekamp-Welch algorithm
37(3)
4.2 Correcting polynomials from overwhelmingly corrupted data
40(1)
4.3 Locally decodable codes
41(3)
4.4 Error-correcting codes and finite-field Nikodym
44(1)
4.5 Conclusion and exercises
45(6)
Chapter 5 On polynomials and linear algebra in combinatorics
51(4)
Chapter 6 The Bezout theorem
55(8)
6.1 Proof of the Bezout theorem
55(3)
6.2 A Bezout theorem about surfaces and lines
58(2)
6.3 Hilbert polynomials
60(3)
Chapter 7 Incidence geometry
63(22)
7.1 The Szemeredi-Trotter theorem
64(3)
7.2 Crossing numbers and the Szemeredi-Trotter theorem
67(4)
7.3 The language of incidences
71(4)
7.4 Distance problems in incidence geometry
75(1)
7.5 Open questions
76(3)
7.6 Crossing numbers and distance problems
79(6)
Chapter 8 Incidence geometry in three dimensions
85(14)
8.1 Main results about lines in R3
85(3)
8.2 Higher dimensions
88(2)
8.3 The Zarankiewicz problem
90(5)
8.4 Reguli
95(4)
Chapter 9 Partial symmetries
99(14)
9.1 Partial symmetries of sets in the plane
99(2)
9.2 Distinct distances and partial symmetries
101(2)
9.3 Incidence geometry of curves in the group of rigid motions
103(1)
9.4 Straightening coordinates on G
104(3)
9.5 Applying incidence geometry of lines to partial symmetries
107(1)
9.6 The lines of (P) don't cluster in a low degree surface
108(3)
9.7 Examples of partial symmetries related to planes and reguli
111(1)
9.8 Other exercises
112(1)
Chapter 10 Polynomial partitioning
113(24)
10.1 The cutting method
113(3)
10.2 Polynomial partitioning
116(1)
10.3 Proof of polynomial partitioning
117(4)
10.4 Using polynomial partitioning
121(1)
10.5 Exercises
122(4)
10.6 First estimates for lines in R3
126(2)
10.7 An estimate for r-rich points
128(1)
10.8 The main theorem
129(8)
Chapter 11 Combinatorial structure, algebraic structure, and geometric structure
137(14)
11.1 Structure for configurations of lines with many 3-rich points
137(2)
11.2 Algebraic structure and degree reduction
139(1)
11.3 The contagious vanishing argument
140(3)
11.4 Planar clustering
143(1)
11.5 Outline of the proof of planar clustering
144(1)
11.6 Flat points
145(3)
11.7 The proof of the planar clustering theorem
148(1)
11.8 Exercises
149(2)
Chapter 12 An incidence bound for lines in three dimensions
151(10)
12.1 Warmup: The Szemeredi-Trotter theorem revisited
152(2)
12.2 Three-dimensional incidence estimates
154(7)
Chapter 13 Ruled surfaces and projection theory
161(34)
13.1 Projection theory
164(8)
13.2 Flecnodes and double flecnodes
172(1)
13.3 A definition of almost everywhere
173(2)
13.4 Constructible conditions are contagious
175(1)
13.5 From local to global
176(7)
13.6 The proof of the main theorem
183(2)
13.7 Remarks on other fields
185(1)
13.8 Remarks on the bound L3/2
186(1)
13.9 Exercises related to projection theory
187(2)
13.10 Exercises related to differential geometry
189(6)
Chapter 14 The polynomial method in differential geometry
195(12)
14.1 The efficiency of complex polynomials
195(2)
14.2 The efficiency of real polynomials
197(1)
14.3 The Crofton formula in integral geometry
198(2)
14.4 Finding functions with large zero sets
200(1)
14.5 An application of the polynomial method in geometry
201(6)
Chapter 15 Harmonic analysis and the Kakeya problem
207(42)
15.1 Geometry of projections and the Sobolev inequality
207(4)
15.2 Lp estimates for linear operators
211(2)
15.3 Intersection patterns of balls in Euclidean space
213(5)
15.4 Intersection patterns of tubes in Euclidean space
218(4)
15.5 Oscillatory integrals and the Kakeya problem
222(10)
15.6 Quantitative bounds for the Kakeya problem
232(2)
15.7 The polynomial method and the Kakeya problem
234(4)
15.8 A joints theorem for tubes
238(2)
15.9 Hermitian varieties
240(9)
Chapter 16 The polynomial method in number theory
249(20)
16.1 Naive guesses about diophantine equations
249(2)
16.2 Parabolas, hyperbolas, and high degree curves
251(3)
16.3 Diophantine approximation
254(4)
16.4 Outline of Thue's proof
258(1)
16.5 Step 1: Parameter counting
259(4)
16.6 Step 2: Taylor approximation
263(2)
16.7 Step 3: Gauss's lemma
265(2)
16.8 Conclusion
267(2)
Bibliography 269
Larry Guth, Massachusetts Institute of Technology, Cambridge, MA, USA.
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