Theorems of the 21st century. Vol.1

Preface......Page 5
Acknowledgements......Page 7
Contents......Page 9
Acronyms......Page 14
 1.1 Moderate Deviations for the Volume of the Wiener Sausage......Page 16
 1.2 The Minimal Average Value of a Bounded Multiplicative Function......Page 21
 1.3 Counting Integer Solutions of Some Inequalities......Page 25
 1.4 On the Arithmetic Difference of Regular Cantor Sets......Page 29
 1.5 The Existence of Different Groups with Isomorphic Group Rings......Page 33
 1.6 Short Representations of Elements of Finite Simple Groups......Page 38
 1.7 The Existence of a Field with u-Invariant 9......Page 41
 2.1 Counting Rational Functions with Given Critical Points......Page 46
 2.2 Representing Braids as Matrices......Page 50
 2.3 Explicit Expander Constructions Using the Zig-Zag Product......Page 53
 2.4 Elliptic Curves Over Function Fields Can Have Arbitrarily Large Rank......Page 57
 2.5 The Optimality of the Standard Double Bubble......Page 61
 2.6 Counting Integer Solutions of Equations in Three Variables......Page 64
 2.7 The Regular-Stochastic Dichotomy for Quadratic Polynomials......Page 68
 2.8 A Finitely Presented Group with an NP-Complete Word Problem......Page 73
 2.9 Finitely Generated Groups with a Word Problem in NP......Page 78
 2.10 Positive Noncommutative Polynomials  are Sums of Squares......Page 81
 2.11 The Only Space Isomorphic to Each of its Subspaces......Page 84
 2.12 Counting Matrices with Some Special Properties......Page 88
 2.13 Transforming Convex Bodies into Balls......Page 92
 3.1 On the Differentiability of Lipschitz Maps  on Infinite-Dimensional Spaces......Page 96
 3.2 Representing 1 as a Sum of Reciprocals of Selected Integers......Page 100
 3.3 The Optimal Hardy–Littlewood Maximal Inequality......Page 103
 3.4 Improved Upper Bounds on Sphere Packings......Page 107
 3.5 Sums Versus Products of Finite Sets of Integers......Page 110
 3.6 The Radius of Integer Points in the Plane......Page 113
 3.7 The Set of Nonergodic Directions Can Have Dimension 1/2......Page 117
 3.8 A Real Number Which Is Far from All Cubic Algebraic Integers......Page 121
 4.1 The Julia Set of Almost All Quadratic Polynomials is Locally Connected......Page 125
 4.2 The Regular Polygons have Minimal Logarithmic Capacity......Page 130
 4.3 On the Growth of the Diffusion Coefficient......Page 133
 4.4 On the Volume of the Intersection of Two Wiener Sausages......Page 137
 4.5 Controlling the Size of the Bilinear Hilbert Transform......Page 141
 4.6 Covering Convex Bodies by Balls......Page 144
 4.7 The Parametrization of Quartic Rings......Page 147
 4.8 The Time it Takes for a Random Walk to Cover the Plane......Page 151
 4.9 On the Geometry of the Uniform Spanning Forest......Page 155
 4.10 A Polynomial Time Algorithm for Primality Testing......Page 158
 5.1 Structured Additive Patterns in Sets of Positive Density......Page 163
 5.2 A Sharp Form of Whitney's Extension Theorem......Page 168
 5.3 Minimal Surfaces in 3-Space I......Page 171
 5.4 Statistical Properties of Quadratic Dynamics......Page 175
 5.5 Every Subset of Primes of Positive Density Contains a 3-Term Progression......Page 178
 5.6 Every Separable Infinite-Dimensional Banach Space Has Infinite Diameter......Page 182
 5.7 The NP-Hardness of the 1.36...-Approximation to the Minimum Vertex Cover......Page 186
 5.8 Embedding Large Subsets of Finite Metric Spaces into Euclidean Space......Page 190
 5.9 On the Number of Quartic Fields with Bounded Discriminant......Page 194
 5.10 Optimal Sphere Packing in Dimension 3......Page 197
 5.11 The Chromatic Number of a Random Graph......Page 201
 6.1 Sufficient Conditions for Completeness of a Set  of Integers......Page 205
 6.2 Counting Number Fields of Bounded Discriminant......Page 208
 6.3 Perfect Powers in Fibonacci and Lucas Sequences......Page 212
 6.4 A Characterization of Perfect Graphs......Page 215
 6.5 Littlewood's Conjecture Holds Outside of a Set of Dimension 0......Page 218
 6.6 The Connection Between Metric Entropy  and Combinatorial Dimension......Page 222
 6.7 On the Approximation of Real Numbers by Irreducible Fractions......Page 225
 7.1 Bounding the Error in Approximation of Smooth Functions by Polynomials......Page 230
 7.2 Superlinear Growth of Digit Patterns in the Decimal Expansion of Algebraic Numbers......Page 234
 7.3 The Existence of Intervals with Too Many and Too Few Primes......Page 237
 7.4 Interval Exchange Transformations Are Almost Always Weakly Mixing......Page 241
 7.5 The Hopf Condition Over Arbitrary Fields......Page 244
 7.6 Any Real Polynomial Can be Approximated by a Hyperbolic Polynomial......Page 248
 7.7 The Schinzel–Zassenhaus Conjecture for Polynomials with Odd Coefficients......Page 252
 7.8 Diophantine Approximation of Points on Smooth Planar Curves......Page 256
 7.9 The Hasse Principle for Systems of Two Diagonal Cubic Equations......Page 260
 7.10 An Effective Multidimensional Szemerédi Theorem......Page 263
 8.1 Minimal Surfaces in 3-Space II......Page 268
 8.2 Arbitrarily Long Arithmetic Progressions  of Prime Numbers......Page 272
 8.3 On Poincaré's Inequality......Page 275
 8.4 Representing Matrices in SL2(mathbbZ/pmathbbZ) Using a Small Number of Generators......Page 279
 8.5 The Cayley Graphs of SL2(mathbbFp) Form a Family  of Expanders......Page 282
 8.6 Determining the Shape of an Infected Region......Page 285
 8.7 A Negative Answer to Littlewood's Question About Zeros of Cosine Polynomials......Page 289
 8.8 The Kissing Number in Dimension Four is 24......Page 292
 8.9 A Criterion for Embedding Lp into Lq Uniformly......Page 296
 8.10 The Distribution of Integers with a Divisor  in a Given Interval......Page 299
 8.11 An Upper Bound for the Norm of the Inverse  of a Random Matrix......Page 304
 8.12 An Isoperimetric Inequality with Optimal Exponent......Page 307
 8.13 A Negative Answer to Maharam's Question......Page 311
 9.1 On De Giorgi's Conjecture in Dimension at Most 8......Page 315
 9.2 An Efficient Algorithm for Fitting a Smooth Function to Data......Page 319
 9.3 A Helicoid-Like Surface with a Hole......Page 322
 9.4 Bounding the Condition Number of Random Discrete Matrices......Page 326
 9.5 Characterizing the Legendre Transform of Convex Analysis......Page 330
 9.6 The Solution of the Ten Martini Problem......Page 333
 9.7 A Linear Time Algorithm for Edge-Deletion Problems......Page 337
 9.8 A Characterization of Stability-Preserving Linear Operators......Page 340
 9.9 On the Gaps Between Primes......Page 344
 9.10 A Proof of the B. and M. Shapiro Conjecture in Real Algebraic Geometry......Page 347
 9.11 Bounding Diagonal Ramsey Numbers......Page 351
 9.12 An Almost Optimal Upper Bound for Moments  of the Riemann Zeta Function......Page 354
 9.13 Optimal Lattice Sphere Packing in Dimension 24......Page 358
 9.14 A Waring-Type Theorem for Large Finite Simple Groups......Page 362
 10.1 Majority Votes Are the Most Stable......Page 366
 10.2 On Divisibility Properties of Dyson's Rank Partition Function......Page 371
 10.3 Exceptional Times in Percolation Theory......Page 375
 10.4 Polynomial Parametrization for the Solutions  of Diophantine Equations......Page 378
 10.5 The Order of Growth of Variance in the Asymmetric Simple Exclusion Process......Page 381
 10.6 Divergent Square Averages in Ergodic Dynamical Systems......Page 385
 10.7 Divisibility Properties of Sums of Digits of Prime Numbers......Page 390
 10.8 Prime Values of Linear Equations......Page 394
 10.9 Entropies of Multidimensional Shifts of Finite Type may be Impossible to Compute......Page 400
 10.10 Subgroups of 2-Generated Groups......Page 404
 10.11 On the Number of Quintic Fields with Bounded Discriminant......Page 409
 10.12 On the Negative Pell Equation......Page 414
 10.13 The Norms of Random Band Matrices......Page 419
11 Further Reading......Page 424
 BookmarkTitle:......Page 426
Author Index......Page 441
Index......Page 446