MATHEMATICS++: SELECTED TOPICS BEYOND THE BASIC COURSES. VOLUME: 75

Mathematics++ is a concise introduction to six selected areas of 20th century mathematics providing numerous modern mathematical tools used in contemporary research in computer science, engineering, and other fields. The areas are: measure theory, high-dimensional geometry, Fourier analysis, representations of groups, multivariate polynomials, and topology. For each of the areas, the authors introduce basic notions, examples, and results. The presentation is clear and accessible, stressing intuitive understanding, and it includes carefully selected exercises as an integral part. Theory is complemented by applications--some quite surprising--in theoretical computer science and discrete mathematics. The chapters are independent of one another and can be studied in any order.

It is assumed that the reader has gone through the basic mathematics courses. Although the book was conceived while the authors were teaching Ph.D. students in theoretical computer science and discrete mathematics, it will be useful for a much wider audience, such as mathematicians specializing in other areas, mathematics students deciding what specialization to pursue, or experts in engineering or other fields.

Readership
Graduate students and research mathematicians interested in theoretical computer science and discrete mathematics.

Authors
Ida Kantor, Charles University, Prague, Czech Republic.
Jirí Matoušek, Charles University, Prague, Czech Republic, and ETH, Zurich, Switzerland.
Robert Šámal, Charles University, Prague, Czech Republic.


Contents
Preface ix
Chapter 1. Measure and Integral 1
§1. Measure 6
§2. The Lebesgue Integral 22
§3. Foundations of Probability Theory 31
§4. Literature 36
Bibliography 37
Chapter 2. High-Dimensional Geometry and
Measure Concentration 39
§1. Peculiarities of Large Dimensions 41
§2. The Brunn–Minkowski Inequality and Euclidean
Isoperimetry 44
§3. The Standard Normal Distribution and the Gaussian
Measure 53
§4. Measure Concentration 61
§5. Literature 81
Bibliography 82
Chapter 3. Fourier Analysis 85
§1. Characters 87
§2. The Fourier Transform 94
§3. Two Unexpected Applications 99
§4. Convolution 106
§5. Poisson Summation Formula 109
§6. Influence of Variables 113
§7. Infinite Groups 124
§8. Literature 137
Bibliography 138
Chapter 4. Representations of Finite Groups 141
§1. Basic Definitions and Examples 142
§2. Decompositions into Irreducible Representations 145
§3. Irreducible Decompositions, Characters, Orthogonality 150
§4. Irreducible Representations of the Symmetric Group 160
§5. An Application in Communication Complexity 163
§6. More Applications and Literature 168
Bibliography 169
Chapter 5. Polynomials 173
§1. Rings, Fields, and Polynomials 173
§2. The Schwartz–Zippel Theorem 175
§3. Polynomial Identity Testing 176
§4. Interpolation, Joints, and Contagious Vanishing 180
§5. Varieties, Ideals, and the Hilbert Basis Theorem 185
§6. The Nullstellensatz 188
§7. B´ezout’s Inequality in the Plane 195
§8. More Properties of Varieties 200
§9. B´ezout’s Inequality in Higher Dimensions 219
§10. Bounding the Number of Connected Components 226
§11. Literature 232
Bibliography 232
Chapter 6. Topology 235
§1. Topological Spaces and Continuous Maps 236
§2. Bits of General Topology 240
§3. Compactness 247
§4. Homotopy and Homotopy Equivalence 253
§5. The Borsuk–Ulam Theorem 257
§6. Operations on Topological Spaces 262
§7. Simplicial Complexes and Relatives 271
§8. Non-embeddability 283
§9. Homotopy Groups 289
§10. Homology of Simplicial Complexes 301
§11. Simplicial Approximation 309
§12. Homology Does Not Depend on Triangulation 314
§13. A Quick Harvest and Two More Theorems 318
§14. Manifolds 321
§15. Literature 330
Bibliography 330
Index 333