Index a priori estimate, 185 active transformation, 54 Archimedean principle, 133 Arzel´ a-Ascoli diagonalisation trick, 139 asymptotic notation, xii Balog-Szemer´ edi-Gowers lemma, 230 barrier, 107 Bayes’ formula, 236 Bayesian probability, 236 Bernoulli numbers, 96 Bolzano-Weierstrass theorem, 160 Burgers’ equation, 185 busy beaver function, 27 Cantor’s theorem, 21, 32 Cartan-Killing form, 211 Cayley graph, 57 cell decomposition, 121 characteristic subgroup, 67 Christoffel symbols, 205 cogeodesic flow, 205 continuity method, 232 coordinate system, 222 decomposition into varieties, 163 difference equation, 186 differentiating the equation, 185 direct product, 66 Duhamel’s formula, 182 elemengary convergence, 154 energy, 197 equipartition of energy, 203 Erd˝ os distance problem, 110 Euclid’s theorem, 19 Euler equations of incompressible fluids, 213 Euler-Arnold equation, 209 Euler-Maclaurin formula, 98 explicit formula, 102 extension problem, 124 Faulhaber formula, 90 finitely generated group, 57 friendship paradox, 229 Furstenberg correspondence principle, 137 Furstenberg recurrence theorem, 143, 164 G-space, 51 G¨ odel incompleteness theorem, 25 G¨ odel sentence, 24 G¨ odel’s universe, 34 Grandi’s series, 91 Gromov’s theorem, 69, 140 growth function, 105 harmonic function, 70 Heine-Borel theorem, 161 hereditary property, 68 homogeneous space, 51 impredicativity of truth, 24 indicator function, xii interesting number paradox, 31 invariant subspace problem, 104 255

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