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