INDEX
abstract surfaces, 147–153
admissibility, 10–11
algebraic geometry, 74, 75, 83, 85, 91, 282–283
algebraic topology, 186
Anderson localization, 117, 118
approximation hardness results, 240–241
Aristotle, 255
arithmetic geometry, 181–183
Atiyah, Michael, 201–208
Avila, Artur, 163–171
Banach–Tarski paradox, 255–256
band gaps, 57
Bell, Eric Temple, 157
Bhargava, Manjul, 173–180
billiard ball trajectories, 152–154, 164
binary quadratic forms, 177
Birch and Swinnerton–Dyer conjecture, 180
bird eyes, 51–57
blind-taste-test protocol, 231
Boltzmann’s ergodic hypothesis, 169–170
Brownian map, 64–72
Calabi–Yau spaces, 83
Cantor, Georg, 247, 248, 255, 256, 262–263
cap sets, 129–133
category theory, 223
central limit theorem, 46
chaos, onset of, 167–168
clock arithmetics, 184–185
cohomology theories, 75
coin tossing, 24–26
commutative algebra, 89
complementarity, 85
complexity theory, 229, 233, 264–265
composition laws, 177–179
computer proof assistants, 219–221, 224
constraint satisfaction problems, 239
continuum hypothesis, 245–252, 263–265
Coq, 219–221
corner cases, 213
Cramér, Harald, 25
decision problem, 126
Defense Advanced Research Projects Agency (DARPA), 211, 216
Deligne, Pierre, 13
dependent-type theories, 215
derivatives, 98
drones, 217
dynamical systems, 152, 166, 169
Dyson, Freeman, 155–161
E8, 139–143
einstein (shape), 125–126
Einstein, Albert, 84
elliptic curves, 173, 179–180, 182
equilibrium systems, 54
Erdős, Paul, 135
Erdős discrepancy problem, 135–138
Erdős–Szemeredi sunflower conjecture, 133
error-free code, 212
Euler’s equation, 204
exact hardness results, 238–240
Fermat’s Last Theorem, 31, 142, 179, 182, 185, 187, 199, 256–258, 271
Feynman diagram, 73–79
Fibonacci sequence, 175
Filoche, Marcel, 118–122
finite–infinite divide, 253–259
forcing axioms, 245–247, 251–252
formal verification/specification, 211–217
Fourier analysis, 131
fractal structures, 181–183
Fractal Wall, 119
Galileo, 81
Galois group, 79
Galois representations, theory of, 271
game theory, 155
gamma distributions, 106
Gauss’ composition law, 177, 178
Gaussian correlation inequality (GCI), 103–107
Gaussian distribution, 44–46, 105–106
geometry of numbers, 179
Gödel, Kurt, 245, 248–250, 257
Goldin, Rebecca, 281–286
GPY (Goldston–Pintz–Yıldırım) paper, 5–7, 13–14, 15
graph isomorphism problem, 229–233
graph theory, 96–97
Grothendieck, Alexander, 75
hacker-proof code, 211–217
Hairer, Martin, 193–200
Harris–Taylor proof, 181
High-assurance building blocks, 216
High-Assurance Cyber Military Systems (HACMS) project, 211, 216
Hilbert’s program, 254–257
Hironaka, Heisuke, 91, 92, 101
Hodge index theorem, 100
Hodge theory, 98–101
homotopy, 222–226
HoTT community, 226
Huh, June, 89–102
hyperbolic geometry, 147–152, 185
hyperbolic three-space, 185
hyperuniformity, 51–57
ideal class group, 178
identities, 271
index theorem, 202–205
infinity
actual infinity, 247, 252, 255, 259
comparing, 261–265
continuum hypothesis and, 245–252
finite–infinite divide and, 253–259
p, 261–265
t, 261–265
uncountably infinite, 263
inner-model axiom, 245, 246, 250
interaction strength, 44
islands of stability, 167, 168
j-function, 29–35
Johnson graph, 233
K3 surfaces, 30–35
Kac–Moody algebras, 30
Kadison–Singer problem, 109–115
Kähler package, 100
Kardar–Parisi–Zhang equation, 197
Katz, Eric, 99–100
Keisler’s order, 264–265
Kepler’s conjecture, 139
Khot, Subhash, 235–242
knot theory, 286
K-theory, 204–205
k-tuple, 10
L (universe of sets), 249–252
landscape function, 117–122
Langlands program, 185
lattice distributions, 52–54
lattice points, 179
Leech lattice, 139–143
Lemke Oliver, Robert, 23–26
Liouville quantum gravity (LQG), 64–72
localization, 117–121
log concavity, 93–100
logistic equation for population growth, 167–168
M12 group, 33
Marcus, Adam, 109, 110, 113–115
Martin-Löf type theory (MLTT), 224–226
Martin’s maximum, 246–247, 251
Mathieu 24 (M24) group, 33
Matomäki, Kaisa, 17–21
matrices, 111
matroids, 97–100
Max Cut, 240
Mayboroda, Svitlana, 117–122
MEGASET, 221
Milnor conjecture, 226
Minimum Vertex Cover, 240
mirror symmetry, 82–85
Mirzakhani, Maryam, 147–154
mock theta functions, 34
model theory, 264–265
modular forms, 32–35, 142, 182
moduli space, 150–153
monster group, 29–35
moonshines, 29–35
motives, 74–79
multihyperuniform patterns, 57, 58
multiplicative functions/sequences, 18–20, 136–138
network sparsification, 109
noncommutative algebra, 201
nonequilibrium systems, 54–55
Ono, Ken, 275–280
operational semantics, 215
Orion spaceship, 159
p-adic numbers, 183
palatial twistor theory, 201
particle collisions, 73–79
PCP theorem, 238
Penrose, Roger, 201–203
Penrose tiling, 126
pentagon tiling, 123–127
perfectoid spaces, 181–186
periods, 74–79
periods of motives, 74–77
perturbative expansion, 73, 74
phase transitions, 46–49
pi, 76
Polignac, Alphonse de, 4
Polymath project, 9–13, 14–15, 137
polynomials
characteristic polynomials, 98–99, 113
chromatic polynomials, 93–96
interlacing polynomials, 113
polynomial method, 131–133
potential, 120
Press, William, 155–156
prime k-tuples conjecture, 26
prime numbers
consecutive, 23–26
distribution of, 41
prime gap, 9–13
prime number theorem, 17–21
twin primes conjecture, 3–8, 187–190
primitive recursive arithmetic, 257, 258
prisoner’s dilemma, iterated, 155
P versus NP, 230–233, 237, 238, 240
quadratic reciprocity law, 185
quantitative literacy, 282–285
quantum chaotic systems, 37
quantum electrodynamics, 155, 158
quantum systems, 40–41
quantum theory, 81–85
quantum well, 121
quasicrystals, 169
quasi-periodic Schrödinger operators, 169
quasi-polynomial-time algorithm, 230–233
quintic, 83
racial equality, 280
Radziwill, Maksym, 17–20
Ramanujan, Srinivasa, 34, 35, 269–273
Ramsey’s theorem for pairs (
), 253, 254, 257–259
Ramsey’s theorem for triplets, 257
Ramsey theory, 130
randomness, 59–72
Rao, Michaël, 123–127
Read’s conjecture, 95–101
reciprocity laws, 184–185
regularity structures, 194, 196, 198–200
relativity, 84
resolution of singularities in characteristic p, 92
Riemann zeta function, 18, 38, 41, 55, 78
Rota conjecture, 90–100
rough path theory, 197
Royen, Thomas, 103–107
Russell, Bertrand, 221
Sanskrit poetry, 174
Scholze, Peter, 181–186
Set (game), 129–133
SET (type), 221
set theory, 220–222, 227, 246–251, 255, 256, 261–265
Sheffield, Scott, 59–72
Sieve of Eratosthenes, 5–6
simultaneous confidence intervals, 105
singularity theory, 92, 96–97, 98, 101
SLE (Schramm–Loewner evolution) curve, 68–71
Socolar–Taylor tile, 126
Soundararajan, Kannan, 23–26
sphere packing, 139–143
Spielman, Daniel, 109, 110, 112–115
STATS, 281–285
stochastic behavior, 167, 168, 193
stochastic partial differential equations (SPDEs), 194–200
string theory, 30–35, 46–47, 63–64, 82, 83, 85, 151, 202
Su, Francis, 275–280
submanifold, 153
Sudoku, 138
supercompacts, 250–251
symplectic geometry, 84, 85, 282–283
Tao, Terence, 9–11, 44, 110, 131, 135–138
Tate motives, 78
Ten Martini Problem, 169
tessellation, 123–127
third-order phase transitions, 48, 49
tiling proof, 123–127
Tracy–Widom distribution, 43–49
translation surface, 153–154
traveling salesman problem, 110, 114, 237
twin primes conjecture, 3–8, 15, 26, 187–190
twistor theory, 201
two-dimensional random spaces, 59–65, 71
type theory, 221–226
ultimate L conjecture, 251
Umbral Moonshine Conjecture, 30, 31, 35
uncertainty principle, 85, 110
UniMath community, 226
unimodal maps, 168–169
Unique Games Conjecture, 235–241
univalence axiom, 225
univalent foundations, 219–220, 223–227
universality
hyperuniformity and, 53–54
in random matrices, 156
in systems, 37–41
Tracy–Widom distribution and, 43–49
universal optimality, 143
Voevodsky, Vladimir, 219–220, 223–227
V = ultimate L, 245–246, 250–252
Wang tiles, 127
waveguides, 57
wavelets, 198–199
Weaver’s conjecture, 112–114
weight (number), 78
weight-monodromy conjecture, 183–184
Wigner, Eugene, 82
Wigner’s hypothesis, 39
Witten, Edward, 35, 48, 151, 202, 205, 207
Witten’s conjecture, 151
worldsheet, 64
zero-knowledge proof, 231