Set. A collection of objects, such as the set {0,1,2,...,N−1} for a given natural number N.
Sheaf. A rule that assigns a vector space to each point of a given manifold, satisfying certain natural properties.
Shimura–Taniyama–Weil conjecture. The statement that there is a one-to-one correspondence between cubic equations and modular forms satisfying certain properties. Under this correspondence, the numbers of solutions of the cubic equation modulo prime numbers are equal to the coefficients of the modular form.
SO(3). The group of rotations of a sphere.
Sphere. A manifold that may be described as the set of all points in a flat three-dimensional space that are equidistant from a given point.
Supersymmetry. A type of symmetry in a quantum field theory that exchanges bosons and fermions.
Symmetry. A transformation of a given object that preserves its properties, such as its shape and position.
Theory. A particular branch of mathematics or physics (such as number theory) or a specific model describing relations between objects (such as gauge theory with gauge group SO(3)).
Vector space. The set of all vectors in a given n-dimensional flat space, carrying operations of addition of vectors and multiplication of vectors by numbers, satisfying natural properties.
Index
3D printing, 3, 23
A-brane, 215–217, 220–222, 225, 280
abelian group, see group, abelian
algebraic closure, 274
analogy, 4, 96, 99, 103, 106, 157, 263
asymmetry, 22
Atiyah, Michael, 209, 242
atom, 9, 237
automorphic function, 80, 93, 105, 154, 161, 173, 222, 275, 283
automorphic representation, see representation, automorphic
automorphic sheaf, see sheaf, automorphic
avatar, 103
B-brane, 215, 217, 220, 280
Beilinson, Alexander, 142, 147, 182
Ben-Zvi, David, 193, 216
Bernstein, Joseph, 142, 146, 147, 166
Betti number, 54, 56, 252
Bhagavad-Gita, 103
Bohr, Niels, 198
borscht, 196
Bott, Raoul, 146, 242
braid group, 47–54, 59, 60, 251
brane, 214, 215
Cantor, Georg, 4, 226
categorification, 156, 157
category, 156, 280
Chern, Shiing-Shen, 242
circle, 19, 33, 36, 103, 113, 118, 120, 156, 206, 217, 277, 283, 285
circle group, 19, 20, 22, 24, 47, 84, 93, 110, 118, 121, 180, 185, 200, 203, 218, 235, 248, 277, 283
classical electromagnetism, see electromagnetism, classical
complex number, 90, 100–103, 160, 177, 251, 256, 269, 276, 277, 283
composition, 8, 248, 250, 283, 284
of braids, 48, 49
of rotations, 18, 20
of symmetries, 19, 21, 247, 248, 255, 275
conjecture, 81
correlation function, 209, 214, 238, 281
correspondence, 8, 283
one-to-one, 52, 91, 92, 100, 102, 218, 257, 268, 279, 283, 286
cross-product, 121, 269
cryptography, 52, 86, 91, 240, 251, 261
cubic equation, see equation, cubic
Cummings, E.E., 182
cup trick, 168, 171, 276
curvature, 117
curve over finite field, 103–105, 154, 160, 161, 216, 222, 262, 263, 283
D-brane, see brane
DARPA, 187, 188
Darwin, Charles, 2, 117
De Concini, Corrado, 183
definition, 251
Deligne, Pierre, 161, 191, 263
Descartes, René, 22, 120, 262
differential equation, 176
dimension, 2, 24, 25, 113–118, 120, 121, 155, 159, 205, 206, 213, 284
dimensional reduction, 205, 209, 213, 214, 280
Dirac, Paul, 197
Drinfeld, Vladimir, 62, 106, 152–154, 161, 166–172, 174–176, 182
duality, 185, 196, 240, 284
electromagnetic, 195, 197–199, 202, 203, 205, 210, 214, 216, 217, 222, 226, 280
Duchamp, Marcel, 115, 116
Eichler, Martin, 88
Einstein, Albert, 2, 4, 27, 71, 99, 116–118, 191, 200, 201, 204, 236, 241
electromagnetic duality, see duality, electromagnetic
electromagnetism
classical, 195, 207, 208
quantum, 195, 197, 199, 208, 235
elliptic curve, 91, 261, 263
Enzensberger, Hans Magnus, 1
equation, 83
cubic, 83, 86, 88, 91, 92, 99, 102, 261, 263, 283
differential, see differential equation
polynomial, see polynomial equation
Esenin-Volpin, Alexander, 63
Euclid, 258
Euler function, 60, 271
Euler, Leonhard, 60, 277
Farber, Thomas, 229, 230
Feigin, Boris, 125–130, 148, 151, 161, 242
Fermat’s Last Theorem, 4, 56, 58, 81–83, 92, 234, 258, 262, 284
Fermat’s little theorem, 158, 271, 274
Fermat, Pierre, 57, 81, 158, 253
Fibonacci numbers, 87, 260
fibration, 217, 219, 253, 276, 284
toric, 219, 220
finite field, 86, 91, 103, 104, 154, 158, 159, 161, 222, 269, 274, 283, 284
flag manifold, 128
flat space, see vector space
fraction, see rational number
free-field model, 124
free-field realization, 124, 129, 130, 182
Frobenius symmetry, 159, 274, 275
Fuchs, Dmitry Borisovich, 45, 46, 54, 55, 60, 68, 69, 108, 109, 122, 125, 129
function, 80, 156, 159, 176, 284
rational, 263, 266
fundamental group, 105, 106, 161, 173, 175–177, 180, 181, 215, 218, 220, 222, 251, 264, 266, 276–279, 284
fundamental lemma, 279
Gödel, Kurt, 191, 234
Galilei, Galileo, 2
Galois group, 75–78, 80, 92, 93, 98, 104, 106, 109, 154, 155, 159, 161, 167, 168, 173, 222, 234, 255, 257, 266, 275, 276, 284
of a finite field, 159, 274
Galois, Évariste, 76, 78, 240, 255
gauge group, 185, 200, 203, 209, 210, 214, 235, 278, 284
abelian, 185
non-abelian, 185
gauge theory, 185, 190, 200, 201, 203, 205, 207, 209, 210, 214, 235, 278, 284
non-abelian, 185, 202
Gelfand’s seminar, 61–67, 125
Gelfand, Israel Moiseevich, 6, 29, 61, 62, 64, 65, 67, 68, 142
Gell-Mann, Murray, 10, 11, 14, 26, 27
generating function, 88, 89, 91
Goddard, Peter, 190–192, 203, 204
golden ratio, 260
Goresky, Mark, 191, 193
Graves, Reine, 231, 232
Gross, Benedict, 146
Gross, David, 226, 278
Grothendieck, Alexander, 157, 158, 160, 226, 263
group, 13, 19–22, 25, 72, 84, 109–112, 261, 270, 272, 284–286
abelian, 52, 283
finite, 111
finite-dimensional, 111
infinite, 111
infinite-dimensional, 111
non-abelian, 52, 285
of permutations, 257
solvable, 77, 257
hadron, 11, 23, 26, 27
Hardy, G.H., 188
harmonic, 80, 90
harmonic analysis, 78, 80, 81, 92, 97, 104, 154, 283, 284
Harvard University, 7, 107, 141, 145, 150, 153, 182
Higgs boson, 22, 198
Hitchin fibration, 219, 223–225
Hitchin moduli space, 209, 210, 214, 215, 219, 222, 224, 279, 280, 285
Hitchin, Nigel, 209, 210
identical symmetry, see symmetry, identical
instanton, 209, 238
Institute for Advanced Study, 4, 71, 95, 186, 190, 191, 204
integer, 51, 52, 7
2, 251, 285
inverse symmetry, see symmetry, inverse
Jaffe, Arthur, 142, 144, 150
Joyce, James, 10
Kac, Victor, 122, 147, 150, 162, 163, 183
Kac–Moody algebra, 97, 122–124, 127, 129, 130, 148, 161, 174, 175, 180–182, 184, 190, 269, 280, 285
Kapustin, Anton, 216, 221
Kazakov, Vladimir, 65, 66
Kazhdan, David, 146, 147
Khurgin, Yakov Isaevich, 133–136, 139, 140
Kirillov, Alexander Alexandrovich, 43
Kontsevich, Maxim, 215
Langlands correspondence, see Langlands relation
Langlands dual group, 167, 168, 170, 172, 173, 175, 181, 185, 203, 210, 219, 285
Langlands Program, 3, 7, 70, 72, 80, 91–93, 95–97, 104, 152, 154, 160, 162, 173, 182, 184, 192, 205, 216, 222, 227, 228, 240, 243
geometric, 105, 106
Langlands relation, 8, 93, 105, 106, 154, 161, 168, 173, 182, 215, 221, 275, 283, 285
Langlands, Robert, 4, 71, 78, 80, 191, 204, 233, 242, 243, 253
Large Hadron Collider, 22, 27, 198
Laumon, Gérard, 154, 161
Letterman, David, 146
Lie algebra, 119, 121, 268, 269, 285
Lie group, 109, 111, 112, 126, 167, 173, 185, 200, 235, 284, 285
finite-dimensional, 118
infinite-dimensional, 118, 119
Lie, Sophus, 109
linear transformation, 248, 270, 275
Logunov, Anatoly, 162–165
loop, 8, 118, 119, 126, 174, 211, 212, 267, 269, 285
loop group, 111, 118, 119, 126, 174, 182, 267, 269
loop space, 118, 119, 267, 269
Losev, Andrei, 238
MacPherson, Robert, 191
manifold, 8, 110, 112, 114, 117–119, 121, 128, 156, 157, 159, 208, 213, 267, 284, 285
Manin, Yuri Ivanovich, 152, 209, 233
Mann, Ben, 188, 190, 191
map, 207–209, 212, 253, 266, 267, 270, 279, 284, 285
Maxwell’s equations, 176, 194, 195, 201, 208, 278
Maxwell, James Clerc, 194
Mazur, Barry, 146
Mekh-Mat, 28, 29, 42, 43, 55, 62, 164, 165, 249, 253
MGU, see Moscow University
Mills, Robert, 200
mirror symmetry, 210, 214, 215, 217, 219–222, 280
Mishima, Yukio, 231, 232, 236
modular arithmetic, see modulo N
modular form, 90–92, 94, 166, 261, 276, 285, 286
modulo N, 20, 83–86, 158, 188, 258, 259, 271
monodromy, 178–181, 266, 277
Moody, Robert, 122
Moscow University, 6, 28, 32, 35, 40, 162, 164, 249
Muchnik Subbotovskaya, Bella, 46
natural number, 51, 57, 72, 81–83, 85, 155, 273, 284–286
Ne’eman, Yuval, 11
Nekrasov, Nikita, 238
neutron, 9, 11, 12, 26, 185
Newton, Isaac, 241
Ngô, Bao Châu, 243, 279
non-abelian group, see group, non-abelian
non-Euclidean geometry, 116, 248
number field, 75, 77, 78, 80, 101, 104, 255, 256, 263, 266, 284, 285
one-to-one correspondence, see correspondence, one-to-one
oper, 181
Pauli, Wolfgang, 11, 198
Penrose, Roger, 234, 235
Petrov, Evgeny Evgenievich, 12–14, 16, 28, 129
Poincaré, Henri, 116
polynomial, 251, 253, 263, 265, 283, 285
polynomial equation, 76, 77, 83, 255, 256, 266, 286
cubic, 257
over finite field, 273, 274, 284
quadratic, 76, 255
quintic, 77, 255, 257
prime number, 85, 86, 89, 100, 103, 158, 258, 259, 261, 270, 272, 275, 284, 286
proton, 9, 11, 12, 26, 185
quantum computing, 52, 273
quantum electromagnetism, see electromagnetism, quantum
quantum field theory, 195, 196, 198, 199, 207, 208, 212, 214, 238, 239, 280, 286
quantum group, 145, 153
quantum physics, 4, 9, 14, 23, 65, 97, 108, 118, 122, 124, 173, 183, 185, 192, 198, 202, 216, 217, 222, 225
quark, 10, 11, 13–15, 23, 26, 27, 185, 195, 198
rational number, 72–75, 77, 98, 254, 256, 263, 266, 273, 275, 285
real number, 100–103, 262, 269
relativity theory, 2, 99, 116–118, 201
representation, 8, 24–26, 80, 109, 112, 275, 279, 286
automorphic, 167, 173, 275
of fundamental group, 174, 176, 180, 181, 215, 218, 220, 222, 277, 279
of Galois group, 92, 104, 154, 274–276
of Kac–Moody algebra, 127, 129, 130, 174, 175, 181
two-dimensional, 24, 93, 112
Reshetikhin, Nicolai, 143, 145, 153
Ribet, Ken, 58, 81, 92, 262
Riemann surface, 6, 98–100, 103–106, 154, 159–161, 173, 174, 176, 180, 181, 205, 207, 210, 212, 216, 220, 222, 262, 263, 266, 276, 279, 285
Riemann, Bernhard, 2, 99, 117
Rites of Love and Math, 7, 232, 236, 239, 281
Rosetta stone, 4, 98, 104, 105, 154, 159–161, 173, 222, 225, 227, 283
RSA algorithm, 60, 86, 188, 272
Séminaire Bourbaki, 226
Serre, Jean-Pierre, 227, 262
set, 17, 19, 21, 84, 85, 102, 103, 250, 284, 286
sheaf, 106, 157–161, 275, 286
automorphic, 161, 173, 175, 181, 215, 221, 222
Shevardnadze, Eduard, 182, 183
Shimura, Goro, 94
Shimura–Taniyama–Weil conjecture, 58, 81–83, 91–95, 166, 258, 262, 286
sigma model, 208–211, 214, 215
Simpson, Homer, 219, 224
SO(3), group, 112–114, 119, 121, 123, 126–128, 155, 167, 168, 170, 180, 200, 223, 224, 267, 269, 275, 276, 285, 286
sphere, 98, 112–114, 117–121, 127, 128, 205, 208, 252, 263, 281, 285, 286
splitting field, 256, 257
string theory, 118, 119, 122, 211–213
strong force, 185, 200, 235
SU(2), group, 200, 224, 235, 276
superstring theory, 213, 214, 279, 280
supersymmetry, 198, 199, 202, 235, 286
symmetry, 15–19, 21, 23, 72, 74–78, 89, 90, 122, 123, 155, 194, 197, 240, 247, 248, 254–256, 268, 275, 286
identical, 17, 19, 255
inverse, 18, 19
symmetry breaking, 22
symmetry group, 14, 23, 76, 90, 119, 200, 224
T–duality, 217
tangent line, 120, 121
tangent plane, 120
tangent space, 120, 121, 268, 285
Taniyama, Yutaka, 94
target manifold, 208, 209, 212, 214
Taylor, Richard, 82
theorem, 251
theory, 8, 286
Thoreau, Henry David, 228, 233
topology, 14, 105, 252
torus, 98, 102, 105, 106, 213, 217, 218, 220, 221, 265, 266, 280
degenerate, 223, 224
mirror dual, 217
Tsygan, Boris, 145
unification, 70, 200, 213
Van Gogh, Vincent, 3, 10
Varchenko, Alexander Nikolaevich, 45
vector, 121, 268, 269, 286
vector field, 207
vector space, 121, 155–157, 159, 207, 268–270, 275, 283, 286
Vilonen, Kari, 187, 191, 193
Virasoro algebra, 123, 269
von Neumann, John, 191, 242
Wakimoto, Minoru, 123
weak force, 185, 200, 235
Weil group, 274
Weil, André, 78, 95, 96, 103, 104, 191, 227
Weil, Simone, 96
whole number, see natural number
Wigner, Eugene, 202
Wiles, Andrew, 58, 82, 83
Witten, Edward, 186, 187, 190–192, 204, 205, 214–216, 221, 225, 226, 242, 280
Yang, Chen Ning, 200, 201
Yang–Mills theory, see gauge theory
&nbs
p; Yau, Shing-Tung, 146, 213, 220, 279
Zagier, Don, 65
zero-brane, 218, 219, 221, 222, 224, 225
Zweig, George, 26
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