954:
684:
949:{\displaystyle \overbrace {\underbrace {\frac {\partial \mathbf {u} }{\partial t}} _{\begin{smallmatrix}{\text{Variation}}\end{smallmatrix}}+\underbrace {(\mathbf {u} \cdot \nabla )\mathbf {u} } _{\begin{smallmatrix}{\text{Convection}}\end{smallmatrix}}} ^{\text{Inertia (per volume)}}\overbrace {{}-\underbrace {\nu \,\nabla ^{2}\mathbf {u} } _{\text{Diffusion}}=\underbrace {-\nabla w} _{\begin{smallmatrix}{\text{Internal}}\\{\text{source}}\end{smallmatrix}}} ^{\text{Divergence of stress}}+\underbrace {\mathbf {g} } _{\begin{smallmatrix}{\text{External}}\\{\text{source}}\end{smallmatrix}}.}
1690:
1006:
1114:
1298:
at the negative even integers; that is, ζ(s) = 0 when s is one of â2, â4, â6, .... These are called its trivial zeros. However, the negative even integers are not the only values for which the zeta function is zero. The other ones are called nontrivial zeros. The
Riemann hypothesis is concerned with
251:
in the 1990s, released his proof in 2002 and 2003. His refusal of the Clay
Institute's monetary prize in 2010 was widely covered in the media. The other six Millennium Prize Problems remain unsolved, despite a large number of unsatisfactory proofs by both amateur and professional mathematicians.
276:
characterized their monetary prize as "show business" representing the "worst manifestations of present-day mass culture", and thought that there are more meaningful ways to invest in public appreciation of mathematics. He viewed the superficial media treatments of
Perelman and his work, with
382:
in 2006. However, he declined to accept the prize. For his proof of the
Poincaré conjecture, Perelman was awarded the Millennium Prize on March 18, 2010. However, he declined the award and the associated prize money, stating that Hamilton's contribution was no less than his own.
1062:
that solution quickly. Since the former describes the class of problems termed NP, while the latter describes P, the question is equivalent to asking whether all problems in NP are also in P. This is generally considered one of the most important open questions in
1269:
277:
disproportionate attention being placed on the prize value itself, as unsurprising. By contrast, Vershik praised the Clay
Institute's direct funding of research conferences and young researchers. Vershik's comments were later echoed by Fields medalist
1539:
418:. The conjecture is that there is a simple way to tell whether such equations have a finite or infinite number of rational solutions. More specifically, the Millennium Prize version of the conjecture is that, if the elliptic curve
614:
1663:
and has a mass gap Î > 0. Existence includes establishing axiomatic properties at least as strong as those cited in
Streater & Wightman (1964), Osterwalder & Schrader (1973), and Osterwalder & Schrader
993:, is to prove either that smooth, globally defined solutions exist that meet certain conditions, or that they do not always exist and the equations break down. The official statement of the problem was given by
1978:
The Clay
Mathematics Institute (CMI) announces today that Dr. Grigoriy Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for resolution of the Poincaré conjecture.
1608:-electromagnetic field itself carries charge. As a classical field theory it has solutions which travel at the speed of light so that its quantum version should describe massless particles (
1142:
363:'s program for the solution of the geometrization conjecture, which he had developed over the course of the preceding twenty years. Hamilton and Perelman's work revolved around Hamilton's
320:
posed the question of whether an analogous statement holds true for three-dimensional shapes. This came to be known as the
Poincaré conjecture, the precise formulation of which states:
1388:. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle.
972:. However, theoretical understanding of their solutions is incomplete, despite its importance in science and engineering. For the three-dimensional system of equations, and given some
218:. Unlike Hilbert's problems, the problems selected by the Clay Institute were already renowned among professional mathematicians, with many actively working towards their resolution.
190:
in 1900 which were highly influential in driving the progress of mathematics in the twentieth century. The seven selected problems span a number of mathematical fields, namely
1661:
1575:
1430:
262:
1 million prize money would popularize, among general audiences, both the selected problems as well as the "excitement of mathematical endeavor". Another board member,
2658:
1418:
470:
dealt with a more general type of equation, and in that case it was proven that there is no algorithmic way to decide whether a given equation even has any solutions.
269:, hoped that the publicity around the unsolved problems would help to combat the "wrong idea" among the public that mathematics would be "overtaken by computers".
166:
To date, the only
Millennium Prize problem to have been solved is the Poincaré conjecture. The Clay Institute awarded the monetary prize to Russian mathematician
2762:
159:
at the
Millennium Meeting held on May 24, 2000. Thus, on the official website of the Clay Mathematics Institute, these seven problems are officially called the
281:, who was additionally critical of the idea of a foundation taking actions to "appropriate" fundamental mathematical questions and "attach its name to them".
2821:
95:
516:
1628:
exists without restriction to low energy scales. The problem is to establish rigorously the existence of the quantum YangâMills theory and a mass gap.
2357:
2171:
2651:
1421:
982:
675:
340:
Although the conjecture is usually stated in this form, it is equivalent (as was discovered in the 1950s) to pose it in the context of
140:
48:
1951:
2891:
2777:
2644:
2718:
1718:
2679:
1585:
and thus the mass gap. This quantity, easy to generalize to other fields, is what is generally measured in lattice computations.
88:
2787:
2453:
2404:
2589:
2527:
2329:
2286:
2131:
2088:
2045:
1371:
152:
69:
1592:
is the current grounding for the majority of theoretical applications of thought to the reality and potential realities of
397:
132:
38:
2816:
2886:
81:
1264:{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }n^{-s}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots }
2615:
1088:
1582:
2306:
2345:
1578:
1092:
2577:
368:
211:
430:
215:
1593:
1344:
961:
20:
1098:
Most mathematicians and computer scientists expect that P â NP; however, it remains unproven.
2581:
2351:
1958:
1713:
467:
117:
1637:
1547:
1534:{\displaystyle \langle \phi (0,t)\phi (0,0)\rangle \sim \sum _{n}A_{n}\exp \left(-\Delta _{n}t\right)}
1117:
The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(
2881:
2689:
2684:
2148:
1625:
423:
352:
222:
2862:
2668:
1732:
1634:
Prove that for any compact simple gauge group G, a non-trivial quantum YangâMills theory exists on
1312:
2522:. Providence, RI: American Mathematical Society and Clay Mathematics Institute. pp. 129â152.
2324:. Providence, RI: American Mathematical Society and Clay Mathematics Institute. pp. 107â124.
2281:. Providence, RI: American Mathematical Society and Clay Mathematics Institute. pp. 87â104.
1291:
1055:
2263:
2126:. Providence, RI: American Mathematical Society and Clay Mathematics Institute. pp. 57â67.
2108:
2083:. Providence, RI: American Mathematical Society and Clay Mathematics Institute. pp. 45â53.
2040:. Providence, RI: American Mathematical Society and Clay Mathematics Institute. pp. 31â44.
1283:
1279:
313:
1394:
295:
248:
156:
58:
2792:
2782:
2741:
1708:
1589:
1316:
1275:
407:
230:
183:
2504:
2022:
2462:
2413:
1962:
1907:
1377:
990:
203:
234:
8:
2065:
1723:
1597:
1045:
372:
360:
195:
171:
144:
53:
2466:
2417:
1925:
1911:
1880:(January 2007). "What is good for mathematics? Thoughts on the Clay Millennium prizes".
1032:-hard set of problems (excluding the empty language and its complement, which belong to
2547:
2478:
2429:
2381:
2241:
1859:
1695:
1621:
1295:
1134:
1122:
503:
451:
301:
199:
191:
148:
64:
258:, as part of the Clay Institute's scientific advisory board, hoped that the choice of
2800:
2611:
2606:
The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time
2604:
2585:
2523:
2482:
2433:
2325:
2282:
2163:
2127:
2104:
2084:
2041:
1851:
1689:
1613:
994:
973:
495:
2846:
2245:
1863:
317:
2636:
2470:
2421:
2233:
1841:
1759:
1601:
1355:
492:
486:
403:
356:
341:
244:
167:
136:
43:
1952:"Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman"
1904:
The shape of a life. One mathematician's search for the universe's hidden geometry
609:{\displaystyle \operatorname {Hdg} ^{k}(X)=H^{2k}(X,\mathbb {Q} )\cap H^{k,k}(X).}
2448:
2399:
2302:
2218:
2214:
1877:
1703:
1359:
1020:
977:
969:
507:
415:
309:
273:
1809:
Jackson, Allyn (September 2000). "Million-dollar mathematics prizes announced".
1795:
920:
867:
773:
722:
2196:
2061:
1899:
1738:
1728:
1287:
1014:
663:
411:
345:
278:
226:
2631:
2875:
2500:
2167:
1675:
1336:
1009:
207:
187:
2237:
1616:
permits only bound states of gluons, forming massive particles. This is the
1050:
The question is whether or not, for all problems for which an algorithm can
378:
For his contributions to the theory of Ricci flow, Perelman was awarded the
2851:
2599:
2567:
2563:
2515:
2511:
2496:
2317:
2313:
2274:
2270:
2259:
2119:
2115:
2076:
2072:
2033:
2029:
2018:
1855:
1671:
1385:
1348:
1340:
1305:
The real part of every nontrivial zero of the Riemann zeta function is 1/2.
1102:
1084:
1005:
474:
379:
329:
266:
263:
255:
2200:
1072:
1068:
1064:
644:
be a non-singular complex projective variety. Then every Hodge class on
499:
113:
1990:
16:
Seven mathematical problems with a US$ 1 million prize for each solution
2767:
2710:
2543:
2474:
2425:
1080:
649:
364:
325:
1354:
The problem has been well-known ever since it was originally posed by
1335:. A proof or disproof of this would have far-reaching implications in
1113:
170:
in 2010. However, he declined the award as it was not also offered to
1846:
1829:
1358:
in 1860. The Clay Institute's exposition of the problem was given by
19:
This article is about the math prizes. For the technology prize, see
2571:
1995:
1929:
1617:
1384:
is the difference in energy between the vacuum and the next lowest
1381:
333:
2772:
2746:
2705:
1091:). A common example of an NP problem not known to be in P is the
1076:
410:
conjecture deals with certain types of equations: those defining
124:
1 million prize for the first correct solution to each problem.
2542:
This article incorporates material from Millennium Problems on
305:
127:
The Clay Mathematics Institute officially designated the title
2201:"Why Philosophers Should Care About Computational Complexity"
1609:
965:
238:
351:
A proof of this conjecture, together with the more powerful
229:
during a ceremony held on May 24, 2000 (at the amphithéùtre
259:
121:
1299:
the locations of these nontrivial zeros, and states that:
648:
is a linear combination with rational coefficients of the
1071:
as it has far-reaching consequences to other problems in
669:
2109:"Existence and smoothness of the NavierâStokes equation"
391:
1830:"Mathematicians chase the seven million-dollar proofs"
1640:
1550:
1433:
1397:
1290:
other than 1, and whose values are also complex. Its
1145:
687:
519:
2666:
1685:
1365:
2446:
2397:
1670:The official statement of the problem was given by
1420:, we can say that the theory has a mass gap if the
1101:The official statement of the problem was given by
662:The official statement of the problem was given by
473:The official statement of the problem was given by
2603:
2380:
1655:
1569:
1533:
1412:
1263:
948:
608:
131:for the seven unsolved mathematical problems, the
2561:
1783:
634:The modern statement of the Hodge conjecture is:
308:is characterized by the fact that it is the only
2873:
2548:Creative Commons Attribution/Share-Alike License
2378:
359:in 2002 and 2003. Perelman's solution completed
221:The seven problems were officially announced by
2213:
1760:"The Millennium Grand Challenge in Mathematics"
2195:
2652:
2149:"P Versus NP: More than just a prize problem"
328:which is closed and simply-connected must be
272:Some mathematicians have been more critical.
89:
1882:Notices of the American Mathematical Society
1811:Notices of the American Mathematical Society
1767:Notices of the American Mathematical Society
1473:
1434:
1315:zeros of the analytical continuation of the
182:The Clay Institute was inspired by a set of
2495:
2449:"Axioms for Euclidean Green's functions II"
2659:
2645:
2023:"The Birch and Swinnerton-Dyer conjecture"
1731:(offered a cash prize for the solution to
989:The problem, restricted to the case of an
976:, mathematicians have not yet proven that
120:in 2000. The Clay Institute has pledged a
96:
82:
2103:
1845:
1751:
1643:
1612:). However, the postulated phenomenon of
815:
568:
2719:Competitions and prizes in biotechnology
2400:"Axioms for Euclidean Green's functions"
2301:
1906:. New Haven, CT: Yale University Press.
1898:
1719:List of unsolved problems in mathematics
1596:. The theory is a generalization of the
1112:
1004:
2060:
1876:
1827:
1808:
1347:, and is still considered an important
2874:
2598:
2454:Communications in Mathematical Physics
2447:Osterwalder, K.; Schrader, R. (1975).
2405:Communications in Mathematical Physics
2398:Osterwalder, K.; Schrader, R. (1973).
2146:
1089:P versus NP problem proof consequences
1054:a given solution quickly (that is, in
983:NavierâStokes existence and smoothness
676:NavierâStokes existence and smoothness
670:NavierâStokes existence and smoothness
326:three-dimensional topological manifold
289:
141:NavierâStokes existence and smoothness
49:NavierâStokes existence and smoothness
2640:
2383:PCT, Spin and Statistics and all That
2360:from the original on 22 November 2015
2017:
1757:
1577:being the lowest energy value in the
1339:, especially for the distribution of
1108:
2817:Gigaton Scale Carbon Removal X Prize
2258:
1991:"ĐĐŸŃĐ»Đ”ĐŽĐœĐ”Đ” "ĐœĐ”Ń" ĐŽĐŸĐșŃĐŸŃĐ° ĐĐ”ŃДлŃĐŒĐ°ĐœĐ°"
398:Birch and Swinnerton-Dyer conjecture
392:Birch and Swinnerton-Dyer conjecture
386:
133:Birch and Swinnerton-Dyer conjecture
39:Birch and Swinnerton-Dyer conjecture
2379:Streater, R.; Wightman, A. (1964).
1758:Jaffe, Arthur M. (JuneâJuly 2006).
1620:. Another aspect of confinement is
1311:The Riemann hypothesis is that all
480:
367:, which is a complicated system of
13:
2555:
1552:
1514:
1177:
853:
817:
754:
707:
697:
316:two-dimensional surface. In 1904,
174:, upon whose work Perelman built.
14:
2903:
2625:
2177:from the original on 17 June 2022
1926:"Maths genius declines top prize"
1784:Carlson, Jaffe & Wiles (2006)
1372:YangâMills existence and mass gap
1366:YangâMills existence and mass gap
1129:) = ±14.135, ±21.022 and ±25.011.
980:always exist. This is called the
919:
866:
772:
721:
491:The Hodge conjecture is that for
284:
153:YangâMills existence and mass gap
70:YangâMills existence and mass gap
2892:Unsolved problems in mathematics
1961:. March 18, 2010. Archived from
1688:
1656:{\displaystyle \mathbb {R} ^{4}}
1624:which makes it conceivable that
1570:{\displaystyle \Delta _{0}>0}
968:, and are one of the pillars of
909:
827:
761:
747:
701:
2489:
2440:
2391:
2372:
2338:
2295:
2252:
2207:
2189:
2140:
2097:
2054:
2011:
1983:
1796:"The Millennium Prize Problems"
186:organized by the mathematician
2680:Science and engineering awards
2546:, which is licensed under the
1944:
1918:
1892:
1870:
1821:
1802:
1788:
1777:
1470:
1458:
1452:
1440:
1407:
1401:
1155:
1149:
1121:) = 1/2. The first nontrivial
1093:Boolean satisfiability problem
1000:
757:
743:
600:
594:
572:
558:
539:
533:
369:partial differential equations
212:partial differential equations
1:
2632:The Millennium Prize Problems
2578:American Mathematical Society
2573:The Millennium Prize Problems
2520:The millennium prize problems
2322:The millennium prize problems
2279:The millennium prize problems
2124:The millennium prize problems
2081:The millennium prize problems
2038:The millennium prize problems
1744:
112:are seven well-known complex
1069:theoretical computer science
247:, who had begun work on the
216:theoretical computer science
7:
2505:"Quantum YangâMills theory"
2147:Rajput, Uday Singh (2016).
1681:
1594:elementary particle physics
652:of complex subvarieties of
177:
21:Millennium Technology Prize
10:
2908:
2582:Clay Mathematics Institute
1959:Clay Mathematics Institute
1714:List of mathematics awards
1369:
1132:
1043:
673:
619:We call this the group of
484:
395:
293:
118:Clay Mathematics Institute
18:
2887:Millennium Prize Problems
2860:
2834:
2809:
2755:
2734:
2727:
2698:
2675:
2669:inducement prize contests
2610:. New York: Basic Books.
2346:"YangâMills and Mass Gap"
2264:"The P versus NP problem"
1626:quantum Yang-Mills theory
1058:), an algorithm can also
353:geometrization conjecture
116:problems selected by the
110:Millennium Prize Problems
30:Millennium Prize Problems
2863:List of challenge awards
2307:"The Riemann hypothesis"
1413:{\displaystyle \phi (x)}
1345:Hilbert's eighth problem
371:defined in the field of
2238:10.1145/1052796.1052804
1902:; Nadis, Steve (2019).
1828:Dickson, David (2000).
1391:For a given real field
1292:analytical continuation
964:describe the motion of
962:NavierâStokes equations
468:Hilbert's tenth problem
2162:. Lucknow, India: 90.
2066:"The Hodge conjecture"
1657:
1571:
1535:
1414:
1265:
1181:
1130:
1041:
950:
610:
338:
2793:DARPA Grand Challenge
2783:UAV Outback Challenge
2742:Centennial Challenges
2510:. In Carlson, James;
2312:. In Carlson, James;
2269:. In Carlson, James;
2114:. In Carlson, James;
2105:Fefferman, Charles L.
2071:. In Carlson, James;
2028:. In Carlson, James;
1733:Fermat's Last Theorem
1658:
1572:
1536:
1415:
1317:Riemann zeta function
1276:Riemann zeta function
1266:
1161:
1116:
1008:
951:
611:
322:
231:Marguerite de Navarre
184:twenty-three problems
2699:Biology and medicine
1638:
1548:
1431:
1395:
1378:quantum field theory
1319:have a real part of
1143:
991:incompressible fluid
898:Divergence of stress
795:Inertia (per volume)
685:
517:
304:, a two-dimensional
204:mathematical physics
2822:Feynman Grand Prize
2467:1975CMaPh..42..281O
2418:1973CMaPh..31...83O
2203:. Technical report.
1912:2019shli.book.....Y
1046:P versus NP problem
504:linear combinations
496:algebraic varieties
450:associated with it
373:Riemannian geometry
296:Poincaré conjecture
290:Poincaré conjecture
249:Poincaré conjecture
196:arithmetic geometry
172:Richard S. Hamilton
161:Millennium Problems
157:Poincaré conjecture
145:P versus NP problem
59:Poincaré conjecture
54:P versus NP problem
2768:Micromouse Contest
2576:. Providence, RI:
2475:10.1007/BF01608978
2426:10.1007/BF01645738
2350:www.claymath.org (
2199:(14 August 2011).
1709:Hilbert's problems
1696:Mathematics portal
1653:
1622:asymptotic freedom
1567:
1531:
1488:
1422:two-point function
1410:
1261:
1135:Riemann hypothesis
1131:
1125:can be seen at Im(
1109:Riemann hypothesis
1042:
974:initial conditions
946:
942:
940:
939:
916:
889:
887:
886:
863:
842:
835:
786:
784:
783:
769:
735:
733:
732:
718:
650:cohomology classes
606:
302:geometric topology
200:geometric topology
192:algebraic geometry
149:Riemann hypothesis
129:Millennium Problem
65:Riemann hypothesis
2869:
2868:
2830:
2829:
2801:Robot competition
2591:978-0-8218-3679-8
2529:978-0-8218-3679-8
2387:. W. A. Benjamin.
2331:978-0-8218-3679-8
2288:978-0-8218-3679-8
2133:978-0-8218-3679-8
2090:978-0-8218-3679-8
2047:978-0-8218-3679-8
1968:on March 31, 2010
1957:(Press release).
1614:color confinement
1590:YangâMills theory
1479:
1424:has the property
1351:a century later.
1253:
1233:
1213:
995:Charles Fefferman
935:
926:
907:
905:
901:
899:
894:
882:
873:
848:
846:
840:
810:
808:
798:
796:
791:
779:
741:
739:
728:
714:
694:
692:
452:vanishes to order
387:Unsolved problems
235:CollĂšge de France
106:
105:
2899:
2882:Challenge awards
2842:Millennium Prize
2732:
2731:
2690:Invention awards
2685:Challenge awards
2661:
2654:
2647:
2638:
2637:
2621:
2609:
2600:Devlin, Keith J.
2595:
2562:Carlson, James;
2534:
2533:
2509:
2493:
2487:
2486:
2444:
2438:
2437:
2395:
2389:
2388:
2386:
2376:
2370:
2369:
2367:
2365:
2342:
2336:
2335:
2311:
2303:Bombieri, Enrico
2299:
2293:
2292:
2268:
2256:
2250:
2249:
2223:
2219:"The P=?NP poll"
2211:
2205:
2204:
2193:
2187:
2186:
2184:
2182:
2176:
2153:
2144:
2138:
2137:
2113:
2101:
2095:
2094:
2070:
2058:
2052:
2051:
2027:
2015:
2009:
2008:
2006:
2004:
1987:
1981:
1980:
1975:
1973:
1967:
1956:
1948:
1942:
1941:
1939:
1937:
1932:. 22 August 2006
1922:
1916:
1915:
1896:
1890:
1889:
1878:Vershik, Anatoly
1874:
1868:
1867:
1849:
1847:10.1038/35013216
1825:
1819:
1818:
1806:
1800:
1799:
1792:
1786:
1781:
1775:
1774:
1764:
1755:
1724:Smale's problems
1698:
1693:
1692:
1662:
1660:
1659:
1654:
1652:
1651:
1646:
1602:electromagnetism
1576:
1574:
1573:
1568:
1560:
1559:
1540:
1538:
1537:
1532:
1530:
1526:
1522:
1521:
1498:
1497:
1487:
1419:
1417:
1416:
1411:
1356:Bernhard Riemann
1334:
1332:
1331:
1328:
1325:
1270:
1268:
1267:
1262:
1254:
1252:
1251:
1239:
1234:
1232:
1231:
1219:
1214:
1212:
1211:
1199:
1194:
1193:
1180:
1175:
978:smooth solutions
955:
953:
952:
947:
941:
936:
933:
927:
924:
917:
912:
900:
897:
895:
890:
888:
883:
880:
874:
871:
864:
859:
841:
838:
836:
831:
830:
825:
824:
804:
801:
799:
797:
794:
792:
787:
785:
780:
777:
770:
765:
764:
750:
734:
729:
726:
719:
713:
705:
704:
695:
690:
688:
615:
613:
612:
607:
593:
592:
571:
557:
556:
529:
528:
508:algebraic cycles
487:Hodge conjecture
481:Hodge conjecture
463:
456:
449:
428:
421:
416:rational numbers
361:Richard Hamilton
357:Grigori Perelman
342:smooth manifolds
314:simply-connected
300:In the field of
245:Grigori Perelman
168:Grigori Perelman
137:Hodge conjecture
98:
91:
84:
44:Hodge conjecture
26:
25:
2907:
2906:
2902:
2901:
2900:
2898:
2897:
2896:
2872:
2871:
2870:
2865:
2856:
2826:
2805:
2788:IGVC challenges
2751:
2723:
2694:
2671:
2665:
2628:
2618:
2592:
2570:, eds. (2006).
2558:
2556:Further reading
2538:
2537:
2530:
2507:
2494:
2490:
2445:
2441:
2396:
2392:
2377:
2373:
2363:
2361:
2344:
2343:
2339:
2332:
2309:
2300:
2296:
2289:
2266:
2257:
2253:
2221:
2215:William Gasarch
2212:
2208:
2194:
2190:
2180:
2178:
2174:
2151:
2145:
2141:
2134:
2111:
2102:
2098:
2091:
2068:
2062:Deligne, Pierre
2059:
2055:
2048:
2025:
2016:
2012:
2002:
2000:
1989:
1988:
1984:
1971:
1969:
1965:
1954:
1950:
1949:
1945:
1935:
1933:
1924:
1923:
1919:
1900:Yau, Shing-Tung
1897:
1893:
1875:
1871:
1826:
1822:
1807:
1803:
1794:
1793:
1789:
1782:
1778:
1762:
1756:
1752:
1747:
1704:Beal conjecture
1694:
1687:
1684:
1647:
1642:
1641:
1639:
1636:
1635:
1555:
1551:
1549:
1546:
1545:
1517:
1513:
1509:
1505:
1493:
1489:
1483:
1432:
1429:
1428:
1396:
1393:
1392:
1374:
1368:
1360:Enrico Bombieri
1329:
1326:
1323:
1322:
1320:
1247:
1243:
1238:
1227:
1223:
1218:
1207:
1203:
1198:
1186:
1182:
1176:
1165:
1144:
1141:
1140:
1137:
1111:
1056:polynomial time
1048:
1028:-complete, and
1003:
970:fluid mechanics
938:
937:
932:
929:
928:
923:
918:
908:
906:
896:
885:
884:
879:
876:
875:
870:
865:
849:
847:
837:
826:
820:
816:
811:
809:
803:
802:
800:
793:
782:
781:
776:
771:
760:
746:
742:
740:
731:
730:
725:
720:
706:
700:
696:
693:
691:
689:
686:
683:
682:
678:
672:
582:
578:
567:
549:
545:
524:
520:
518:
515:
514:
489:
483:
458:
454:
436:
426:
419:
412:elliptic curves
408:Swinnerton-Dyer
400:
394:
389:
355:, was given by
346:diffeomorphisms
298:
292:
287:
274:Anatoly Vershik
264:Fields medalist
180:
102:
24:
17:
12:
11:
5:
2905:
2895:
2894:
2889:
2884:
2867:
2866:
2861:
2858:
2857:
2855:
2854:
2849:
2847:ErdĆs problems
2844:
2838:
2836:
2832:
2831:
2828:
2827:
2825:
2824:
2819:
2813:
2811:
2807:
2806:
2804:
2803:
2796:
2795:
2790:
2785:
2780:
2775:
2770:
2765:
2759:
2757:
2753:
2752:
2750:
2749:
2744:
2738:
2736:
2729:
2725:
2724:
2722:
2721:
2714:
2713:
2708:
2702:
2700:
2696:
2695:
2693:
2692:
2687:
2682:
2676:
2673:
2672:
2664:
2663:
2656:
2649:
2641:
2635:
2634:
2627:
2626:External links
2624:
2623:
2622:
2616:
2596:
2590:
2557:
2554:
2553:
2552:
2536:
2535:
2528:
2501:Witten, Edward
2488:
2461:(3): 281â305.
2439:
2390:
2371:
2337:
2330:
2294:
2287:
2251:
2206:
2197:Scott Aaronson
2188:
2139:
2132:
2096:
2089:
2053:
2046:
2010:
1999:. July 1, 2010
1982:
1943:
1917:
1891:
1869:
1820:
1801:
1787:
1776:
1749:
1748:
1746:
1743:
1742:
1741:
1739:abc conjecture
1736:
1729:Paul Wolfskehl
1726:
1721:
1716:
1711:
1706:
1700:
1699:
1683:
1680:
1668:
1667:
1666:
1665:
1650:
1645:
1566:
1563:
1558:
1554:
1542:
1541:
1529:
1525:
1520:
1516:
1512:
1508:
1504:
1501:
1496:
1492:
1486:
1482:
1478:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1409:
1406:
1403:
1400:
1370:Main article:
1367:
1364:
1309:
1308:
1307:
1306:
1288:complex number
1272:
1271:
1260:
1257:
1250:
1246:
1242:
1237:
1230:
1226:
1222:
1217:
1210:
1206:
1202:
1197:
1192:
1189:
1185:
1179:
1174:
1171:
1168:
1164:
1160:
1157:
1154:
1151:
1148:
1133:Main article:
1110:
1107:
1044:Main article:
1002:
999:
958:
957:
945:
931:
930:
922:
921:
915:
911:
904:
893:
878:
877:
869:
868:
862:
858:
855:
852:
845:
834:
829:
823:
819:
814:
807:
790:
775:
774:
768:
763:
759:
756:
753:
749:
745:
738:
724:
723:
717:
712:
709:
703:
699:
674:Main article:
671:
668:
664:Pierre Deligne
660:
659:
658:
657:
617:
616:
605:
602:
599:
596:
591:
588:
585:
581:
577:
574:
570:
566:
563:
560:
555:
552:
548:
544:
541:
538:
535:
532:
527:
523:
485:Main article:
482:
479:
396:Main article:
393:
390:
388:
385:
318:Henri Poincaré
294:Main article:
291:
288:
286:
285:Solved problem
283:
279:Shing-Tung Yau
227:Michael Atiyah
179:
176:
104:
103:
101:
100:
93:
86:
78:
75:
74:
73:
72:
67:
62:
56:
51:
46:
41:
33:
32:
15:
9:
6:
4:
3:
2:
2904:
2893:
2890:
2888:
2885:
2883:
2880:
2879:
2877:
2864:
2859:
2853:
2850:
2848:
2845:
2843:
2840:
2839:
2837:
2833:
2823:
2820:
2818:
2815:
2814:
2812:
2808:
2802:
2798:
2797:
2794:
2791:
2789:
2786:
2784:
2781:
2779:
2776:
2774:
2771:
2769:
2766:
2764:
2761:
2760:
2758:
2754:
2748:
2745:
2743:
2740:
2739:
2737:
2733:
2730:
2726:
2720:
2716:
2715:
2712:
2709:
2707:
2704:
2703:
2701:
2697:
2691:
2688:
2686:
2683:
2681:
2678:
2677:
2674:
2670:
2662:
2657:
2655:
2650:
2648:
2643:
2642:
2639:
2633:
2630:
2629:
2619:
2617:0-465-01729-0
2613:
2608:
2607:
2601:
2597:
2593:
2587:
2583:
2579:
2575:
2574:
2569:
2568:Wiles, Andrew
2565:
2564:Jaffe, Arthur
2560:
2559:
2551:
2549:
2545:
2540:
2539:
2531:
2525:
2521:
2517:
2516:Wiles, Andrew
2513:
2512:Jaffe, Arthur
2506:
2502:
2498:
2497:Jaffe, Arthur
2492:
2484:
2480:
2476:
2472:
2468:
2464:
2460:
2456:
2455:
2450:
2443:
2435:
2431:
2427:
2423:
2419:
2415:
2412:(2): 83â112.
2411:
2407:
2406:
2401:
2394:
2385:
2384:
2375:
2359:
2355:
2353:
2347:
2341:
2333:
2327:
2323:
2319:
2318:Wiles, Andrew
2315:
2314:Jaffe, Arthur
2308:
2304:
2298:
2290:
2284:
2280:
2276:
2275:Wiles, Andrew
2272:
2271:Jaffe, Arthur
2265:
2261:
2260:Cook, Stephen
2255:
2247:
2243:
2239:
2235:
2231:
2227:
2220:
2217:(June 2002).
2216:
2210:
2202:
2198:
2192:
2173:
2169:
2165:
2161:
2157:
2150:
2143:
2135:
2129:
2125:
2121:
2120:Wiles, Andrew
2117:
2116:Jaffe, Arthur
2110:
2106:
2100:
2092:
2086:
2082:
2078:
2077:Wiles, Andrew
2074:
2073:Jaffe, Arthur
2067:
2063:
2057:
2049:
2043:
2039:
2035:
2034:Wiles, Andrew
2031:
2030:Jaffe, Arthur
2024:
2020:
2019:Wiles, Andrew
2014:
1998:
1997:
1992:
1986:
1979:
1964:
1960:
1953:
1947:
1931:
1927:
1921:
1913:
1909:
1905:
1901:
1895:
1887:
1883:
1879:
1873:
1865:
1861:
1857:
1853:
1848:
1843:
1839:
1835:
1831:
1824:
1817:(8): 877â879.
1816:
1812:
1805:
1797:
1791:
1785:
1780:
1773:(6): 652â660.
1772:
1768:
1761:
1754:
1750:
1740:
1737:
1734:
1730:
1727:
1725:
1722:
1720:
1717:
1715:
1712:
1710:
1707:
1705:
1702:
1701:
1697:
1691:
1686:
1679:
1677:
1676:Edward Witten
1673:
1648:
1633:
1632:
1631:
1630:
1629:
1627:
1623:
1619:
1615:
1611:
1607:
1603:
1599:
1595:
1591:
1586:
1584:
1580:
1564:
1561:
1556:
1527:
1523:
1518:
1510:
1506:
1502:
1499:
1494:
1490:
1484:
1480:
1476:
1467:
1464:
1461:
1455:
1449:
1446:
1443:
1437:
1427:
1426:
1425:
1423:
1404:
1398:
1389:
1387:
1383:
1379:
1373:
1363:
1361:
1357:
1352:
1350:
1346:
1342:
1341:prime numbers
1338:
1337:number theory
1318:
1314:
1304:
1303:
1302:
1301:
1300:
1297:
1293:
1289:
1285:
1281:
1277:
1258:
1255:
1248:
1244:
1240:
1235:
1228:
1224:
1220:
1215:
1208:
1204:
1200:
1195:
1190:
1187:
1183:
1172:
1169:
1166:
1162:
1158:
1152:
1146:
1139:
1138:
1136:
1128:
1124:
1120:
1115:
1106:
1104:
1099:
1096:
1094:
1090:
1086:
1082:
1078:
1074:
1070:
1066:
1061:
1057:
1053:
1047:
1039:
1035:
1031:
1027:
1023:
1022:
1017:
1016:
1011:
1010:Euler diagram
1007:
998:
996:
992:
987:
985:
984:
979:
975:
971:
967:
963:
956:
943:
913:
902:
891:
860:
856:
850:
843:
832:
821:
812:
805:
788:
766:
751:
736:
715:
710:
680:
679:
677:
667:
665:
655:
651:
647:
643:
639:
638:
637:
636:
635:
632:
630:
626:
622:
621:Hodge classes
603:
597:
589:
586:
583:
579:
575:
564:
561:
553:
550:
546:
542:
536:
530:
525:
521:
513:
512:
511:
509:
505:
502:are rational
501:
497:
494:
488:
478:
476:
471:
469:
465:
461:
453:
447:
443:
439:
435:
433:
425:
417:
413:
409:
405:
399:
384:
381:
376:
374:
370:
366:
362:
358:
354:
349:
347:
343:
337:
335:
331:
327:
321:
319:
315:
311:
307:
303:
297:
282:
280:
275:
270:
268:
265:
261:
257:
253:
250:
246:
242:
240:
236:
232:
228:
224:
219:
217:
213:
209:
208:number theory
205:
201:
197:
193:
189:
188:David Hilbert
185:
175:
173:
169:
164:
162:
158:
154:
150:
146:
142:
138:
134:
130:
125:
123:
119:
115:
111:
99:
94:
92:
87:
85:
80:
79:
77:
76:
71:
68:
66:
63:
60:
57:
55:
52:
50:
47:
45:
42:
40:
37:
36:
35:
34:
31:
28:
27:
22:
2852:Hutter Prize
2841:
2605:
2572:
2541:
2519:
2491:
2458:
2452:
2442:
2409:
2403:
2393:
2382:
2374:
2362:. Retrieved
2349:
2340:
2321:
2297:
2278:
2254:
2232:(2): 34â47.
2229:
2225:
2209:
2191:
2179:. Retrieved
2159:
2155:
2142:
2123:
2099:
2080:
2056:
2037:
2013:
2001:. Retrieved
1994:
1985:
1977:
1970:. Retrieved
1963:the original
1946:
1934:. Retrieved
1920:
1903:
1894:
1885:
1881:
1872:
1840:(383): 383.
1837:
1833:
1823:
1814:
1810:
1804:
1790:
1779:
1770:
1766:
1753:
1672:Arthur Jaffe
1669:
1605:
1587:
1543:
1390:
1386:energy state
1375:
1353:
1349:open problem
1310:
1273:
1126:
1118:
1103:Stephen Cook
1100:
1097:
1085:cryptography
1059:
1051:
1049:
1037:
1036:but are not
1033:
1029:
1025:
1019:
1013:
988:
981:
959:
681:
661:
653:
645:
641:
633:
628:
624:
620:
618:
500:Hodge cycles
490:
475:Andrew Wiles
472:
466:
459:
445:
441:
437:
431:
401:
380:Fields Medal
377:
350:
339:
330:homeomorphic
323:
299:
271:
267:Alain Connes
256:Andrew Wiles
254:
243:
220:
181:
165:
160:
128:
126:
114:mathematical
109:
107:
29:
2835:Mathematics
2728:Engineering
2226:SIGACT News
1888:(1): 45â47.
1583:Hamiltonian
1343:. This was
1286:may be any
1073:mathematics
1065:mathematics
1001:P versus NP
623:of degree 2
429:, then the
155:, and the
2876:Categories
2799:See also:
2717:See also:
2711:Prize4Life
2544:PlanetMath
2003:25 January
1745:References
1604:where the
1600:theory of
1313:nontrivial
1278:ζ(s) is a
1081:philosophy
1040:-complete)
986:problem.
778:Convection
493:projective
365:Ricci flow
2602:(2003) .
2483:119389461
2434:189829853
2168:0046-5402
1972:March 18,
1553:Δ
1515:Δ
1511:−
1503:
1481:∑
1477:∼
1474:⟩
1456:ϕ
1438:ϕ
1435:⟨
1399:ϕ
1284:arguments
1259:⋯
1188:−
1178:∞
1163:∑
1147:ζ
914:⏟
892:⏞
861:⏟
854:∇
851:−
839:Diffusion
833:⏟
818:∇
813:ν
806:−
789:⏞
767:⏟
755:∇
752:⋅
727:Variation
716:⏟
708:∂
698:∂
576:∩
531:
434:-function
414:over the
233:) in the
223:John Tate
2756:Robotics
2518:(eds.).
2503:(2006).
2358:Archived
2352:Claymath
2320:(eds.).
2305:(2006).
2277:(eds.).
2262:(2006).
2246:18759797
2172:Archived
2122:(eds.).
2107:(2006).
2079:(eds.).
2064:(2006).
2036:(eds.).
2021:(2006).
1996:Interfax
1930:BBC News
1864:31169641
1856:10839504
1682:See also
1618:mass gap
1588:Quantum
1579:spectrum
1382:mass gap
1280:function
925:External
872:Internal
334:3-sphere
178:Overview
61:(solved)
2773:RoboCup
2747:N-Prize
2706:M-prize
2667:Active
2463:Bibcode
2414:Bibcode
2364:29 June
2181:17 June
1936:16 June
1908:Bibcode
1664:(1975).
1598:Maxwell
1581:of the
1333:
1321:
1083:and to
1077:biology
332:to the
2614:
2588:
2526:
2481:
2432:
2328:
2285:
2244:
2166:
2156:Ganita
2130:
2087:
2044:
1862:
1854:
1834:Nature
1610:gluons
1606:chromo
1380:, the
1282:whose
1052:verify
966:fluids
934:source
881:source
310:closed
306:sphere
214:, and
2810:Other
2763:MAGIC
2735:Space
2508:(PDF)
2479:S2CID
2430:S2CID
2310:(PDF)
2267:(PDF)
2242:S2CID
2222:(PDF)
2175:(PDF)
2152:(PDF)
2112:(PDF)
2069:(PDF)
2026:(PDF)
1966:(PDF)
1955:(PDF)
1860:S2CID
1763:(PDF)
1544:with
1296:zeros
1123:zeros
1087:(see
1075:, to
404:Birch
239:Paris
2778:IARC
2612:ISBN
2586:ISBN
2580:and
2524:ISBN
2366:2021
2326:ISBN
2283:ISBN
2183:2022
2164:ISSN
2128:ISBN
2085:ISBN
2042:ISBN
2005:2024
1974:2010
1938:2011
1852:PMID
1674:and
1562:>
1294:has
1274:The
1067:and
1060:find
1012:for
960:The
640:Let
424:rank
422:has
406:and
402:The
344:and
324:Any
312:and
260:US$
225:and
122:US$
108:The
2471:doi
2422:doi
2234:doi
1842:doi
1838:405
1500:exp
1376:In
627:on
522:Hdg
506:of
462:= 1
457:at
237:in
2878::
2584:.
2566:;
2514:;
2499:;
2477:.
2469:.
2459:42
2457:.
2451:.
2428:.
2420:.
2410:31
2408:.
2402:.
2356:.
2348:.
2316:;
2273:;
2240:.
2230:33
2228:.
2224:.
2170:.
2160:66
2158:.
2154:.
2118:;
2075:;
2032:;
1993:.
1976:.
1928:.
1886:54
1884:.
1858:.
1850:.
1836:.
1832:.
1815:47
1813:.
1771:53
1769:.
1765:.
1678:.
1362:.
1105:.
1095:.
1079:,
1038:NP
1030:NP
1026:NP
1024:,
1021:NP
1018:,
997:.
666:.
631:.
510:.
498:,
477:.
464:.
444:,
375:.
348:.
241:.
210:,
206:,
202:,
198:,
194:,
163:.
151:,
147:,
143:,
139:,
135:,
2660:e
2653:t
2646:v
2620:.
2594:.
2550:.
2532:.
2485:.
2473::
2465::
2436:.
2424::
2416::
2368:.
2354:)
2334:.
2291:.
2248:.
2236::
2185:.
2136:.
2093:.
2050:.
2007:.
1940:.
1914:.
1910::
1866:.
1844::
1798:.
1735:)
1649:4
1644:R
1565:0
1557:0
1528:)
1524:t
1519:n
1507:(
1495:n
1491:A
1485:n
1471:)
1468:0
1465:,
1462:0
1459:(
1453:)
1450:t
1447:,
1444:0
1441:(
1408:)
1405:x
1402:(
1330:2
1327:/
1324:1
1256:+
1249:s
1245:3
1241:1
1236:+
1229:s
1225:2
1221:1
1216:+
1209:s
1205:1
1201:1
1196:=
1191:s
1184:n
1173:1
1170:=
1167:n
1159:=
1156:)
1153:s
1150:(
1127:s
1119:s
1034:P
1015:P
944:.
910:g
903:+
857:w
844:=
828:u
822:2
762:u
758:)
748:u
744:(
737:+
711:t
702:u
656:.
654:X
646:X
642:X
629:X
625:k
604:.
601:)
598:X
595:(
590:k
587:,
584:k
580:H
573:)
569:Q
565:,
562:X
559:(
554:k
551:2
547:H
543:=
540:)
537:X
534:(
526:k
460:s
455:r
448:)
446:s
442:E
440:(
438:L
432:L
427:r
420:E
336:.
97:e
90:t
83:v
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.