Cov teeb meem Unsolvable yog 7 feem ntau nthuav cov teeb meem lej. Txhua tus ntawm lawv tau thov ib zaug los ntawm cov kws tshawb fawb paub zoo, raws li txoj cai, nyob rau hauv daim ntawv ntawm hypotheses. Tau ntau xyoo lawm, cov lej lej thoob plaws ntiaj teb tau rack lawv lub hlwb los ntawm lawv cov kev daws teeb meem. Cov neeg ua tiav yuav tau txais txiaj ntsig nrog ib lab US las uas muab los ntawm Clay Institute.
Backstory
Hauv xyoo 1900, tus kws lej German zoo David Hilbert tau nthuav tawm cov npe ntawm 23 qhov teeb meem.
Kev tshawb fawb los daws lawv tau cuam tshuam loj rau kev tshawb fawb ntawm xyoo pua 20th. Tam sim no, feem ntau ntawm lawv tau tso tseg tsis paub tsis meej. Ntawm cov tsis tau daws los yog daws ib nrab yog:
- teeb meem ntawm qhov sib xws ntawm cov lej lej;
- txoj cai dav dav ntawm kev sib koom ua ke ntawm qhov chaw ntawm txhua qhov chaw;
- kev kawm lej ntawm lub cev axioms;
- Kev kawm ntawm cov ntawv plaub rau cov lej algebraic arbitraryodds;
- qhov teeb meem ntawm kev nruj me ntsis ntawm kev suav geometry ntawm Fyodor Schubert;
- etc.
Unexplored yog: qhov teeb meem ntawm kev nthuav qhov zoo-paub Kronecker theorem rau ib cheeb tsam algebraic ntawm rationality thiab Riemann hypothesis.
The Clay Institute
Nov yog lub npe ntawm lub koom haum ntiag tug tsis muaj txiaj ntsig hauv lub hauv paus hauv Cambridge, Massachusetts. Nws tau tsim muaj nyob rau hauv 1998 los ntawm Harvard lej A. Jeffey thiab businessman L. Clay. Lub hom phiaj ntawm lub koom haum yog kom nrov thiab txhim kho kev paub txog lej. Txhawm rau ua tiav qhov no, lub koom haum muab khoom plig rau cov kws tshawb fawb thiab cov neeg txhawb nqa kev cog lus rau kev tshawb fawb.
Nyob rau xyoo pua 21st, Clay Institute of Mathematics tau muab khoom plig rau cov neeg uas daws qhov paub tias yog cov teeb meem nyuaj tshaj plaws uas tsis tuaj yeem daws tau, hu rau lawv cov npe cov teeb meem nqi zog ib txhiab xyoo. Tsuas yog Riemann qhov kev xav tau suav nrog hauv Hilbert List.
Millennium Challenges
Lub Tsev Kawm Ntawv Clay cov npe thawj zaug suav nrog:
- Hodge cycle hypothesis;
- quantum Yang-Mills theory equations;
- Poincaré hypothesis;
- qhov teeb meem ntawm kev sib npaug ntawm cov chav kawm P thiab NP;
- Riemann hypothesis;
- Navier-Stokes sib npaug, ntawm qhov muaj nyob thiab smoothness ntawm nws cov kev daws teeb meem;
- Birch-Swinnerton-Dyer teeb meem.
Cov teeb meem kev ua lej qhib no yog qhov txaus siab heev, vim lawv tuaj yeem muaj ntau yam kev siv.
Grigory Perelman ua pov thawj dab tsi
Hauv xyoo 1900, tus kws tshawb fawb nto moo Henri Poincaré tau hais tias ib qho yooj yim txuas nrog 3-manifold tsis muaj ciam teb yog homeomorphic rau 3-sphere sphere. Nws cov pov thawj hauv cov ntaub ntawv dav dav tsis pom nyob rau ib puas xyoo. Tsuas yog nyob rau hauv 2002-2003, St. Petersburg mathematician G. Perelman luam tawm ib tug xov tooj ntawm cov khoom nrog ib tug daws rau cov teeb meem Poincaré. Lawv muaj cov nyhuv ntawm lub foob pob tawg. Xyoo 2010, Poincaré qhov kev xav tau raug tshem tawm los ntawm cov npe "Cov Teeb Meem Tsis Muaj Teeb Meem" ntawm Clay Institute, thiab Perelman nws tus kheej tau txais kev them nyiaj ntau heev vim nws, uas tom kawg tsis kam tsis piav qhia vim li cas rau nws qhov kev txiav txim siab.
Qhov kev nkag siab zoo tshaj plaws ntawm qhov uas tus kws lej Lavxias tau tswj hwm los ua pov thawj tuaj yeem muab los ntawm kev xav tias cov roj hmab disk rub mus rau lub khob noom cookie (torus), thiab tom qab ntawd lawv sim rub cov npoo ntawm nws lub voj voog rau hauv ib qho taw tes. Obviously qhov no tsis tau. Lwm qhov, yog tias koj ua qhov kev sim no nrog lub pob. Nyob rau hauv cov ntaub ntawv no, ib tug zoo li peb-dimensional sphere, uas tshwm sim los ntawm ib tug disk uas nws ncig tau rub mus rau ib tug taw tes los ntawm ib tug hypothetical qaum, yuav yog peb-dimensional nyob rau hauv kev nkag siab ntawm ib tug neeg zoo tib yam, tab sis ob-dimensional nyob rau hauv cov nqe lus ntawm kev ua lej.
Poincare tau hais tias qhov peb-dimensional sphere yog tib qhov "khoom" peb sab uas nws qhov chaw tuaj yeem cog lus rau ib qho, thiab Perelman tau tswj xyuas nws. Yog li, daim ntawv teev "Cov teeb meem tsis daws teeb meem" hnub no muaj 6 qhov teeb meem.
Yang-Mills theory
Qhov teeb meem lej no tau hais los ntawm nws cov kws sau ntawv hauv xyoo 1954. Cov qauv kev tshawb fawb ntawm txoj kev xav yog raws li hauv qab no:rau txhua pawg kev ntsuas qhov yooj yim, quantum spatial txoj kev xav tsim los ntawm Yang thiab Mills muaj nyob, thiab tib lub sijhawm muaj qhov tsis xws luag.
Hais lus rau tib neeg nkag siab, kev sib cuam tshuam ntawm cov khoom ntuj (particles, lub cev, nthwv dej, thiab lwm yam) tau muab faib ua 4 hom: electromagnetic, gravitational, qaug zog thiab muaj zog. Tau ntau xyoo, physicists tau sim los tsim ib qho kev tshawb xav dav dav. Nws yuav tsum dhau los ua ib qho cuab yeej los piav qhia tag nrho cov kev sib cuam tshuam no. Yang-Mills txoj kev xav yog lus lej uas nws tau los piav txog 3 ntawm 4 lub zog tseem ceeb ntawm xwm. Nws tsis siv rau lub ntiajteb txawj nqus. Yog li ntawd, nws tsis tuaj yeem suav hais tias Yang thiab Mills ua tiav hauv kev tsim txoj kev xav.
Dhau li ntawd, qhov tsis yog-linearity ntawm cov lus sib npaug tau ua rau lawv nyuaj heev los daws. Rau cov me me sib txuas tsis tu ncua, lawv tuaj yeem kwv yees li kev daws teeb meem nyob rau hauv daim ntawv ntawm kev cuam tshuam kev xav. Txawm li cas los xij, nws tseem tsis tau paub meej tias cov kev sib npaug no tuaj yeem daws tau nrog kev sib txuas zoo li cas.
Navier-Stokes equations
Cov kab lus no piav qhia txog cov txheej txheem xws li huab cua tam sim no, cov kua dej ntws, thiab turbulence. Rau qee qhov xwm txheej tshwj xeeb, kev tshuaj ntsuam xyuas ntawm Navier-Stokes kab zauv twb pom lawm, tab sis txog tam sim no tsis muaj leej twg ua tiav qhov no rau feem ntau. Nyob rau tib lub sijhawm, cov lej simulations rau cov txiaj ntsig tshwj xeeb ntawm kev ceev, ceev, siab, lub sijhawm, thiab lwm yam tuaj yeem ua tiav cov txiaj ntsig zoo. Nws tseem yuav tsum tau cia siab tias ib tus neeg yuav tuaj yeem siv qhov sib npaug ntawm Navier-Stokes rov qab.kev taw qhia, piv txwv li xam cov tsis siv lawv, lossis ua pov thawj tias tsis muaj kev daws teeb meem.
Birch-Swinnerton-Dyer teeb meem
qeb ntawm "Cov Teeb Meem Tsis Muaj Teeb Meem" kuj suav nrog qhov kev xav pom los ntawm cov kws tshawb fawb Askiv los ntawm University of Cambridge. Txawm tias 2300 xyoo dhau los, tus kws tshawb fawb Greek thaum ub Euclid tau muab cov lus piav qhia tag nrho ntawm cov kev daws teeb meem rau qhov sib npaug x2 + y2=z2.
Yog tias rau txhua tus lej tseem ceeb peb suav tus lej ntawm cov ntsiab lus ntawm qhov nkhaus modulo nws, peb tau txais cov lej tsis kawg. Yog tias koj tshwj xeeb "kua nplaum" nws rau hauv 1 txoj haujlwm ntawm qhov sib txawv ntawm qhov sib txawv, tom qab ntawd koj tau txais Hasse-Weil zeta muaj nuj nqi rau qhov kev txiav txim thib peb, qhia los ntawm tsab ntawv L. Nws muaj cov ntaub ntawv hais txog tus cwj pwm modulo txhua tus lej tseem ceeb ib zaug.
Brian Birch thiab Peter Swinnerton-Dyer tau pom zoo txog elliptic nkhaus. Raws li nws, cov qauv thiab tus naj npawb ntawm cov txheej txheem ntawm nws cov kev daws teeb meem muaj feem xyuam nrog tus cwj pwm ntawm L-function ntawm tus kheej. Qhov tam sim no unproven Birch-Swinnerton-Dyer conjecture nyob ntawm qhov kev piav qhia ntawm 3rd degree algebraic equations thiab yog tib txoj kev yooj yim heev los xam cov qeb ntawm elliptic curves.
Yuav kom nkag siab txog qhov tseem ceeb ntawm txoj haujlwm no, nws txaus los hais tias nyob rau hauv niaj hnub cryptography tag nrho cov chav kawm ntawm asymmetric systems yog raws li elliptic curves, thiab domestic digital kos npe qauv yog raws li lawv daim ntawv thov.
kev sib npaug ntawm cov chav kawm p thiab np
Yog tias qhov seem ntawm Kev Sib Tw Millennium yog kev ua lej nkaus xwb, ces qhov no muajmuaj feem xyuam rau qhov tseeb txoj kev xav ntawm algorithms. Qhov teeb meem hais txog qhov sib npaug ntawm cov chav kawm p thiab np, tseem hu ua qhov teeb meem Cooke-Levin, tuaj yeem tsim cov lus nkag siab raws li hauv qab no. Piv txwv tias cov lus teb zoo rau qee cov lus nug tuaj yeem tshawb xyuas sai txaus, piv txwv li, nyob rau hauv lub sijhawm polynomial (PT). Tom qab ntawd cov nqe lus puas yog tias cov lus teb rau nws tuaj yeem pom sai sai? Txawm tias yooj yim dua qhov teeb meem no suab zoo li no: puas yog nws tsis yooj yim dua los xyuas qhov kev daws teeb meem ntawm kev nrhiav nws? Yog tias qhov sib npaug ntawm cov chav kawm p thiab np tau ua pov thawj, ces txhua qhov teeb meem xaiv tuaj yeem daws tau rau PV. Tam sim no, ntau tus kws tshaj lij tsis ntseeg qhov tseeb ntawm cov lus no, txawm hais tias lawv tsis tuaj yeem ua pov thawj qhov txawv.
Riemann Hypothesis
Txog xyoo 1859, tsis pom muaj tus qauv uas yuav piav qhia tias tus lej tseem ceeb raug faib li cas ntawm cov lej ntuj. Tej zaum qhov no yog vim qhov tseeb tias kev tshawb fawb tau cuam tshuam nrog lwm yam teeb meem. Txawm li cas los xij, los ntawm nruab nrab ntawm lub xyoo pua 19th, qhov xwm txheej tau hloov pauv, thiab lawv tau dhau los ua ib qho tseem ceeb tshaj plaws uas kev lej pib cuam tshuam nrog.
Lub Riemann Kev Ntseeg, uas tau tshwm sim thaum lub sijhawm no, yog qhov kev xav tias muaj qee yam qauv hauv kev faib cov lej tseem ceeb.
Hnub no, ntau tus kws tshawb fawb niaj hnub ntseeg tias yog muaj pov thawj, ces nws yuav tsum tau hloov kho ntau lub hauv paus ntsiab lus ntawm niaj hnub cryptography, uas yog lub hauv paus ntawm ib feem tseem ceeb ntawm cov txheej txheem ntawm kev lag luam hluav taws xob.
Raws li Riemann kev xav, tus cwj pwmkev faib tawm ntawm primes yuav txawv txawv ntawm qhov tam sim no xav. Qhov tseeb yog tias txog tam sim no tsis muaj qhov system tau pom nyob rau hauv kev faib tawm ntawm tus lej tseem ceeb. Piv txwv li, muaj qhov teeb meem ntawm "cov ntxaib", qhov txawv ntawm qhov uas yog 2. Cov lej no yog 11 thiab 13, 29. Lwm cov lej tseem ceeb hauv pawg. Cov no yog 101, 103, 107, thiab lwm yam. Cov kws tshawb fawb tau xav ntev ntev tias cov pawg no muaj nyob ntawm cov lej tseem ceeb heev. Yog tias lawv pom, ces lub zog ntawm niaj hnub crypto yuam sij yuav nug.
Hodge cycle hypothesis
Qhov no tseem tsis tau daws teeb meem tau tsim nyob rau xyoo 1941. Hodge qhov kev xav pom tau tias muaj peev xwm kwv yees cov duab ntawm txhua yam khoom los ntawm "gluing" ua ke yooj yim lub cev ntawm qhov ntev dua. Txoj kev no tau paub thiab ua tiav siv tau ntev. Txawm li cas los xij, nws tsis paub tias yuav ua kom yooj yim npaum li cas.
Tam sim no koj paub tias muaj teeb meem dab tsi uas tsis tuaj yeem daws tau tam sim no. Lawv yog cov kev tshawb fawb los ntawm ntau txhiab tus kws tshawb fawb thoob ntiaj teb. Nws tseem yuav tsum tau cia siab tias lawv yuav raug daws nyob rau yav tom ntej, thiab lawv daim ntawv thov siv tau yuav pab tib neeg nkag mus rau qhov tshiab ntawm kev txhim kho thev naus laus zis.
