Timeline of Mathematics and Theoritical Physics

8/13/2019 Timeline of Mathematics and Theoritical Physics

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A Timeline of Mathematics and Theoretical Physics

-1500 Babylonians establish the metric of flat 2-dimensional space by observation, in their effortsto keep track of land for legal and economic purposes.

-518

Pythagoras, a Greek educated by mystics in Egypt and Babylon, founds community of menand women calling themselves mathematikoi,in southern taly. !hey believe that reality isin essence mathematical. Pythagoras noted that vibrating lyre strings with harmoniousnotes have lengths that are proportional by a whole number. !he Pythagorean theoremproves by reasoning what the Babylonians figured out by measurement "### years earlier.

-387

Plato, after traveling to taly and learning about the Pythagoreans, founds his $cademy in$thens, and continues to develop the idea that reality must be e%pressible in mathematicalterms. But $thens at that time has developed a notoriously misogynist culture. &nlike hisrole model Pythagoras, whose school developed many women mathematikoi, Plato does not

allow women to participate.

-300Euclid of $le%andria, a gifted teacher, produces Elements, one of the top mathematicste%tbooks of recorded history, which organi'es the e%isting (editerranean understanding ofgeometry into a coherent logical framework.

-225onian mathematician $pollonius writes Conics, and introduces the terms ellipse, parabolaand hyperbola to describe conic sections.

-140)icaean mathematician and astronomer *ipparchus develops what will be known astrigonometry.

150

The Almagestby $le%andrian astronomer and mathematician +laudius Ptolemy asserts thatthe un and planets orbit around the Earth in perfect circles. Ptolemys work is so

influential that will become official church doctrine when the +hristians later come to ruleEurope.

415

$s a glorious 2### years of ancient (editerranean mathematics and science comes to aclose, *ypatia of $le%andria, a renowned teacher, mathematician, astronomer, andpriestess of sis, is kidnapped from a public religious procession and brutally murdered by amob of angry +hristian monks.

628

*indu mathematician-astronomer Brahmagupta writes Brahma- sphuta- siddhanta!he/pening of the &niverse0. *indu mathematicians develop numerals and start investigatingnumber theory.

830

!he spread of slam leads to the spread of written $rabic language. $s ancient Greek and*indu works are translated into $rabic, a culture of mathematics and astronomy develops.

!he peak of this cultural flowering is represented by $rabic mathematician $l-1hwori'mi,working at the *ouse of isdom in Baghdad, who develops what will be known as algebrain his work Hisab al-jabr w'al-muqabala.

1070

ranian poet, mathematician and astronomer /mar 1hayyam begins his Treatise onemonstration o! "roblems o! Algebra, classifying cubic e3uations that could be solved byconic sections. 1hayyam was such a brilliant poet that history has nearly forgotten that hewas also a brilliant scientist and mathematician. The mo#ing !inger writes$$$
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1120$delard of Bath translates works of Euclid and $l-1hwori'mi into 4atin and introduces themto European scholars.

1482Euclids Elementsis published using the revolutionary new technology of the printing press,leading to a revolution in education and scholarship as information becomes more difficultfor authorities to control.

1543

+opernicus publishes e re#olutionibus orbium coelestium/n the revolutions of theheavenly spheres0 asserting that the Earth and planets revolve about the un. !he +atholic+hurch has accorded an official holy status to Ptolemys geocentric &niverse. +opernicusavoids prosecution as a heretic by waiting until the end of his own life to publish hiscontroversial claims.

1589Pisa &niversity mathematics instructor Galileo Galilei studies the motion of ob5ects andbegins a book e %otu/n (otion0 which he never finishes.

1602Galileo observes that the period of a swinging pendulum is independent of the amplitude ofthe swing.

1609

6ohannes 1epler claims in the 5ournalAstronomia &o#athat the orbit of (ars is an ellipse

with the un at one focus, and sweeps out e3ual areas in e3ual time. *e will latergenerali'e these into his famous !hree 4aws of Planetary (otion.

1609

Galileo makes his first telescope. *is observations of the moon show that it looks like avery large lumpy rock, not a divinely smooth and perfect shining Platonic heavenly orb.!his discovery has enormously distressing cultural reverberations for estern culture andreligion.

1614cottish theologian 6ohn )apier, who does mathematics as a hobby, publishes his discoveryof the logarithm in his work %iri!ici logarithmorum canonis descriptio$

1615

1eplers mother, 7rau 1atharina 1epler, is accused of witchcraft by a local prostitute.European witch hunting was at its peak during 1eplers career, and witch hunting wassupported by all levels of society, including secular officials and intellectuals in universities.

1epler spends the ne%t several years making legal appeals and hiding his mother from legalauthorities seeking to torture her into confessing to witchcraft. E%amining an accused witchad torturamwas a standard court procedure during this era.

1620&nder court order, 1eplers mother is kidnapped in the middle of the night from herdaughters home and taken to prison. 1epler spends the ne%t year appealing to the duke of8rttemberg to prevent his imprisoned mother from being e%amined ad torturam.

1621

/n eptember 29, 1atharina 1epler is taken from her prison cell into the torture room,shown the instruments of torture and ordered to confess. he replies "Do with me whatyou want. Even if you were to pu one vein after another out of my !oy# $ wouhave nothin% to amit#"and says the 4ords Prayer. he is taken back to prison. he isfreed on /ctober : upon order of the duke, who rules that her refusal to confess under

threat of torture proves her innocence. *e also orders her accusers to pay the cost of hertrial and imprisonment.

1622$fter having spent most of the last seven years under the legal threat of imminent torture,1atharina 1epler dies on $pril ";, still being threatened with violence from those who insistshe is a witch.

1624Pope &rban


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16281epler uses )apiers logarithms to compute a set of astronomical tables, the udolphineTables, whose accuracy is so impressive that it leads to the 3uiet acceptance of theheliocentric solar system by everyone in the shipping industry.

1629

Bas3ue mathematician Pierre de 7ermat, the founder of modern number theory, begins hisbrilliant career by reconstructing the work of $pollonius on conic sections . 7ermat and

=escartes pioneer the application of algebraic methods to solving problems in geometry.

1632Galileo publishes ialogue concerning the two greatest world s(stems, which arguesconvincingly for the +opernican view that the Earth and planets revolve around the un.

1633!he n3uisition calls Galieo to >ome to answer charges of heresy against the +atholic+hurch.

1637

=escartes publishes his revolutionary iscours de la m)thode=iscourse on (ethod0containing three essays on the use of reason to search for the truth. n the third essay=escartes describes analytic geometry, and uses the letters %,y,'0 for the coordinatesystem that will later bear his name.

1642 Galileo dies at his villa in 7lorence, still under house arrest from charges of heresy.

1663+ambridge mathematician saac Barrow delivers lectures on modern methods of computingtangents that inspire his student saac )ewton towards developing calculus

1665

)ewtons ?miraculous years? in math and physics, when he discovers the derivative, whichhe sees as a ratio of velocities called !lu*ions, and the integral, which he sees as a !luentofthe !lu*ions. )ewton shows that the fluent and flu%ion are inversely related, a result nowcalled the Fundamental Theorem of Calculus. )ewton also develops his ideas on opticsand gravitation. *e tries to publish his work in "@A", but the publisher goes bankrupt.

1683

6acob Bernoulli, who studied mathematics and astronomy against the wishes of his career-minded parents, teaches )ewtonian mechanics at the &niversity of Basel, and turnsmathematical physics into a family business.

1684

4eibni' publishes the beginning of his work on differential and integral calculus. *ediscovers the 7undamental !heorem of +alculus in his own way. 4eibni' originates most ofthe current calculus notation including the integral sign. *e portrays an integral as a sum ofinfinitesimals, a concept re5ected by )ewton.

1687

)ewton publishes "rincipia %echanicaafter Edmund *alley convinces )ewton to write uphis alleged proof that an inverse s3uare force law leads to elliptical orbits. )ewtons 4aws of(otion and 4aw of Gravitation lead to the development of theoretical physics itself. !hisevent marks a permanent change in the relationship between human beings and the&niverse.

1693)ewton has a nervous breakdown after his close companion 7atio =e =uillier becomes illand has to return to wit'erland.

1696 Brachistochrone problem solved by 6acob and 6ohann Bernoulli, an early result in thecalculus of variations.

1712 !hanks to a campaign waged by )ewton, a commission appointed by >oyal ociety of4ondon President saac )ewton rules that 4eibni' is guilty of plagiarism against )ewton inthe discovery of calculus. English mathematics and theoretical physics go into declinebecause those loyal to )ewton are hesitant to adopt 4eibni' infinitesimal and his clean,


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intuitively appealing notation.

17364eonhard Euler begins the field of topology when he publishes his solution of the1onigsberg Bridge problem.

1738 H(drod(namicsby =aniel Bernoulli

1748 !he multitalented Euler begins the fields of mathematical analysis and analytical mechanicswith +ntroductio in anal(sin in!initorum. Euler introduces the formula ei* cos % C i sin %

1758

6oseph-4ouis 4agrange finds the complete general solution to the )ewtonian e3uations ofmotion for a vibrating string, which e%plains the harmonic relations observed by Pythagoras22 centuries ago.

1770 *yperbolic trigonometry -- cosh, sinh -- is developed.

1772

*enry +avendish, a wealthy but paranoid recluse, discovers that the electrostatic force isdescribed by an inverse s3uare law similar to gravity, but doesnt tell anyone in the sciencecommunity.

17884agrange further develops the analytical mechanics of Euler when he publishes %)caniqueAnal(tique, revealing )ewtonian mechanics to be a rich field of e%ploration formathematicians.

1789

$ristocrat +harles-$ugustin de +oulomb, hiding from the 7rench >evolution after thestorming of the Bastille, shows that the electrostatic force between electric charges wasvery well described by an inverse s3uare law, in full analogy with )ewtonian gravity. !hisbecomes known as +oulombs 4aw, even though *enry +avendish was the first one todemonstrate it.

17934agrange is arrested during the >eign of !error, but is rescued by $ntoine-4aurent4avoisier, the founder of modern chemistry. &nfortunately, 4avoisiers career in chemistryis ended when he is taken to meet (adame Guillotine on (ay 9, "AD:.

1799Pierre-imon 4aplace publishes his work Trait) du %)canique C)leste!reatise on +elestial(echanics0 using differential e3uations to solve problems in planetary motion and fluidflow.

1807

$fter serving as a member of the >evolutionary +ommittee that terrori'ed 7rance, sent+oulomb into hiding, arrested 4agrange and guillotined 4avoisier, a repentant 6eanBaptiste 6oseph 7ourier causes controversy with his memoir n the "ropagation o!Heat in olid Bodies. *is former teachers 4aplace and 4agrange ob5ect to his use ofinfinite trigonometric series, which we now call 7ourier series. 7ourier later wins theParis nstitute (athematics Pri'e for solving the problem of heat propagation, over the

repeated ob5ections of 4aplace and 4agrange.

1817

6ohann 1arl 7riedrich Gauss begins working on non-Euclidean geometry, and lays thefoundations of differential geometry, but doesnt publish because he is afraid of thecontroversy that would result.

1820

=anish physicist *ans +hristian /ersted studied the way an electric current in a wirecould move the magnetic needle of a compass, which strongly suggested thatelectricity and magnetism were related somehow.


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1823

!ransylvanian mathematician 6nos Bolyai, despite being warned against it by hisfather, tosses out Euclids 7ifth $%iom and shows that non-Euclidean geometry ispossible. Gauss calls him a genius of the first order, but then crushes the young manby telling him he discovered it years ago but failed to publish due to his own fear ofcontroversy.

1826 Elliptic functions are developed by Gauss, 6acobi and $bel.

1826

n his book %emoir on the %athematical Theor( o! Electrod(namic "henomena,.niquel( educed !rom E*perience. $ndrF (arie $mpre gave a mathematicalderivation of the magnetic force between two parallel wires with electric current, whatwe now call $mpres 4aw.

1827/hms 4aw of electrical resistance is published in his book ie gal#anische /ette,mathematisch bearbeitet$

1827$ugustin-4ouis +auchy develops the calculus of residues, beginning his work inmathematics that made comple% analysis one of the most important analytical tools ofmodern theoretical physics, including string theory.

1828elf-educated English mill worker George Green publishes his work on the use ofpotential theory to solve partial differential e3uations, and develops one of the mostpowerful mathematical technologies in theoretical physics -- the &reen fun'tion.

1829>ussian mathematician )ikolai vanovich 4obachevsky publishes his independentdiscovery of non-Euclidean geometry in the /a0an %essenger. Hears later, one of hisphysics students will become known to history as 4enins father.

1831Evariste Galois develops the nascent group theory with his work on the permutationgroup.

1831(ichael 7araday discovers magnetic induction, now known as 7aradays 4aw, wheremoving magnetism creates electricity, and this result increases support for the idea ofa unified theory of electricity and magnetism.

1829

7rench mathematician 6oseph 4iouville begins to work on boundary value problems inpartial differential e3uations, leading to turm-4iouville theory. *e then develops thestudy of conformal transformations, and later proves the 4iouville !heorem regardingthe invariance of the measure of phase space under what will later be called*amiltonian flow.

1834

illiam >owan *amilton applies his mathematical development of characteristicfunctions in optics to mechanics and the enormous and potent mathematicaltechnology of *amiltonian dynamics is born.

1840 1arl eierstrass begins his work on elliptic functions.

1843

$fter a period of emotional distress and alcohol abuse, *amilton finally deduces the

noncommutative multiplication rule for 3uaternions. *is first publication on the sub5ectis to carve the 3uaternion formula into a bridge.

1844 *ermann Grassmann develops e%terior algebra and the Grassmannian.

1851

Bernhard >iemann submits his Ph.=. thesis to his supervisor Gauss. n his thesis hedescribes what is now called a >iemann surface, an essential element in understandingstring theory.


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1854 George Boole develops Boolean logic in 1aws o! Thought$

1871

)orwegian mathematician (arius ophus 4ie publishes work on 4ie algebras, openingup the field of differential topology and paving the way for gauge field theory "##years later.

1873

6ames +lerk (a%well publishes a set of e3uations from which all of the observed lawsof electromagnetism could be derived through mathematics. !hese e3uations turn outto have solutions that describe waves traveling through space with a speed that agreeswith the measured speed of light.(a%well makes the bold conclusion that light therefore must consist of electromagneticwaves, writing that he could "('ar'ey avoi the inferen'e that i%ht 'on(i(t( inthe tran(ver(e unuation( of the (ame meium whi'h i( the 'au(e of ee'tri'

an ma%neti' phenomena."

1874 +antor invents set theory.

1878 illiam +lifford develops +lifford algebras from the work of Grassmann and *amilton.

1878 $rthur +ayley writes The theor( o! groups, where he proved that every finite groupcan be represented as a group of permutations.

1883ilhelm 1illing works on n-dimensional non-Euclidean geometry and 4ie algebras,work that later results in the concept of a 1illing vector, a powerful tool in differentialgeometry, 3uantum gauge field theory, supergravity and and string theory.

1884

*einrich *ert' rewrites )a*we+( E,uation(in a more elegant notation where thesymmetry between electricity and magnetism was obvious. *ert' then creats the firstradio waves and microwaves in his laboratory and shows that these electromagneticwaves behaved 5ust as observable optical light behaved, proving that i%ht wa(ee'troma%neti' raiation, as (a%well had predicted.

1884

4udwig Bolt'mann makes a theoretical derivation of black body radiation using(a%wells e3uations and thermodynamics, confirming the "9AD result measurede%perimentally by 6osef tefan. !heir result, the tefan-Bolt'mann 4aw, is not 3uiteright, and the correct solution in the ne%t century will mark the beginning of 3uantumtheory.

1887

(ichelson and (orley measure the Earths velocity through the ether to be 'ero,strongly suggesting that there is no ether, and that the velocity of light is the same forall observers, a result whose full implications have changed the world forever.

1894 Elie +artan classifies simple 4ie algebras

1895*enri PoincarF publishesAnal(sis itus, and gives birth to the field of algebraictopology.

1897 Electron discovered by 6.6. !hompson.

1899

*endrik 4orent' becomes the third person, after


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1901

(a% Planck makes his quantum h(pothesis-- that energy is carried by indistinguishableunits called quanta, rather than flowing in a pure continuum. !his hypothesis leads to asuccessful derivation of the black body radiation law, now called Plancks 4aw, although in"D#" the 3uantum hypothesis as yet had no e%perimental support. !he unit of 3uantumaction is now called Plancks constant.

1905

wiss patent clerk $lbert Einstein proposes Plancks 3uantum hypothesis as the physicsunderlying the photoelectric effect. Planck wins the )obel Pri'e in "D"9, and Einstein in"D2", for developing 3uantum theory, one of the two most important developments in2#th century physics.

1905

Einstein publishes his simple, elegant pecial !heory of >elativity, making mincemeat ofhis competition by relying on only two ideasI ". !he laws of physics are the same in allinertial frames, and 2. !he speed of light is the same for all inertial observers.

1905

PoincarF shows that 4orent' transformations in space and time plus rotations in spaceform a group, which comes to be known as the 4orent' group. !he 4orent' group plustranslations in space form a group called the PoincarF group.

1907(inkowski publishes >aum und Jeit pace and !ime0, and establishes the idea of aspacetime continuum

1909*ilberts work on integral e3uations later leads to the concept of a i!ert (pa'ein3uantum mechanics

1915

Emmy )oether publishes oether+( /heorem, discovering the relationship betweensymmetries and conserved currents that was crucial to the later development of 3uantumgauge field theory and string theory

1915

Einstein, with *ilbert in stiff competition, publishes his stunning General !heory of>elativity, and is lucky enough to be able to find observational support for his theory rightaway, in the perihelial advance of (ercury, and the deflection of starlight by the un.

1916

German astrophysicist 1arl chwar'schild, serving on the >ussian front in , mailsEinstein his paper on the chwar'schild metric and Einstein presents it at a meeting ofthe Prussian $cademy of ciences. i% months and another ma5or paper later,chwar'schild dies of illness on the front.

1921

!heodor 1alu'a follows Einsteins advice and publishes his highly unorthodo% ideas aboutunifying gravity with electromagnetism by adding an e%tra dimension of space that iscompactified into a small circle. 1alu'a-1lein compactification will become a rich sub5ectof e%ploration in particle theory @# years later.

1925

erner *eisenberg shows that his 3uanti'ed probability operators form a non-commutative algebra. Born and 6ordan point out to him that this is a matri% algebra, andthe matri% formulation of 3uantum mechanics is born. *e gets the )obel Pri'e in "D;2.

19244ouis duc de Broglie proposes the particle-wave duality of the electron in his doctoralthesis at the orbonne. *e gets the )obel Pri'e in "D2D.

1926$fter learning of the work of de Broglie, Erwin chrKdinger develops his wave e3uationversion of 3uantum mechanics, and unravels its relationship to the matri% formulation of3uantum mechanics by *eisenberg. *e shares the )obel Pri'e with =irac in "D;;.

1926Houng +ambridge math student Paul =irac discovers the operator algebra behind*eisenbergs &ncertainty Principle for his doctoral thesis.


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1927 *eisenberg discovers the &ncertainty Principle that bears his name.

1928

=irac introduces a relativistic 3uantum e3uation for the electron, an e3uation now knownas the Dira' e,uation. *is e3uation predicts the discovery of the positron, and heshares the )obel Pri'e with chrodinger in "D;;.

1928 erner *eisenberg, *ermann eyl and Eugene igner begin an e%ploration of symmetrygroups in 3uantum mechanics that has far-reaching conse3uences.

1929Edwin *ubble, with the help of his mule driver *umason, observes the redshift of distantgala%ies and concludes that the &niverse is e%panding.

1931 Einstein stops using the cosmological constant to keep the &niverse from e%panding.

1931=irac shows that the e%istence of magnetic monopoles would lead to electric charge3uanti'ation.

1931Georges =e >ham goes to work on his famous theorem in cohomology and characteristicclasses, results that would become very important in string theory.

1935

Houng physicist ubramahnyan +handrasekhar is attacked by famous astronomer $rthur

Eddington for his report that there is a stellar mass limit beyond which collapse to whatwe now call a black hole is inevitable. +handrasekhar wins the )obel Pri'e in "D9; for hiswork on stellar evolution.

1938 igner constructs a class of irreducible unitary representations of the 4orent' group

1939

Elements de %athematique,by )icholas Bourbaki, pseudonym for a group of youngmathematicians at the Ecole )ormale in Paris, is begun. !his e%tended set of works aimsto set down in writing what is no longer in doubt, but rather on a boring and rigorousfooting, in modern mathematics.

1943+hinese mathematician hiing-hen +hern begins his work on characteristic classes andfiber bundles that will become an important tool for understanding 3uantum gauge

theories and string theory.

1948

>ichard 7eynman, 6ulian chwinger and !omonaga hinichiro report that the divergentintegrals that plague the 3uantum gauge field theory of electrodynamics LE=0 can besensibly dealt with through the process of renormali'ation.

1953

Based on particle scattering data, (urray Gell-(ann suggests that there is a new3uantum number, called hypercharge, which we now call stangeness and recogni'e as apart of the 3uark model coming from the strange 3uark. Gell-(ann receives the )obelPri'e in "D@D for his work on the 3uark model.

1954

Gell-(ann and 7rancis 4ow develop the idea that the physical content of a 3uantumtheory should be invariant under a change of scale in the theory. !his is calledrenormali'ation group, and it turns out to constrain 3uantum field theories enough to

make it a very powerful tool for analy'ing asymptotic behavior of 3uantum theories.

1954+.). Hang and >. (ills develop non-$belian gauge invariance, an idea that takes "A yearsto gain acceptance, and then revolutioni'es particle physics.

1954

Eugenio +alabi con5ectures the e%istence of a 1Mhler manifold with a >icci-flat metric witha vanishing first +hern class, and a given comple% structure and 1Mhler class. !his funny-sounding stuff will eventually become of ma5or importance in understanding superstring


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theory.

1964+ambridge mathematician >oger Penrose proves that a black hole spacetime mustcontain behind the black hole event hori'on a singularity where spacetime physics ceasesto make good sense.

1964Gell-(ann and George Jweig independently propose fundamental particles that Gell-(annsucceeds in naming ?3uarks?.

1964Peter *iggs, 7rancois Englert and >. Brout suggest a method of breaking 3uantum gaugesymmetry that is later called the *iggs mechanism.

1967

n his paperA %odel o! 1eptons, teven einberg relies on 4ie group theory combinedwith 3uantum field theory to e%plain the weak nuclear and electromagnetic forces in asingle theory, using the *iggs mechanism to give mass to the weak bosons. $dbus alamand heldon Glashow share the )obel Pri'e with einberg in "DAD for Electroweak!heory.

1967

idney +oleman and 6effrey (andula prove that well-behaved particle scattering theories

cant have symmetry algebras that relate particles of different spin. But the strictconse3uences of the +oleman-(andula !heorem were avoided by the supersymmetryalgebras that were discovered a few years later.

1968

(ichael $tiyah and sadore inger begin their work on The +nde* o! Elliptic perators.!hey prove the $tiyah-inger inde% theorem, a powerful mathematical result that willlater be used e%tensively in theoretical physics.

1968Gabriele


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discovered by 6ulius ess and Bruno Jumino.

1974

tephen *awking combines 3uantum field theory with classical general relativity andpredicts that black holes radiate through particle emission, behave as thermodynamicob5ects, and decay with a finite lifetime into ob5ects that we dont yet understand.

1974 (agnetic monopole solutions of non-$belian gauge field theories are found separately byt *ooft and (oscow physicist $le%ander Polyakov.

19746oel cherk and 6ohn chwar' propose string theory as a theory of 3uantum gravity, anidea that takes ten years to be widely appreciated.

1974*oward Georgi and heldon Glashow propose &N0 for a ?Grand &nified !heory? G&!0of all forces e%cept gravity, the theory predicts that protons could decay.

1975

nstanton solutions of Hang-(ills e3uations are discovered by Belavin, Polyakov, $.chwar' and !yupkin. !his is e%citing because instantons can tell us about non-perturbative physics that is not approachable by other means of calculation.

1976hing-!ung Hau proves the +alabi con5ecture and discovers the +alabi-Hau space, an

important development for later progress in string theory.

1980$lan Guth puts forward the idea of an inflationary phase of the early &niverse, before theBig Bang.

1981 (ichael Green and 6ohn chwar' develop superstring theory.

1981$fter choen and Hau do it in a more traditional manner, Ed itten uses supersymmetryto prove the positive mass con5ecture.

1982(athematician 1aren &hlenbeck shows that Hang-(ills instantons discovered byphysicists can be used as a powerful analytical tool in abstract mathematics.

1983

itten and 4uis $lvare'-GaumF derive general formulas for gauge and gravitationalanomalies in 3uantum field theories in any dimension. !hey show that the gravitational

anomalies cancel in type B superstring theory.

1983(athematics graduate student imon =onaldson discovers e%otic :-manifolds, usinginstanton techni3ues learned in part from &hlenbeck.

1984(ichael Green and 6ohn chwar' show that superstring theory is free from 3uantumanomalies if the spacetime dimension is "# and the 3uantum gauge symmetry is /;20or E9 times E9.

1984Gross, *arvey, (artinec and >ohm find another class of anomaly-free superstringtheories, and call it the heterotic string theory.

1985+andelas, trominger, *orowit' and itten propose the use of +alabi-Hau spaces for thee%tra dimensions in heterotic string theory.

1991+onnes and 4ott develop non-commutative geometry, which will find its way into theheart of string theorists at the turn of the millennium.

1993

n search of an understanding of black hole entropy, t *ooft suggests the idea that theinformation in a ;C"-dimensional system cannot be greater than what is need to store itas an image in 2C" dimensions. usskind generali'es this idea and applies it to stringtheory in his paper !he orld as a *ologram, and the *olographic Principle is born.
http://arxiv.org/abs/gr-qc/9310026http://arxiv.org/abs/hep-th/9409089http://arxiv.org/abs/gr-qc/9310026http://arxiv.org/abs/hep-th/9409089


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1994)athan eiberg and Ed ittendiscover electric-magnetic duality in )2 supersymmetricgauge theory in four spacetime dimensions, with very important applications in bothmathematics and string theory.

1995

ittenand !ownsendintroduce the idea of !ype $ superstring theory as a special limitof ""-dimensional supergravity theory with 3uanti'ed membranes. !his begins the (-

theory revolution in superstring theory, and leads people to ponder the role of spacetimein string theory.

1995$ndrew iles, with help from >ichard !aylor, completes a rigorous proof of 7ermats 4ast!heorem.

19956oseph Polchinskiignites the =-brane revolution in string theory with his paper describinge%tended ob5ects in string theory formed by dual open strings with =irichlet boundaryconditions.

1996

n their paper (icroscopic /rigin of Black *ole Entropy, $ndy trominger and +umrun

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Timeline of Mathematics and Theoritical Physics