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== Istorija ==
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The index problem for elliptic differential operators was posed by Israel Gelfand.<ref>{{harvs|txt|first=Israel |last=Gel'fand|authorlink=Israel Gelfand|year=1960}}</ref> He noticed the homotopy invariance of the index, and asked for a formula for it by means of [[topological invariant]]s. Some of the motivating examples included the [[Riemann–Roch theorem]] and its generalization the [[Hirzebruch–Riemann–Roch theorem]], and the [[Hirzebruch signature theorem]]. [[Friedrich Hirzebruch]] and [[Armand Borel]] had proved the integrality of the [[Â genus]] of a spin manifold, and Atiyah suggested that this integrality could be explained if it were the index of the [[Dirac operator]] (which was rediscovered by Atiyah and Singer in 1961).
 
[[Izrael Gelfand]] je postulirao indeksni problem za eliptične diferencijalne operatore.<ref>{{harvs|txt|first=Israel |last=Gel'fand|authorlink=Israel Gelfand|year=1960}}</ref> On je uočio homotopnu invarijansu indeksa, i zatražio formulu za njeno izražavanje pomoću [[Topological property|topoloških invarijanti]]. Neki od motivirajućih primera obuhvaćali su [[Riemann–Roch theorem|teoremu Riman-Roča]] i njenu generalizaciju [[Hirzebruch–Riemann–Roch theorem|teoremu Hirzebruč-Riman-Roča]] i [[Hirzebruch signature theorem|Hirzebručovu teoremu potpisa]]. [[Friedrich Hirzebruch|Fridrih Hirzebruč]] i [[Armand Borel]] su dokazali integralnost [[Genus of a multiplicative sequence|Â vrste]] spinske mnogostukosti, a Ati je sugerisao da se ovaj integritet može objasniti ako on predstavlja indeks [[Dirac operator|Dirakovog operatora]] (koji su ponovo otkrili Ati i Zinger 1961. godine).
The Atiyah–Singer theorem was announced by {{harvtxt|Atiyah|Singer|1963}}. The proof sketched in this announcement was never published by them, though it appears in the book {{harv|Palais|1965}}. It appears also in the "Séminaire Cartan-Schwartz 1963/64" {{harv|Cartan-Schwartz|1965}} that was held in Paris simultaneously with the seminar led by [[Richard Palais]] at [[Princeton University]]. The last talk in Paris was by Atiyah on manifolds with boundary. Their first published proof {{harv|Atiyah|Singer|1968a}} replaced the [[cobordism]] theory of the first proof with [[K-theory]], and they used this to give proofs of various generalizations in the papers {{harvs|txt|last=Atiyah|last2=Singer|year1=1968a|year2=1968b|year3=1971a|year4=1971b}}.
 
Ati–Zingerovu teoremu su objavili Ati i Zinger<ref>{{harvtxt|Atiyah|Singer|1963}}</ref>. Oni nisu objavili dokaze skicirane u ovoj najavi, iako se dokazi pojavljuju u knjizi objavljenoj par godina kasnije.<ref>(Palais 1965)</ref> Dokazi su takođe predstavljeni na naučnom skupu „Séminaire Cartan-Schwartz 1963/64”<ref>{{harv|Cartan-Schwartz|1965}}</ref> koji je održan u Parizu istovremeno sa seminarom koji je na [[Универзитет Принстон|Univerzitetu Prinston]] vodio [[Richard Palais|Ričard Palais]]. Ati je održao poslednje predavanje u Parizu o mnogostrukostima sa granicama. Njihov prvi objavljeni dokaz<ref>{{harv|Atiyah|Singer|1968a}}</ref> je zamenio teoriju [[cobordism|kobordizma]] prvog dokaza sa [[K-theory|K-teorijom]], i oni su to koristili za dokaz raznih generalizacija u naknadnim radovima.<ref>{{harvs|txt|last=Atiyah|last2=Singer|year1=1968a|year2=1968b|year3=1971a|year4=1971b}}</ref>
*'''1965:''' [[Sergei Novikov (mathematician)|Sergey P. Novikov]] {{harv|Novikov|1965}} published his results on the topological invariance of the rational Pontrjagin classes on smooth manifolds.
 
* [[Robion Kirby]] and [[Laurent C. Siebenmann]]'s results {{harv|Kirby|Siebenmann|1969}}, combined with [[René Thom]]'s paper {{harv|Thom|1956}} proved the existence of rational Pontryagin classes on topological manifolds. The rational Pontryagin classes are essential ingredients of the index theorem on smooth and topological manifolds.
*'''1965:''' [[Sergei Novikov (mathematician)|Sergej P. Novikov]]<ref>{{harv|Novikov|1965}}</ref> je objavio svoje rezultate o topološkoj invarijansi racionalnih [[Лав Понтрјагин|Pontrjaginovih]] klasa na glatnim mnogostrukostima.
*'''1969:'''{{harvs|txt|first=Michael F.|last=Atiyah|authorlink=Michael Atiyah|year=1970}} defines abstract elliptic operators on arbitrary metric spaces. Abstract elliptic operators became protagonists in Kasparov's theory and Connes's noncommutative differential geometry.
*'''1969:''' Rezultati [[Robion Kirby|Robina Kirbija]] i [[Laurent C. Siebenmann|Lorenta Sibermana]]<ref>{{harv|Kirby|Siebenmann|1969}}</ref> u kombinaciji sa [[René Thom|Rene Tomovom]] publikacijom <ref>{{harv|Thom|1956}}</ref> dokazali su postojanje racionalnih Pontrjaginovih klasa na topološkim mnogostrukostima. Racionalne Pontrjaginove klase su esencijalni sastojci indeksne teoreme na glatkim i topološkim mnogostrukostima.
*'''1971:''' {{harvs|txt|first=Isadore M.|last=Singer|authorlink=Isadore Singer|year=1971}} proposes a comprehensive program for future extensions of index theory.
*'''1969:''' Mičel Ati je definisao apstraktne eliptične operatore na proizvoljnim metričkim prostorima.<ref>{{harvs|txt|first=Michael F.|last=Atiyah|authorlink=Michael Atiyah|year=1970}}</ref> Apstraktni eliptični operatori su postali protagonisti u Kasparovoj teoriji i Konesovoj nekomutativnoj diferencijalnoj geometriji.
*'''1972:''' {{harvs|txt|first=Gennadi G.|last=Kasparov|year=1972}} publishes his work on the realization of K-homology by abstract elliptic operators.
*'''1971:''' Isador Zinger je predložio sveobuhvatni program za buduća proširenja indeksne teorije.<ref>{{harvs|txt|first=Isadore M.|last=Singer|authorlink=Isadore Singer|year=1971}}</ref>
*'''1973:''' {{harvs|txt=yes|last1=Atiyah |author2-link=Raoul Bott|first2=Raoul|last2=Bott|author3-link=Vijay Kumar Patodi|first3=Vijay |last3=Patodi|year=1973}} gave a new proof of the index theorem using the [[heat equation]], described in {{harv|Melrose|1993}}.▼
*'''1972:''' Genadi Kasparov je objavio svoj rad o realizaciji K-homologije pomoću apstraktnih eliptičkih operatora.<ref>{{harvs|txt|first=Gennadi G.|last=Kasparov|year=1972}}</ref>
*'''1977:''' {{harvs|txt|authorlink=Dennis Sullivan|first=Dennis|last=Sullivan|year=1979}} establishes his theorem on the existence and uniqueness of Lipschitz and quasiconformal structures on topological manifolds of dimension different from 4.
▲*'''1973:''' Ati, Bot i Raoul su dali novi dokaz indeksne teoreme koristeći [[heat equation|jednačinu toplote]],<ref>{{harvs|txt=yes|last1=Atiyah |author2-link=Raoul Bott|first2=Raoul|last2=Bott|author3-link=Vijay Kumar Patodi|first3=Vijay |last3=Patodi|year=1973}}</ref> gaveopisan au new proof of the index theorem using the [[heat equation]], described inMelrozovoj knjizi.<ref>{{harv|Melrose|1993}}.</ref>
*{{harvs|txt|first=Ezra|last=Getzler|authorlink=Ezra Getzler|year=1983}} motivated by ideas of {{harvs|txt=yes|authorlink=Edward Witten|first=Edward|last=Witten|year=1982}} and [[Luis Alvarez-Gaume]], gave a short proof of the local index theorem for operators that are locally [[Dirac operator]]s; this covers many of the useful cases.▼
*'''1977:''' Salivan je uspostavio svoju teoremu o postojanju i jedinstvenosti Lipšicovih i kvazikonformalnih struktura na topološkim mnogostrukostima s dimenzijama različitim od 4.<ref>{{harvs|txt|authorlink=Dennis Sullivan|first=Dennis|last=Sullivan|year=1979}}</ref>
*'''1983:''' {{harvs|txt|first=Nicolae|last= Teleman|year=1983}} proves that the analytical indices of signature operators with values in vector bundles are topological invariants.
▲*'''1983:''' Gecler<ref>{{harvs|txt|first=Ezra|last=Getzler|authorlink=Ezra Getzler|year=1983}}</ref> motivatedje bymotivisan ideas ofidejama Vitena<ref>{{harvs|txt=yes|authorlink=Edward Witten|first=Edward|last=Witten|year=1982}}</ref> andi [[Luis Alvarez-Gaume|Lisa Alvareza-Gauma]], gavedao akratak shortdokaz prooflokalne indeksne ofteoreme theza localoperatore indexkoji theoremsu for operators that are locallylokalni [[Dirac operator|Dirakovi operatori]]s; this coverstime manysu ofpokriveni themnogi usefulkorisni casesslučajevi.
*'''1984:''' {{harv|Teleman|1984}} establishes the index theorem on topological manifolds.
*'''1983:''' Teleman je dokazao da su analitički indeksi potpisnih operatora sa vrednostima u vektorskim svežnjevima topološke invarijante.<ref>{{harvs|txt|first=Nicolae|last= Teleman|year=1983}}</ref>
*'''1986:''' {{harvs|txt|first=Alain|last=Connes|authorlink=Alain Connes|year=1986}} publishes his fundamental paper on [[noncommutative geometry]].
*'''1984:''' Teleman je uspostavio indeksnu teoremu na topološkim mnogostrukostima.<ref>{{harv|Teleman|1984}}</ref>
*'''1989:''' {{harvs|txt|first=Simon K.|last1=Donaldson|author1-link=Simon Donaldson|last2=Sullivan|year=1989}} study Yang–Mills theory on quasiconformal manifolds of dimension 4. They introduce the signature operator ''S'' defined on differential forms of degree two.
*'''1986:''' Kons je objavio svoju fundamentalnu publikaciju o [[noncommutative geometry|nekomutativnoj geometriji]].<ref>{{harvs|txt|first=Alain|last=Connes|authorlink=Alain Connes|year=1986}}</ref>
*'''1990:''' {{harvs|txt|last1=Connes|first2=Henri|last2=Moscovici|year=1990}} prove the local index formula in the context of non-commutative geometry.
*'''1989:''' Donalsonova i Salivanova su objavili studiju Jang-Milsove teorije kvazikonformalnih mnogostrukosti dimenzije 4. Oni su uveli potpini operator -{''S''}- definisan na diferencijalnim formama drugog stepena.<ref>{{harvs|txt|first=Simon K.|last1=Donaldson|author1-link=Simon Donaldson|last2=Sullivan|year=1989}}</ref>
*'''1994:''' {{harvs|txt|last1=Connes|last2=Sullivan|last3=Teleman|year=1994}} prove the index theorem for signature operators on quasiconformal manifolds.
*'''1990:''' Kons i Moskovici su dokazali lokalnu indeksnu formulu u kontekstu nekomutativne geometrije.<ref>{{harvs|txt|last1=Connes|first2=Henri|last2=Moscovici|year=1990}}</ref>
*'''1994:''' Kons, Salivan i Teleman su dokazali indeksnu teoremu za potpisne operatore na kvazikonformalnim mnogostrukostima.<ref>{{harvs|txt|last1=Connes|last2=Sullivan|last3=Teleman|year=1994}}</ref>
 
== Reference ==
Преузето из „https://sr.wikipedia.org/wiki/Ati–Zingerova_indeksna_teorema”