20.500.12556/RUL-184515 Atiyah-Singerjev izrek o indeksu magistrsko delo Atiyah-Singer index theorem V delu obravnavamo Atiyah-Singerjev izrek o indeksu in potrebno matematično ozadje. Najprej se posvetimo $K$-teoriji, kjer uvedemo reducirane, relativne in negativne $K$-grupe, $K$-teorijo s kompaktnimi nosilici, zunanje produkte, komplekse vektorskih sveženjev, Bottovo periodičnost in Thomov izomorfizem. S tem lahko konstruiramo topološki indeks $t$-ind, ki je ena izmed dveh ključnih sestavin Atiyah-Singerjevega izreka o indeksu. Analitični del naloge se začne s teorijo psevdo-diferencialnih operatorjev na evklidskem prostoru ${\mathbb R}^n$, ki jo nato razširimo na operatorje med prerezi vektorskih svežnjev nad mnogoterostjo. Po uvedbi prostorov Soboljeva in eliptičnosti operatorjev na mnogoterostih pokažemo, da so eliptični psevdo-diferencialni operatorji nad sklenjenimi mnogoterostmi Fredholmovi. S pomočjo glavnega simbola osvetlimo korespondenco med psevdo-diferencialnimi operatorji na $M$ in elementi $K_c(T^\vee M)$, s pomočjo katere definiramo še analitični indeks $a$-ind, drugo ključno sestavino Atiyah-Singerjevega izreka o indeksu. Ta izrek formuliramo kot $t$-ind $=a$-ind. Formalno strukturo dokaza pojasnimo, za podrobnosti pa navedemo ustrezno literaturo. Za konec s pomočjo Chernovega karakterja $K$-teoretično enakost $t$-ind $=a$-ind prevedemo v kohomološko formulo, s katero pokažemo, da sta znani enakosti $\chi(M)=\langle e(TM),[M]\rangle$ in ${\rm sign}(M)=\langle L(TM),[M]\rangle$ posledici Atiyah-Singerjevega izreka o indeksu. In this work, we present the Atiyah-Singer index theorem together with the necessary mathematical background. We begin by developing $K$-theory, including reduced, relative, and negative $K$-groups, compactly supported $K$-theory, external products, complexes of vector bundles, Bott periodicity, and the Thom isomorphism. These tools are used to construct the first of two key ingredients of the Atiyah-Singer index theorem, the topological index $t$-ind. The analytic side begins with pseudodifferential operators on ${\mathbb R}^n$, which are then extended to operators between sections of vector bundles over manifolds. After introducing Sobolev spaces and ellipticity, we prove that elliptic pseudodifferential operators on closed manifolds are Fredholm. The principal symbol determines a correspondence between such operators and classes in $K_c(T^\vee M)$, which allows us to define the second key ingredient of the Atiyah-Singer index theorem, the analytic index $a$-ind. We then formulate the theorem as the equality $t$-ind $=a$-ind. We explain the formal structure of the proof and refer to the relevant literature for the technical details. Finally, using the Chern character, we translate the $K$-theoretic statement into a cohomological formula and derive the well-known formulas $\chi(M)=\langle e(TM),[M]\rangle$ and $ {\rm sign}(M)=\langle L(TM),[M]\rangle$ as corollaries. K-teorija psevdo-diferencialni operatorji teorija indeksa K-theory pseudo-differential operators index theory true false false Slovenski jezik Angleški jezik Magistrsko delo/naloga 2026-07-09 08:16:03 2026-07-09 08:16:10 2026-07-14 14:49:46 0000-00-00 00:00:00 2026 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 517 160628 284760579 22018.pdf 22018.pdf 1 09CABCA7E41AE451145F6E9C03CF4BE2 8381c979722407308307997d10658ed8ca8b03a961ae4fcc03b4924df41a1c97 8e81f719-7b5d-11f1-9b0d-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=237385 Fakulteta za matematiko in fiziko 0 0 0