
This article is cited in 3 scientific papers (total in 3 papers)
Noncommutative geometry and classification of elliptic operators
V. E. Nazaikinskii^{a}, A. Yu. Savin^{bc}, B. Yu. Sternin^{bc} ^{a} A. Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences
^{b} Independent University of Moscow
^{c} Russian State Social University
Abstract:
The computation of a stable homotopic classification of elliptic operators is an important problem of elliptic theory. The classical solution of this problem is given by Atiyah and Singer for the case of smooth compact manifolds. It is formulated in terms of $K$theory for a cotangent fibering of the given manifold. It cannot be extended for the case of nonsmooth manifolds because their cotangent fiberings do not contain all necessary information. Another Atiyah definition might fit in such a case: it is based on the concept of abstract elliptic operators and is given in term of $K$homologies of the manifold itself (instead of its fiberings). Indeed, this theorem is recently extended for manifolds with conic singularities, ribs, and general socalled stratified manifolds: it suffices just to replace the phrase “smooth manifold” by the phrase “stratified manifold” (of the corresponding class). Thus, stratified manifolds is a strange phenomenon in a way: the algebra of symbols of differential (pseudodifferential) operators is quite noncommutative on such manifolds (the symbol components corresponding to strata of positive codimensions are operatorvalued functions), but the solution of the classification problem can be found in purely geometric terms. In general, it is impossible for other classes of nonsmooth manifolds. In particular, the authors recently found that, for manifolds with angles, the classification is given by a $K$group of a noncommutative $C^*$algebra and it cannot be reduced to a commutative algebra if normal fiberings of faces of the considered manifold are nontrivial. Note that the proofs are based on noncommutative geometry (more exactly, the Ktheory of $C^*$algebras) even in the case of stratified manifolds though the results are “classical.” In this paper, we provide a review of the abovementioned classification results for elliptic operators on manifolds with singularities and corresponding methods of noncommutative geometry (in particular, the localization principle in $C^*$algebras).
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Journal of Mathematical Sciences, 2010, 164:4, 603–636
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515.168.5+517.986.32
Citation:
V. E. Nazaikinskii, A. Yu. Savin, B. Yu. Sternin, “Noncommutative geometry and classification of elliptic operators”, Proceedings of the Crimean autumn mathematical schoolsymposium, CMFD, 29, PFUR, M., 2008, 131–164; Journal of Mathematical Sciences, 164:4 (2010), 603–636
Citation in format AMSBIB
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\paper Noncommutative geometry and classification of elliptic operators
\inbook Proceedings of the Crimean autumn mathematical schoolsymposium
\serial CMFD
\yr 2008
\vol 29
\pages 131164
\publ PFUR
\publaddr M.
\mathnet{http://mi.mathnet.ru/cmfd127}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=2472267}
\elib{http://elibrary.ru/item.asp?id=15317220}
\transl
\jour Journal of Mathematical Sciences
\yr 2010
\vol 164
\issue 4
\pages 603636
\crossref{https://doi.org/10.1007/s1095801097658}
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Yu. A. Kordyukov, V. A. Pavlenko, “Singular integral operators on a manifold with a distinguished submanifold”, Ufa Math. J., 6:3 (2014), 35–68

A. Yu. Savin, B. Yu. Sternin, “Homotopy classification of elliptic problems associated with discrete group actions on manifolds with boundary”, Ufa Math. J., 8:3 (2016), 122–129

A. Yu. Savin, “O gomotopicheskoi klassifikatsii ellipticheskikh zadach so szhatiyami i $K$gruppakh sootvetstvuyuschikh $C^*$algebr”, Differentsialnye i funktsionalnodifferentsialnye uravneniya, SMFN, 64, no. 1, Rossiiskii universitet druzhby narodov, M., 2018, 164–179

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