TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris A1 - Savin, Anton T1 - The homotopy classification and the index of boundary value problems for general elliptic operators N2 - We give the homotopy classification and compute the index of boundary value problems for elliptic equations. The classical case of operators that satisfy the Atiyah-Bott condition is studied first. We also consider the general case of boundary value problems for operators that do not necessarily satisfy the Atiyah-Bott condition. T3 - Preprint - (1999) 20 KW - elliptic boundary value problems KW - Atiyah-Bott condition KW - index theory KW - K-theory KW - homotopy classification Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25568 ER - TY - INPR A1 - Savin, Anton A1 - Sternin, Boris T1 - Pseudodifferential subspaces and their applications in elliptic theory N2 - The aim of this paper is to explain the notion of subspace defined by means of pseudodifferential projection and give its applications in elliptic theory. Such subspaces are indispensable in the theory of well-posed boundary value problems for an arbitrary elliptic operator, including the Dirac operator, which has no classical boundary value problems. Pseudodifferential subspaces can be used to compute the fractional part of the spectral Atiyah–Patodi–Singer eta invariant, when it defines a homotopy invariant (Gilkey’s problem). Finally, we explain how pseudodifferential subspaces can be used to give an analytic realization of the topological K-group with finite coefficients in terms of elliptic operators. It turns out that all three applications are based on a theory of elliptic operators on closed manifolds acting in subspaces. T3 - Preprint - (2005) 17 KW - elliptic operator KW - boundary value problem KW - pseudodifferential subspace KW - dimension functional KW - η-invariant KW - index KW - modn-index KW - parity condition Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29937 ER - TY - INPR A1 - Savin, Anton A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - On the invariant index formulas for spectral boundary value problems N2 - In the paper we study the possibility to represent the index formula for spectral boundary value problems as a sum of two terms, the first one being homotopy invariant of the principal symbol, while the second depends on the conormal symbol of the problem only. The answer is given in analytical, as well as in topological terms. T3 - Preprint - (1998) 18 KW - spectral boundary value problems KW - Fredholm property KW - index of elliptic operator KW - Chern character KW - Atiyah-Patodi-Singer theory Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25285 ER - TY - INPR A1 - Nazaikinskii, Vladimir E. A1 - Savin, Anton A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - On the homotopy classification of elliptic operators on manifolds with edges N2 - We obtain a stable homotopy classification of elliptic operators on manifolds with edges. T3 - Preprint - (2004) 16 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26769 ER - TY - INPR A1 - Savin, Anton A1 - Sternin, Boris T1 - Index defects in the theory of nonlocal boundary value problems and the η-invariant N2 - The paper deals with elliptic theory on manifolds with boundary represented as a covering space. We compute the index for a class of nonlocal boundary value problems. For a nontrivial covering, the index defect of the Atiyah-Patodi-Singer boundary value problem is computed. We obtain the Poincaré duality in the K-theory of the corresponding manifolds with singularities. T3 - Preprint - (2001) 31 KW - elliptic operator KW - boundary value problem KW - finiteness theorem KW - nonlocal problem KW - covering KW - relative η-invariant KW - index KW - modn-index Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26146 ER - TY - INPR A1 - Savin, Anton A1 - Sternin, Boris T1 - Eta-invariant and Pontrjagin duality in K-theory N2 - The topological significance of the spectral Atiyah-Patodi-Singer η-invariant is investigated. We show that twice the fractional part of the invariant is computed by the linking pairing in K-theory with the orientation bundle of the manifold. The Pontrjagin duality implies the nondegeneracy of the linking form. An example of a nontrivial fractional part for an even-order operator is presented. T3 - Preprint - (2000) 08 KW - eta-invariant KW - K-theory KW - Pontrjagin duality KW - linking coefficients KW - Atiyah-Patodi-Singer theory KW - modulo n index Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25747 ER - TY - INPR A1 - Savin, Anton A1 - Sternin, Boris T1 - Eta invariant and parity conditions N2 - We give a formula for the η-invariant of odd order operators on even-dimensional manifolds, and for even order operators on odd-dimensional manifolds. Geometric second order operators are found with nontrivial η-invariants. This solves a problem posed by P. Gilkey. T3 - Preprint - (2000) 21 KW - eta invariant KW - parity conditions KW - K-theory KW - linking coefficients KW - Dirac operators KW - spectral flow KW - elliptic operators Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25869 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Savin, Anton A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Elliptic theory on manifolds with nonisolated singularities : V. Index formulas for elliptic problems on manifolds with edges N2 - For elliptic problems on manifolds with edges, we construct index formulas in form of a sum of homotopy invariant contributions of the strata (the interior of the manifold and the edge). Both terms are the indices of elliptic operators, one of which acts in spaces of sections of finite-dimensional vector bundles on a compact closed manifold and the other in spaces of sections of infinite-dimensional vector bundles over the edge. T3 - Preprint - (2003) 02 KW - manifold with edge KW - elliptic problem KW - index formula KW - symmetry conditions Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26500 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Savin, Anton A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Elliptic theory on manifolds with nonisolated singularities : IV. Obstructions to elliptic problems on manifolds with edges N2 - The obstruction to the existence of Fredholm problems for elliptic differentail operators on manifolds with edges is a topological invariant of the operator. We give an explicit general formula for this invariant. As an application we compute this obstruction for geometric operators. T3 - Preprint - (2002) 24 KW - manifolds with edges KW - edge-degenerate operators KW - elliptic families KW - edge symbol KW - Atiyah-Bott obstruction KW - parameter-dependent ellipticity Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26415 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Savin, Anton A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Elliptic theory on manifolds with nonisolated singularities : III. The spectral flow of families of conormal symbols N2 - When studyind elliptic operators on manifolds with nonisolated singularities one naturally encounters families of conormal symbols (i.e. operators elliptic with parameter p ∈ IR in the sense of Agranovich-Vishik) parametrized by the set of singular points. For homotopies of such families we define the notion of spectral flow, which in this case is an element of the K-group of the parameter space. We prove that the spectral flow is equal to the index of some family of operators on the infinite cone. T3 - Preprint - (2002) 20 KW - elliptic family KW - conormal symbol KW - spectral flow KW - relative index Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26386 ER -