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- K-theory (7)
- elliptic operators (6)
- relative index (6)
- Atiyah-Patodi-Singer theory (5)
- Fredholm property (5)
- index theory (5)
- boundary value problems (4)
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- index (4)
- manifold with singularities (4)
- surgery (4)
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- eta invariant (3)
- index of elliptic operators in subspaces (3)
- Atiyah-Bott condition (2)
- Atiyah-Bott obstruction (2)
- Fredholm operators (2)
- Lefschetz fixed point formula (2)
- Mellin transform (2)
- boundary value problem (2)
- edge-degenerate operators (2)
- elliptic boundary value problems (2)
- elliptic families (2)
- elliptic family (2)
- ellipticity (2)
- eta-invariant (2)
- homotopy classification (2)
- index formulas (2)
- linking coefficients (2)
- manifolds with conical singularities (2)
- modn-index (2)
- pseudodiferential operators (2)
- quantization (2)
- regularizer (2)
- regularizers (2)
- spectral flow (2)
- symmetry conditions (2)
- (co)boundary operator (1)
- APS problem (1)
- Atiyah-Singer theorem (1)
- Calderón projections (1)
- Cauchy Riemann operator (1)
- Chern character (1)
- Dirac operators (1)
- Euler operator (1)
- Green operator (1)
- Pontrjagin duality (1)
- Riemann-Roch theorem (1)
- Sobolev problem (1)
- analytic index (1)
- contact transformations (1)
- covering (1)
- dimension functional (1)
- edge symbol (1)
- elliptic morphism (1)
- elliptic operators in subspaces (1)
- elliptic problem (1)
- exterior tensor product (1)
- finiteness theorem (1)
- index formula (1)
- index of elliptic operator (1)
- manifold with edge (1)
- manifolds with edges (1)
- mod k index (1)
- modulo n index (1)
- nonhomogeneous boundary value problems (1)
- nonlocal problem (1)
- parameter-dependent ellipticity (1)
- parity condition (1)
- parity conditions (1)
- problem of classification (1)
- pseudo-diferential operators (1)
- pseudodifferential subspace (1)
- pseudodifferential subspaces (1)
- relative η-invariant (1)
- spectral boundary value problems (1)
- spectral resolution (1)
- symplectic (canonical) transformations (1)
- η-invariant (1)
Institute
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.
Contents: Chapter 5: Manifolds with Isolated Singularities 5.1. Differential Operators and the Geometry of Singularities 5.1.1. How do isolated singularities arise? Examples 5.1.2. Definition and methods for the description of manifolds with isolated singularities 5.1.3. Bundles. The cotangent bundle 5.2. Asymptotics of Solutions, Function Spaces,Conormal Symbols 5.2.1. Conical singularities 5.2.2. Cuspidal singularities 5.3. A Universal Representation of Degenerate Operators and the Finiteness Theorem 5.3.1. The cylindrical representation 5.3.2. Continuity and compactness 5.3.3. Ellipticity and the finiteness theorem 5.4. Calculus of ΨDO 5.4.1. General ΨDO 5.4.2. The subalgebra of stabilizing ΨDO 5.4.3. Ellipticity and the finiteness theorem
Contents: Chapter 7: The Index Problemon Manifolds with Singularities Preface 7.1. The Simplest Index Formulas 7.1.1. General properties of the index 7.1.2. The index of invariant operators on the cylinder 7.1.3. Relative index formulas 7.1.4. The index of general operators on the cylinder 7.1.5. The index of operators of the form 1 + G with a Green operator G 7.1.6. The index of operators of the form 1 + G on manifolds with edges 7.1.7. The index on bundles with smooth base and fiber having conical points 7.2. The Index Problem for Manifolds with Isolated Singularities 7.2.1. Statement of the index splitting problem 7.2.2. The obstruction to the index splitting 7.2.3. Computation of the obstruction in topological terms 7.2.4. Examples. Operators with symmetries 7.3. The Index Problem for Manifolds with Edges 7.3.1. The index excision property 7.3.2. The obstruction to the index splitting 7.4. Bibliographical Remarks