Anton Yu. Savin

Elliptic theory on singular manifolds

I Singular Manifolds and Differential Operators GEOMETRY OF SINGULARITIES Preliminaries Manifolds with conical singularities Manifolds with edges ELLIPTIC OPERATORS ON SINGULAR MANIFOLDS Operators on manifolds with conical singularities Operators on manifolds with edges Examples of elliptic edge operators II Analytical Tools PSEUDODIFFERENTIAL OPERATORS Preliminary remarks Classical theory Operators in sections of Hilbert bundles Operators on singular manifolds Ellipticity and finiteness theorems Index theorems on smooth closed manifolds LOCALIZATION (SURGERY) IN ELLIPTIC THEORY The index locality principle Localization in index theory on smooth manifolds Surgery for the index of elliptic operators on singular manifolds Relative index formulas on manifolds with isolated singularities III Topological Problems INDEX THEORY Statement of the problem Invariants of interior symbol and symmetries Invariants of the edge symbol Index theorems Index on manifolds with isolated singularities Supplement. Classification of elliptic symbols with symmetry and K-theory Supplement. Proof of Proposition 5.16 ELLIPTIC EDGE PROBLEMS Morphisms The obstruction to ellipticity A formula for the obstruction in topological terms Examples. Obstructions for geometric operators IV Applications and Related Topics FOURIER INTEGRAL OPERATORS ON SINGULAR MANIFOLDS Homogeneous canonical (contact) transformations Definition of Fourier integral operators Properties of Fourier integral operators The index of elliptic Fourier integral operators Application to quantized contact transformations Example RELATIVE ELLIPTIC THEORY Analytic aspects of relative elliptic theory Topological aspects of relative elliptic theory INDEX OF GEOMETRIC OPERATORS ON MANIFOLDS WITH CYLINDRICAL ENDS Operators on manifolds with cylindrical ends Index formulas HOMOTOPY CLASSIFICATION OF ELLIPTIC OPERATORS The homotopy classification problem Classification on smooth manifolds Atiyah-de Rham duality Abstract elliptic operators and analytic K-homology Classification on singular manifolds Some applications LEFSCHETZ FORMULAS Main result Proof of the theorem Contributions of conical points as sums of residues Supplement. The Lefschetz number Supplement. The Sternin-Shatalov method APPENDICES Spectral Flow Eta Invariants Index of Parameter-Dependent Elliptic Families Bibliographic Remarks Bibliography Index

Elliptic Operators in Subspaces and the Eta Invariant

The spectral eta-invariant of a self-adjoint elliptic differential operator on a closed manifold is rigid, provided that the parity of the order is opposite to the parity of dimension of the manifold. The paper deals with the calculation of the fractional part of the eta-invariant in this case. The method used to obtain the corresponding formula is based on the index theorem for elliptic operators in subspaces. It also utilizes K-theory with coefficients Z_n. In particular, it is shown that the group K(T^*M,Z_n) is realized by elliptic operators (symbols) acting in appropriate subspaces.

The Eta-Invariant and Pontryagin Duality in K-Theory

AbstractThe topological significance of the spectral Atiyah--Patodi--Singer n

Elliptic theory and noncommutative geometry : nonlocal elliptic operators

eta

Homotopy Classification of Elliptic Operators

n-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. Pontryagin duality implies the nondegeneracy of the linking form. An example of a nontrivial fractional part for an even-order operator is presented.

Guillemin Transform and Toeplitz Representations for Operators on Singular Manifolds

The present paper deals with the homotopy classification problem of boundary value problems for elliptic operators. We start with classical boundary value problems. The ellipticity condition allows us to reduce classical problems to the Dirichlet problem for the Laplace operator and also to obtain the homotopy classification. We then study general case of operators, that do not necessarily satisfy the Atiyah-Bott condition. The boundary value problem reduces then to the so-called spectral boundary value problem.