Complexes of Differential Operators

Complexes of Differential Operators by Nikolai Tarkhanov.

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Bibliographic Details
Main Author: Tarkhanov, Nikolai (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Dordrecht : Springer Netherlands : Imprint: Springer, 1995.
Edition:1st ed. 1995.
Series:Mathematics and Its Applications ; 340
Springer eBook Collection.
Subjects:
Global analysis (Mathematics).
Manifolds (Mathematics).
Partial differential equations.
Functions of complex variables.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 0.0.1 Timeliness
  • 0.0.2 Directions
  • 0.0.3 Purpose
  • 0.0.4 Methods
  • 0.0.5 Approach
  • 0.0.6 Results
  • 0.0.7 Authorship
  • List of Main Notations
  • 1 Resolution of Differential Operators
  • 1.1 Differential Complexes and Their Cohomology
  • 1.2 The Hilbert Resolution of a Differential Operator with Constant Coefficients
  • 1.3 The Spencer Resolution of a Formally Integrable Differential Operator
  • 1.4 Tensor products of differential complexes and Künneth’s formula
  • 1.5 Cochain mappings of differential complexes
  • 2 Parametrices and Fundamental Solutions of Differential Complexes
  • 2.1 Parametrices of Differential Complexes
  • 2.2 Hodge Theory for Elliptic Complexes on Compact Manifolds
  • 2.3 Fundamental Solutions of Differential Complexes
  • 2.4 Green Operators for Differential Operators and the Homotopy Formula on Manifolds with Boundary
  • 2.5 The Most Immediate Corollaries and Examples
  • 3 Sokhotskii-Plemelj Formulas for Elliptic Complexes
  • 3.1 Formally Non-characteristic Hypersurfaces for Differential Complexes. The Tangential Complex
  • 3.2 Sokhotskii-Plemelj Formulas for Elliptic Complexes of First Order Differential Operators
  • 3.3 Generalization of the Sokhotskii-Plemelj Formulas to the Case of Arbitrary Elliptic Complexes
  • 3.4 Integral Formulas for Elliptic Complexes. Morera’s Theorem
  • 3.5 Multiplication of Currents via Their Harmonic Representations
  • 4 Boundary Problems for Differential Complexes
  • 4.1 The Neumann-Spencer Problem
  • 4.2 The L2-Cohomologies of Differential Complexes and the Bergman Projector
  • 4.3 The Mayer-Vietoris sequence
  • 4.4 The Cauchy problem for cohomology classes of differential complexes
  • 4.5 The Kernel Approach to Solving the Equation Pu = f
  • 5 Duality Theory for Cohomologies of Differential Complexes
  • 5.1 The Poincaré Duality and the Alexander-Pontryagin Duality
  • 5.2 The Weil Homomorphism
  • 5.3 Integral Formulas Connected by the Weil Homomorphism
  • 5.4 Grothendieck’s Theorem on Cohomology Classes Regular at Infinity
  • 5.5 Grothendieck Duality for Elliptic Complexes
  • 6 The Atiyah-Bott-Lefschetz Theorem on Fixed Points for Elliptic Complexes
  • 6.1 The Argument Principle for Elliptic Complexes
  • 6.2 An Integral Formula for the Lefschetz Number
  • 6.3 The Atiyah-Bott Formula for Simple Fixed Points
  • 6.4 Isolated Components of the Set of Fixed Points
  • 6.5 Some Examples for the Classical Complexes
  • Name Index
  • Index of Notation.

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