Geometric analysis on singular spaces
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Francesco Bei
Abstract
We are interested in the analysis of Dirac and Schrödinger-type operators associated to certain stratified spaces known as Smooth Thom-Mather Spaces. These are topological spaces that consist of a smooth manifold as dense open subset to which manifolds of lower dimension are attached in a suitableway; they are briefly described in Section 2.2. Prominent examples are polyhedra, projective varieties, and connected orbit spaces of proper Lie group actions. They all come equipped with canonical metrics that are of a rather different character. We therefore discuss in Section 3 the geometric analysis of Dirac and Schrödinger-type operators on arbitrary Riemannian manifolds and the construction of resovents and heat semigroups. In Section 4, we describe the construction of certain geometric invariants which are expressed in terms of the spectral data of suitable self-adjoint extensions of these operators.
Abstract
We are interested in the analysis of Dirac and Schrödinger-type operators associated to certain stratified spaces known as Smooth Thom-Mather Spaces. These are topological spaces that consist of a smooth manifold as dense open subset to which manifolds of lower dimension are attached in a suitableway; they are briefly described in Section 2.2. Prominent examples are polyhedra, projective varieties, and connected orbit spaces of proper Lie group actions. They all come equipped with canonical metrics that are of a rather different character. We therefore discuss in Section 3 the geometric analysis of Dirac and Schrödinger-type operators on arbitrary Riemannian manifolds and the construction of resovents and heat semigroups. In Section 4, we describe the construction of certain geometric invariants which are expressed in terms of the spectral data of suitable self-adjoint extensions of these operators.
Kapitel in diesem Buch
- Frontmatter I
- Contents V
- Introduction VII
- Algebraic K-theory, assembly maps, controlled algebra, and trace methods 1
- Lorentzian manifolds with special holonomy – Constructions and global properties 51
- Contributions to the spectral geometry of locally homogeneous spaces 69
- On conformally covariant differential operators and spectral theory of the holographic Laplacian 90
- Moduli and deformations 116
- Vector bundles in algebraic geometry and mathematical physics 150
- Dyson–Schwinger equations: Fix-point equations for quantum fields 186
- Hidden structure in the form factors of N = 4 SYM 197
- On regulating the AdS superstring 221
- Constraints on CFT observables from the bootstrap program 245
- Simplifying amplitudes in Maxwell-Einstein and Yang-Mills-Einstein supergravities 266
- Yangian symmetry inmaximally supersymmetric Yang-Mills theory 288
- Wave and Dirac equations on manifolds 324
- Geometric analysis on singular spaces 349
- Singularities and long-time behavior in nonlinear evolution equations and general relativity 417
- Index 491
Kapitel in diesem Buch
- Frontmatter I
- Contents V
- Introduction VII
- Algebraic K-theory, assembly maps, controlled algebra, and trace methods 1
- Lorentzian manifolds with special holonomy – Constructions and global properties 51
- Contributions to the spectral geometry of locally homogeneous spaces 69
- On conformally covariant differential operators and spectral theory of the holographic Laplacian 90
- Moduli and deformations 116
- Vector bundles in algebraic geometry and mathematical physics 150
- Dyson–Schwinger equations: Fix-point equations for quantum fields 186
- Hidden structure in the form factors of N = 4 SYM 197
- On regulating the AdS superstring 221
- Constraints on CFT observables from the bootstrap program 245
- Simplifying amplitudes in Maxwell-Einstein and Yang-Mills-Einstein supergravities 266
- Yangian symmetry inmaximally supersymmetric Yang-Mills theory 288
- Wave and Dirac equations on manifolds 324
- Geometric analysis on singular spaces 349
- Singularities and long-time behavior in nonlinear evolution equations and general relativity 417
- Index 491
