Manifolds with Group Actions and Elliptic Operators
Title | Manifolds with Group Actions and Elliptic Operators PDF eBook |
Author | Vladimir I︠A︡kovlevich Lin |
Publisher | American Mathematical Soc. |
Pages | 90 |
Release | 1994 |
Genre | Mathematics |
ISBN | 0821826042 |
This work studies equivariant linear second order elliptic operators [italic capital]P on a connected noncompact manifold [italic capital]X with a given action of a group [italic capital]G. The action is assumed to be cocompact, meaning that [italic capitals]GV = [italic capital]X for some compact subset of [italic capital]V of [italic capital]X. The aim is to study the structure of the convex cone of all positive solutions of [italic capital]P[italic]u = 0.
Group Actions on Manifolds
Title | Group Actions on Manifolds PDF eBook |
Author | Reinhard Schultz |
Publisher | American Mathematical Soc. |
Pages | 586 |
Release | 1985 |
Genre | Mathematics |
ISBN | 0821850385 |
Presents an understanding of the sorts of problems one studies in group actions and the methods used to study such problems. This book features articles based upon lectures at the 1983 AMS-IMS-SIAM Joint Summer Research Conference, Group Actions on Manifolds, held at the University of Colorado.
Essays on Topology and Related Topics
Title | Essays on Topology and Related Topics PDF eBook |
Author | Andre Haefliger |
Publisher | Springer Science & Business Media |
Pages | 267 |
Release | 2012-12-06 |
Genre | Mathematics |
ISBN | 3642491979 |
Elliptic Operators and Compact Groups
Title | Elliptic Operators and Compact Groups PDF eBook |
Author | M.F. Atiyah |
Publisher | Springer |
Pages | 100 |
Release | 2006-08-01 |
Genre | Mathematics |
ISBN | 3540378111 |
Analysis, Geometry and Topology of Elliptic Operators
Title | Analysis, Geometry and Topology of Elliptic Operators PDF eBook |
Author | Bernhelm Booss |
Publisher | World Scientific |
Pages | 553 |
Release | 2006 |
Genre | Mathematics |
ISBN | 9812773606 |
Modern theory of elliptic operators, or simply elliptic theory, has been shaped by the Atiyah-Singer Index Theorem created 40 years ago. Reviewing elliptic theory over a broad range, 32 leading scientists from 14 different countries present recent developments in topology; heat kernel techniques; spectral invariants and cutting and pasting; noncommutative geometry; and theoretical particle, string and membrane physics, and Hamiltonian dynamics. The first of its kind, this volume is ideally suited to graduate students and researchers interested in careful expositions of newly-evolved achievements and perspectives in elliptic theory. The contributions are based on lectures presented at a workshop acknowledging Krzysztof P Wojciechowski''s work in the theory of elliptic operators. Sample Chapter(s). Contents (42 KB). Contents: On the Mathematical Work of Krzysztof P Wojciechowski: Selected Aspects of the Mathematical Work of Krzysztof P Wojciechowski (M Lesch); Gluing Formulae of Spectral Invariants and Cauchy Data Spaces (J Park); Topological Theories: The Behavior of the Analytic Index under Nontrivial Embedding (D Bleecker); Critical Points of Polynomials in Three Complex Variables (L I Nicolaescu); Chern-Weil Forms Associated with Superconnections (S Paycha & S Scott); Heat Kernel Calculations and Surgery: Non-Laplace Type Operators on Manifolds with Boundary (I G Avramidi); Eta Invariants for Manifold with Boundary (X Dai); Heat Kernels of the Sub-Laplacian and the Laplacian on Nilpotent Lie Groups (K Furutani); Remarks on Nonlocal Trace Expansion Coefficients (G Grubb); An Anomaly Formula for L 2- Analytic Torsions on Manifolds with Boundary (X Ma & W Zhang); Conformal Anomalies via Canonical Traces (S Paycha & S Rosenberg); Noncommutative Geometry: An Analytic Approach to Spectral Flow in von Neumann Algebras (M-T Benameur et al.); Elliptic Operators on Infinite Graphs (J Dodziuk); A New Kind of Index Theorem (R G Douglas); A Note on Noncommutative Holomorphic and Harmonic Functions on the Unit Disk (S Klimek); Star Products and Central Extensions (J Mickelsson); An Elementary Proof of the Homotopy Equivalence between the Restricted General Linear Group and the Space of Fredholm Operators (T Wurzbacher); Theoretical Particle, String and Membrane Physics, and Hamiltonian Dynamics: T-Duality for Non-Free Circle Actions (U Bunke & T Schick); A New Spectral Cancellation in Quantum Gravity (G Esposito et al.); A Generalized Morse Index Theorem (C Zhu). Readership: Researchers in modern global analysis and particle physics.
Manifolds and Modular Forms
Title | Manifolds and Modular Forms PDF eBook |
Author | Friedrich Hirzebruch |
Publisher | Springer Science & Business Media |
Pages | 216 |
Release | 2013-06-29 |
Genre | Technology & Engineering |
ISBN | 3663107264 |
This book provides a comprehensive introduction to the theory of elliptic genera due to Ochanine, Landweber, Stong, and others. The theory describes a new cobordism invariant for manifolds in terms of modular forms. The book evolved from notes of a course given at the University of Bonn. After providing some background material elliptic genera are constructed, including the classical genera signature and the index of the Dirac operator as special cases. Various properties of elliptic genera are discussed, especially their behaviour in fibre bundles and rigidity for group actions. For stably almost complex manifolds the theory is extended to elliptic genera of higher level. The text is in most parts self-contained. The results are illustrated by explicit examples and by comparison with well-known theorems. The relevant aspects of the theory of modular forms are derived in a seperate appendix, providing also a useful reference for mathematicians working in this field.
Index Theory for Invariant Elliptic Operators on Manifolds with Proper Cocompact Group Actions
Title | Index Theory for Invariant Elliptic Operators on Manifolds with Proper Cocompact Group Actions PDF eBook |
Author | Gong Cheng (Mathematician) |
Publisher | |
Pages | 54 |
Release | 2018 |
Genre | Electronic dissertations |
ISBN |
In this thesis, we study G-invariant elliptic operators, and in particular Dirac operators, on the space of invariant sections of a Hermitian bundle over a (non-compact) manifold with a proper and cocompact Lie group action. We provide a canonical way to define the Hilbert space of invariant sections for proper and cocompact actions and prove that the G-invariant Dirac operators, and more generally, elliptic operators, are Fredholm for the Hilbert space we constructed. Using the framework developed in this thesis, we give a new proof of a generalized Lichnerowicz Vanishing Theorem for proper cocompact group actions as an application.