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Published
**1998** by Cambridge University Press in Cambridge, New York .

Written in English

Read online- Spectral geometry.,
- Differential operators.,
- Mathematical physics.

**Edition Notes**

Includes bibliographical references (p. 191-206) and index.

Statement | Giampiero Esposito. |

Series | Cambridge lecture notes in physics ;, 12 |

Classifications | |
---|---|

LC Classifications | QC20.7.S64 E78 1998 |

The Physical Object | |

Pagination | xiii, 209 p. ; |

Number of Pages | 209 |

ID Numbers | |

Open Library | OL377471M |

ISBN 10 | 0521648629 |

LC Control Number | 98039477 |

**Download Dirac operators and spectral geometry**

The Dirac operator has many useful applications in theoretical physics and mathematics. This book provides a clear, concise and self-contained introduction to the global theory of the Dirac operator and to the analysis of spectral asymptotics with local or non-local boundary by: The Dirac operator has many useful applications in theoretical physics and mathematics.

This book provides a clear, concise and self-contained introduction to the global theory of the Dirac Author: Giampiero Esposito. The Dirac operator; 2. Differential operators on manifolds; 3.

Index problems; 4. Spectral asymmetry; 5. Spectral geometry with operators of Laplace type; 6. New Cited by: The Dirac operator has many useful applications in theoretical physics and mathematics.

This book provides a clear, concise and self-contained introduction to the global theory of the Dirac operator and to the analysis of spectral asymptotics with local or nonlocal boundary conditions. D/ is an (unbounded) operator on a Hilbert space H = L2 (M, S) of “square-integrable spinors” and C ∞ (M) also acts on H by multiplication / f ]k = k grad f k∞.

operators with k [D, Noncommutative geometry generalizes (C ∞ (M), L2 (M, S), D) / to a spectral. Noncommutative geometry generalizes (C∞(M),L2(M,S),D/) to a spectral triple of the form (A,H,D), where Ais a “smooth” algebra acting on a Hilbert space H, Dis an (unbounded) selfadjoint operator on H, subject to certain conditions: in particular that [D,a] be a bounded operator for each a∈ by: 6.

The eigenvalues of the Laplacian are naturally linked to the geometry of the manifold. For manifolds that admit spin (or) structures, one obtains further information from equations involving Dirac operators and spinor fields.

In the case of four-manifolds, for example, one has the remarkable Seiberg-Witten. DIRAC OPERATORS AND SPECTRAL TRIPLES FOR SOME FRACTALS 3 Hausdorﬀmeasure canberecovered fromoperatoralgebraicdata.

Work in this direction was pursued by Daniele Guido and Tommaso Isola in several papers, [16, 17, 18]. Earlier, in [27], using the results and methods of [21] and [6] (includ. Elliptic operators self-adjoint extensions Sobolev spaces p-Laplacian spectral theory Schauder estimates regularity theory Dirac operator Authors and affiliations D.

Edmunds. We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space. Riemannian geometry in completely algebraic terms, using a -algebra A represented on a Hilbert space H (which capture the topological aspects), and a not necessarily bounded generalized Dirac operator D(which captures the metric aspects).

These elements form a spectral triple and are at the basis of the construction. These ingredients are naturallyCited by: Asymptotic Formulae in Spectral Geometry Peter B. Gilkey A great deal of progress has been made recently in the field of asymptotic formulas that arise in the theory of Dirac and Laplace type operators.

Provides a clear, concise and self-contained Dirac operators and spectral geometry book to the global theory of the Dirac operator and to the analysis of spectral asymptotics with local or non-local boundary conditions. Ideal for graduate students and researchers working in theoretical physics and mathematics.

The basics of different ingredients will be presented and studied like, Dirac operators, heat equation asymptotics, zeta functions and then, how to get within the framework of operators Author: Oussama Hijazi.

Dirac operators and spectral geometry book the first edition of this book, simple proofs of the Atiyah-Singer Index Theorem for Dirac operators on compact Riemannian manifolds and its generalizations (due to the authors and J.-M.

Bismut) were presented, using an explicit geometric construction of the heat kernel of a generalized Dirac operator; the new edition makes this popular book available to students. In particular, investigating some of the open problems or conjectures proposed within that framework in [27,28], and aimed at merging aspects of fractal, spectral and noncommutative geometry; (iv) studying the differential operators (including â€˜Laplaciansâ€™) connected to the Dirac-type op- erators constructed in this paper, as well Cited by: An appropriate version of a Dirac operator is a part of a concept of the spectral triple in noncommutative geometry a la Alain Connes.

Properties Eta invariant and functional determinant. The eta function (see there for more) of a Dirac operator D D expresses the functional determinant of its Laplace operator H = D 2 H = D^2.

Index and. Abstract: The goal of these lectures is to present the few fundamentals of noncommutative geometry looking around its spectral approach. Strongly motivated by physics, in particular by relativity and quantum mechanics, Chamseddine and Connes have defined an action based on spectral considerations, the so-called spectral action.

A Dirac Operator for Extrinsic Shape Analysis Spectral geometry processing encodes shape in terms of the eigen- Dirac operator is highly sensitive to features like surface texture and sharp creases, nicely complementing existing operators that emphasize global, intrinsic features.

In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces.

It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations.

The theory is connected to. (iii) Dirac operators and spectral geometry by Giampiero Esposito. (iv) Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem by Peter Gilkey.

and (v) The laplacian on a Riemannian Manifold by Rosenberg, however I am having difficulty deciding which one or two to study between these. Scattering Theory and Spectral Stability under a Ricci Flow for Dirac Operators.

With Batu Güneysu. Preprint; Contributions to the spectral geometry of locally homogeneous spaces. With Dorothee Schüth. Book chapter in Space - Time - Matter. Analytic and Geometric Structures.

Brüning, J. (Ed.), Staudacher, M. (Ed.), Fiedler, B., et al. ( This item: Some Problems in the Spectral Theory of Separated Dirac Operators: A representation of the spectral function of a Dirac operator and a bound for points of Author: Joshua T. Eggenberger.

1 Introduction. The ﬁeld of spectral geometry is a vibrant and active one. In these brief notes, we will sketch some of the recent developments in this area.

Our choice is somewhat idiosyncratic and owing to constraints of space necessarily by: 8. This makes it clear why the Dirac operator is the most fundamental, in the theory of elliptic operators on manifolds. The topic of spectral geometry is developed by studying non-local boundary conditions of the Atiyah-Patodi-Singer type, and heat-kernel asymptotics for operators of Laplace type on manifolds with : Giampiero Esposito.

This volume surveys the spectral properties of the spin Dirac operator. After a brief introduction to spin geometry, we present the main known estimates for Dirac eigenvalues on compact manifolds with or without boundaries.

We give examples where the spectrum can be made explicit and present a chapter dealing with the non-compact setting. Dirac operators and spectral geometry. [Giampiero Esposito] -- A clear, concise and up-to-date introduction to the theory of the Dirac operator and its wide range of applications in theoretical physics for graduate students and researchers.

CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Abstract. This paper is devoted to mathematical and physical properties of the Dirac operator and spectral geometry. Spin-structures in Lorentzian and Riemannian manifolds, and the global theory of the Dirac operator, are first analyzed.

Elliptic boundary-value problems, index problems for. Lesch, The uniqueness of the spectral flow on spaces of unbounded self-adjoint Fredholm operators, in Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds, Contemporary Mathematics, Vol.

(American Mathematical Society, Providence, ), pp. Author: Jianqing Yu. celebrated Atiyah-Singer index theorem for Dirac operators. We shall also see that the study of Dirac operators reveals some interconnections between the geometry and the topology of the underlying manifold.

Perhaps one of the most well-known results of this type is the Gauss-Bonnet theorem: c(M) = 1 4p Z M kdx. Symplectic spinors form an infinite-rank vector bundle.

Dirac operators on this bundle were constructed recently by Habermann, K. [“The Dirac operator on symplectic spinors,” Ann. Global Anal.

Geom. 13, ()]. Here we study the spectral geometry aspects of these operators. In particular, we define the associated distance function and compute the Cited by: 1.

Full Description: "This book gives a detailed and self-contained introduction into the theory of spectral functions, with an emphasis on their applications to quantum field theory. All methods are illustrated with applications to specific physical problems from the forefront of current research, such as finite-temperature field theory, D-branes, quantum solitons and.

Dirac Operators and Spectral Geometry Written examination Each problem is worth 5 points, out of a possible total of 1. Let Cl p,q:= Cl(Rp+q,g) be the real Cliﬀord algebra for the symmetric bilinear form g(x,x) = x 2 1 + + x2 p − x p+1 − − x 2.

Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian manifold has been most intensively studied, although other Laplace operators in.

The goal of the present paper is to extend the results of to the case of arbitrary Dirac operators on compact even dimensional manifolds with boundary. Our main result, Theoremexpresses the spectral flow of the Dirac operator with local boundary conditions in terms of the spectral flow of the boundary Dirac by: 8.

In this text, Friedrich examines the Dirac operator on Riemannian manifolds, especially its connection with the underlying geometry and topology of the manifold.

The presentation includes a review of Clifford algebras, spin groups and the spin representation, as well as a review of spin structures and $\textrm{spin}^\mathbb{C}$ by: The Dirac operator has many useful applications in theoretical physics and mathematics.

This book provides a clear, concise and self-contained introduction to the global theory of the Dirac operator and to the analysis of spectral asymptotics with local or non-local boundary conditions. In mathematics and physics, the spectral asymmetry is the asymmetry in the distribution of the spectrum of eigenvalues of an operator.

In mathematics, the spectral asymmetry arises in the study of elliptic operators on compact manifolds, and is given a deep meaning by the Atiyah-Singer index theorem. Dirac Operators and Spectral Geometry (Cambridge Lecture Notes in Physics) By Giampiero Esposito The Dirac operator has many useful applications in theoretical physics and mathematics.

This book provides a clear, concise and self-contained introduction to the global theory of the Dirac operator and to the analysis of spectral. The spectral flow, the Maslov index and decompositions of manifolds, Duke Univ.J, 80(), p This is essentially my PhD dissertation.

I prove a gluing formula for the spectral flow of a path of selfadjoint Dirac operators (on a manifold cut in two parts by a hypersurface) in terms of an infinite dimensional Maslov index.

Author: Nicole Berline,Ezra Getzler,Michèle Vergne; Publisher: Springer Science & Business Media ISBN: Category: Mathematics Page: View: DOWNLOAD NOW» In the first edition of this book, simple proofs of the Atiyah-Singer Index Theorem for Dirac operators on compact Riemannian manifolds and its generalizations (due to the authors and .A great deal of progress has been made recently in the field of asymptotic formulas that arise in the theory of Dirac and Laplace type operators.

Asymptotic Formulae in Spectral Geometry collects these results and computations into one book. Written by a .Book Description. A great deal of progress has been made recently in the field of asymptotic formulas that arise in the theory of Dirac and Laplace type operators.

Asymptotic Formulae in Spectral Geometry collects these results and computations into one book.