Author: Suhyoung Choi
Publisher: World Scientific Publishing Company Incorporated
ISBN: 9784931469686
Category : Mathematics
Languages : en
Pages : 183
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Book Description
This book exposes the connection between the low-dimensional orbifold theory and geometry that was first discovered by Thurston in 1970s providing a key tool in his proof of the hyperbolization of Haken 3-manifolds. Our main aims are to explain most of the topology of orbifolds but to explain the geometric structure theory only for 2-dimensional orbifolds, including their Teichmüller (Fricke) spaces. We tried to collect the theory of orbifolds scattered in various literatures for our purposes. Here, we set out to write down the traditional approach to orbifolds using charts, and we include the categorical definition using groupoids. We will also maintain a collection of illustrative MathematicaTM packages at our homepages.
Author: Michel Boileau
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 180
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Book Description
Orbifolds locally look like quotients of manifolds by finite group actions. They play an important role in the study of proper actions of discrete groups on manifolds. This monograph presents recent fundamental results on the geometry and topology of 3-dimensional orbifolds, with an emphasis on their geometric properties. It is suitable for graduate students and research mathematicians interested in geometry and topology.
Author: Gunther Cornelissen
Publisher: Springer Nature
ISBN: 303127704X
Category : Mathematics
Languages : en
Pages : 120
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Book Description
The question of reconstructing a geometric shape from spectra of operators (such as the Laplace operator) is decades old and an active area of research in mathematics and mathematical physics. This book focusses on the case of compact Riemannian manifolds, and, in particular, the question whether one can find finitely many natural operators that determine whether two such manifolds are isometric (coverings). The methods outlined in the book fit into the tradition of the famous work of Sunada on the construction of isospectral, non-isometric manifolds, and thus do not focus on analytic techniques, but rather on algebraic methods: in particular, the analogy with constructions in number theory, methods from representation theory, and from algebraic topology. The main goal of the book is to present the construction of finitely many “twisted” Laplace operators whose spectrum determines covering equivalence of two Riemannian manifolds. The book has a leisure pace and presents details and examples that are hard to find in the literature, concerning: fiber products of manifolds and orbifolds, the distinction between the spectrum and the spectral zeta function for general operators, strong isospectrality, twisted Laplacians, the action of isometry groups on homology groups, monomial structures on group representations, geometric and group-theoretical realisation of coverings with wreath products as covering groups, and “class field theory” for manifolds. The book contains a wealth of worked examples and open problems. After perusing the book, the reader will have a comfortable working knowledge of the algebraic approach to isospectrality. This is an open access book.
Author: Ara S. Basmajian
Publisher: American Mathematical Soc.
ISBN: 147042651X
Category : Mathematics
Languages : en
Pages : 264
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Book Description
The Ahlfors–Bers Colloquia commemorate the mathematical legacy of Lars Ahlfors and Lipman Bers. The core of this legacy lies in the fields of geometric function theory, Teichmüller theory, hyperbolic geometry, and partial differential equations. Today we see the influence of Ahlfors and Bers on algebraic geometry, mathematical physics, dynamics, probability, geometric group theory, number theory and topology. Recent years have seen a flowering of this legacy with an increased interest in their work. This current volume contains articles on a wide variety of subjects that are central to this legacy. These include papers in Kleinian groups, classical Riemann surface theory, Teichmüller theory, mapping class groups, geometric group theory, and statistical mechanics.
Author: Jan Gregorovic̊
Publisher:
ISBN: 9788021060104
Category : Geometry
Languages : en
Pages : 91
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Book Description
Author: Clay Mathematics Institute. Summer School
Publisher: American Mathematical Soc.
ISBN: 9780821837153
Category : Mathematics
Languages : en
Pages : 396
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Book Description
Contains selection of expository and research article by lecturers at the school. Highlights current interests of researchers working at the interface between string theory and algebraic supergravity, supersymmetry, D-branes, the McKay correspondence andFourer-Mukai transform.
Author: Ana Cannas da Silva
Publisher: Springer
ISBN: 354045330X
Category : Mathematics
Languages : en
Pages : 240
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Book Description
The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.
Author: Kentaro Hori
Publisher: American Mathematical Soc.
ISBN: 0821829556
Category : Mathematics
Languages : en
Pages : 954
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Book Description
This thorough and detailed exposition is the result of an intensive month-long course on mirror symmetry sponsored by the Clay Mathematics Institute. It develops mirror symmetry from both mathematical and physical perspectives with the aim of furthering interaction between the two fields. The material will be particularly useful for mathematicians and physicists who wish to advance their understanding across both disciplines. Mirror symmetry is a phenomenon arising in string theory in which two very different manifolds give rise to equivalent physics. Such a correspondence has significant mathematical consequences, the most familiar of which involves the enumeration of holomorphic curves inside complex manifolds by solving differential equations obtained from a ``mirror'' geometry. The inclusion of D-brane states in the equivalence has led to further conjectures involving calibrated submanifolds of the mirror pairs and new (conjectural) invariants of complex manifolds: the Gopakumar-Vafa invariants. This book gives a single, cohesive treatment of mirror symmetry. Parts 1 and 2 develop the necessary mathematical and physical background from ``scratch''. The treatment is focused, developing only the material most necessary for the task. In Parts 3 and 4 the physical and mathematical proofs of mirror symmetry are given. From the physics side, this means demonstrating that two different physical theories give isomorphic physics. Each physical theory can be described geometrically, and thus mirror symmetry gives rise to a ``pairing'' of geometries. The proof involves applying $R\leftrightarrow 1/R$ circle duality to the phases of the fields in the gauged linear sigma model. The mathematics proof develops Gromov-Witten theory in the algebraic setting, beginning with the moduli spaces of curves and maps, and uses localization techniques to show that certain hypergeometric functions encode the Gromov-Witten invariants in genus zero, as is predicted by mirror symmetry. Part 5 is devoted to advanced topi This one-of-a-kind book is suitable for graduate students and research mathematicians interested in mathematics and mathematical and theoretical physics.
Author: Alejandro Adem
Publisher: Cambridge University Press
ISBN: 1139464485
Category : Mathematics
Languages : en
Pages : 138
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Book Description
An introduction to the theory of orbifolds from a modern perspective, combining techniques from geometry, algebraic topology and algebraic geometry. One of the main motivations, and a major source of examples, is string theory, where orbifolds play an important role. The subject is first developed following the classical description analogous to manifold theory, after which the book branches out to include the useful description of orbifolds provided by groupoids, as well as many examples in the context of algebraic geometry. Classical invariants such as de Rham cohomology and bundle theory are developed, a careful study of orbifold morphisms is provided, and the topic of orbifold K-theory is covered. The heart of this book, however, is a detailed description of the Chen-Ruan cohomology, which introduces a product for orbifolds and has had significant impact. The final chapter includes explicit computations for a number of interesting examples.
Author: William P. Thurston
Publisher: Princeton University Press
ISBN: 9780691083049
Category : Mathematics
Languages : en
Pages : 340
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Book Description
Every mathematician should be acquainted with the basic facts about the geometry of surfaces, of two-dimensional manifolds. The theory of three-dimensional manifolds is much more difficult and still only partly understood, although there is ample evidence that the theory of three-dimensional manifolds is one of the most beautiful in the whole of mathematics. This excellent introductory work makes this mathematical wonderland remained rather inaccessible to non-specialists. The author is both a leading researcher, with a formidable geometric intuition, and a gifted expositor. His vivid descriptions of what it might be like to live in this or that three-dimensional manifold bring the subject to life. Like Poincaré, he appeals to intuition, but his enthusiasm is infectious and should make many converts for this kind of mathematics. There are good pictures, plenty of exercises and problems, and the reader will find a selection of topics which are not found in the standard repertoire. This book contains a great deal of interesting mathematics.
