Branched Standard Spines of 3-manifolds

Branched Standard Spines of 3-manifolds
Author: Riccardo Benedetti
Publisher: Springer
Total Pages: 140
Release: 2006-11-14
Genre: Mathematics
ISBN: 3540683453
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This book provides a unified combinatorial realization of the categroies of (closed, oriented) 3-manifolds, combed 3-manifolds, framed 3-manifolds and spin 3-manifolds. In all four cases the objects of the realization are finite enhanced graphs, and only finitely many local moves have to be taken into account. These realizations are based on the notion of branched standard spine, introduced in the book as a combination of the notion of branched surface with that of standard spine. The book is intended for readers interested in low-dimensional topology, and some familiarity with the basics is assumed. A list of questions, some of which concerning relations with the theory of quantum invariants, is enclosed.

European Congress of Mathematics

European Congress of Mathematics
Author: Antal Balog
Publisher: Springer Science & Business Media
Total Pages: 364
Release: 1998-07-21
Genre: Mathematics
ISBN: 9783764354978
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This is the first volume of the procedings of the second European Congress of Mathematics. Volume I presents the speeches delivered at the Congress, the list of lectures, and short summaries of the achievements of the prize winners. Together with volume II it contains a collection of contributions by the invited lecturers. Finally, volume II also presents reports on some of the Round Table discussions. This two-volume set thus gives an overview of the state of the art in many fields of mathematics and is therefore of interest to every professional mathematician. Contributors: Vol. I: N. Alon, L. Ambrosio, K. Astala, R. Benedetti, Ch. Bessenrodt, F. Bethuel, P. Bjørstad, E. Bolthausen, J. Bricmont, A. Kupiainen, D. Burago, L. Caporaso, U. Dierkes, I. Dynnikov, L.H. Eliasson, W.T. Gowers, H. Hedenmalm, A. Huber, J. Kaczorowski, J. Kollár, D.O. Kramkov, A.N. Shiryaev, C. Lescop, R. März. Vol. II: J. Matousek, D. McDuff, A.S. Merkurjev, V. Milman, St. Müller, T. Nowicki, E. Olivieri, E. Scoppola, V.P. Platonov, J. Pöschel, L. Polterovich , L. Pyber, N. Simányi, J.P. Solovej, A. Stipsicz, G. Tardos, J.-P. Tignol, A.P. Veselov, E. Zuazua

Canonical Wick Rotations in 3-Dimensional Gravity

Canonical Wick Rotations in 3-Dimensional Gravity
Author: R. Benedetti
Publisher: American Mathematical Soc.
Total Pages: 181
Release: 2009-03-06
Genre: Mathematics
ISBN: 0821842811
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The authors develop a canonical Wick rotation-rescaling theory in $3$-dimensional gravity. This includes (a) A simultaneous classification: this shows how maximal globally hyperbolic spacetimes of arbitrary constant curvature, which admit a complete Cauchy surface and canonical cosmological time, as well as complex projective structures on arbitrary surfaces, are all different materializations of ``more fundamental'' encoding structures. (b) Canonical geometric correlations: this shows how spacetimes of different curvature, that share a same encoding structure, are related to each other by canonical rescalings, and how they can be transformed by canonical Wick rotations in hyperbolic $3$-manifolds, that carry the appropriate asymptotic projective structure. Both Wick rotations and rescalings act along the canonical cosmological time and have universal rescaling functions. These correlations are functorial with respect to isomorphisms of the respective geometric categories.

Morse Theory Of Gradient Flows, Concavity And Complexity On Manifolds With Boundary

Morse Theory Of Gradient Flows, Concavity And Complexity On Manifolds With Boundary
Author: Katz Gabriel
Publisher: World Scientific
Total Pages: 516
Release: 2019-08-21
Genre: Mathematics
ISBN: 9814719684
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This monograph is an account of the author's investigations of gradient vector flows on compact manifolds with boundary. Many mathematical structures and constructions in the book fit comfortably in the framework of Morse Theory and, more generally, of the Singularity Theory of smooth maps.The geometric and combinatorial structures, arising from the interactions of vector flows with the boundary of the manifold, are surprisingly rich. This geometric setting leads organically to many encounters with Singularity Theory, Combinatorics, Differential Topology, Differential Geometry, Dynamical Systems, and especially with the boundary value problems for ordinary differential equations. This diversity of connections animates the book and is the main motivation behind it.The book is divided into two parts. The first part describes the flows in three dimensions. It is more pictorial in nature. The second part deals with the multi-dimensional flows, and thus is more analytical. Each of the nine chapters starts with a description of its purpose and main results. This organization provides the reader with independent entrances into different chapters.

Homotopy Quantum Field Theory

Homotopy Quantum Field Theory
Author: Vladimir G. Turaev
Publisher: European Mathematical Society
Total Pages: 300
Release: 2010
Genre: EMS tracts in mathematics
ISBN: 9783037190869
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Homotopy Quantum Field Theory (HQFT) is a branch of Topological Quantum Field Theory founded by E. Witten and M. Atiyah. It applies ideas from theoretical physics to study principal bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a fixed target space. This book is the first systematic exposition of Homotopy Quantum Field Theory. It starts with a formal definition of an HQFT and provides examples of HQFTs in all dimensions. The main body of the text is focused on $2$-dimensional and $3$-dimensional HQFTs. A study of these HQFTs leads to new algebraic objects: crossed Frobenius group-algebras, crossed ribbon group-categories, and Hopf group-coalgebras. These notions and their connections with HQFTs are discussed in detail. The text ends with several appendices including an outline of recent developments and a list of open problems. Three appendices by M. Muger and A. Virelizier summarize their work in this area. The book is addressed to mathematicians, theoretical physicists, and graduate students interested in topological aspects of quantum field theory. The exposition is self-contained and well suited for a one-semester graduate course. Prerequisites include only basics of algebra and topology.

Geometric topology

Geometric topology
Author: William Hilal Kazez
Publisher: American Mathematical Soc.
Total Pages: 500
Release: 1997
Genre: Mathematics
ISBN: 9780821806531
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Covers the proceedings of the 1993 Georgia International Topology Conference held at the University of Georgia during the month of August. This work includes Kirby's problem list, which contains a description of the progress made on each of the problems and includes a bibliography. It is suitable for those interested in the many areas of topology.

Integral Geometry, Radon Transforms and Complex Analysis

Integral Geometry, Radon Transforms and Complex Analysis
Author: Carlos A. Berenstein
Publisher: Springer Science & Business Media
Total Pages: 178
Release: 1998-04-16
Genre: Mathematics
ISBN: 9783540642077
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This book contains the notes of five short courses delivered at the "Centro Internazionale Matematico Estivo" session "Integral Geometry, Radon Transforms and Complex Analysis" held in Venice (Italy) in June 1996: three of them deal with various aspects of integral geometry, with a common emphasis on several kinds of Radon transforms, their properties and applications, the other two share a stress on CR manifolds and related problems. All lectures are accessible to a wide audience, and provide self-contained introductions and short surveys on the subjects, as well as detailed expositions of selected results.

Approximation of Free-Discontinuity Problems

Approximation of Free-Discontinuity Problems
Author: Andrea Braides
Publisher: Springer Science & Business Media
Total Pages: 176
Release: 1998-09-17
Genre: Mathematics
ISBN: 9783540647713
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Functionals involving both volume and surface energies have a number of applications ranging from Computer Vision to Fracture Mechanics. In order to tackle numerical and dynamical problems linked to such functionals many approximations by functionals defined on smooth functions have been proposed (using high-order singular perturbations, finite-difference or non-local energies, etc.) The purpose of this book is to present a global approach to these approximations using the theory of gamma-convergence and of special functions of bounded variation. The book is directed to PhD students and researchers in calculus of variations, interested in approximation problems with possible applications.