New Topological Invariants For Real And Angle Valued Maps An Alternative To Morse Novikov Theory PDF Download
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Author | : Dan Burghelea |
Publisher | : World Scientific |
Total Pages | : 259 |
Release | : 2017-08-16 |
Genre | : Mathematics |
ISBN | : 9814618268 |
Download New Topological Invariants For Real- And Angle-valued Maps: An Alternative To Morse-novikov Theory Book in PDF, ePub and Kindle
This book is about new topological invariants of real- and angle-valued maps inspired by Morse-Novikov theory, a chapter of topology, which has recently raised interest outside of mathematics; for example, in data analysis, shape recognition, computer science and physics. They are the backbone of what the author proposes as a computational alternative to Morse-Novikov theory, referred to in this book as AMN-theory.These invariants are on one side analogues of rest points, instantons and closed trajectories of vector fields and on the other side, refine basic topological invariants like homology and monodromy. They are associated to tame maps, considerably more general than Morse maps, that are defined on spaces which are considerably more general than manifolds. They are computable by computer implementable algorithms and have strong robustness properties. They relate the dynamics of flows that admit the map as 'Lyapunov map' to the topology of the underlying space, in a similar manner as Morse-Novikov theory does.
Author | : Dan Burghelea |
Publisher | : |
Total Pages | : |
Release | : 2013 |
Genre | : |
ISBN | : |
Download Topology of Angle Valued Maps, Bar Codes and Jordan Blocks Book in PDF, ePub and Kindle
In this paper one presents a collection of results relating the \bar codes" and \Jordan blocks", a new class of invariants for a tame angle valued map, with the topology of underlying space (and map). As a consequence one proposes refinements of Betti numbers and Novikov{Betti numbers provided by a continuous real or angle valued map defined on a compact ANR. These refinements can be interpreted as monic polynomials of degree the Betti numbers or Novikov{Betti numbers. One shows that these polynomials depend continuously on the real or the angle valued map and satisfy a Poincaré duality property in case the underlying space is a closed manifold. Our work offers an alternative perspective on Morse{Novikov theory which can be applied to a considerably larger class of spaces and maps and provides features inexistent in classical Morse{Novikov theory.
Author | : Alain Connes |
Publisher | : Springer |
Total Pages | : 364 |
Release | : 2003-12-15 |
Genre | : Mathematics |
ISBN | : 3540397027 |
Download Noncommutative Geometry Book in PDF, ePub and Kindle
Noncommutative Geometry is one of the most deep and vital research subjects of present-day Mathematics. Its development, mainly due to Alain Connes, is providing an increasing number of applications and deeper insights for instance in Foliations, K-Theory, Index Theory, Number Theory but also in Quantum Physics of elementary particles. The purpose of the Summer School in Martina Franca was to offer a fresh invitation to the subject and closely related topics; the contributions in this volume include the four main lectures, cover advanced developments and are delivered by prominent specialists.
Author | : Ana Cannas da Silva |
Publisher | : Springer |
Total Pages | : 240 |
Release | : 2004-10-27 |
Genre | : Mathematics |
ISBN | : 354045330X |
Download Lectures on Symplectic Geometry Book in PDF, ePub and Kindle
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 | : Danny Calegari |
Publisher | : Oxford University Press on Demand |
Total Pages | : 378 |
Release | : 2007-05-17 |
Genre | : Mathematics |
ISBN | : 0198570082 |
Download Foliations and the Geometry of 3-Manifolds Book in PDF, ePub and Kindle
This unique reference, aimed at research topologists, gives an exposition of the 'pseudo-Anosov' theory of foliations of 3-manifolds. This theory generalizes Thurston's theory of surface automorphisms and reveals an intimate connection between dynamics, geometry and topology in 3 dimensions. Significant themes returned to throughout the text include the importance of geometry, especially the hyperbolic geometry of surfaces, the importance of monotonicity, especially in1-dimensional and co-dimensional dynamics, and combinatorial approximation, using finite combinatorical objects such as train-tracks, branched surfaces and hierarchies to carry more complicated continuous objects.
Author | : S. Lefschetz |
Publisher | : Springer Science & Business Media |
Total Pages | : 190 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 1468493671 |
Download Applications of Algebraic Topology Book in PDF, ePub and Kindle
This monograph is based, in part, upon lectures given in the Princeton School of Engineering and Applied Science. It presupposes mainly an elementary knowledge of linear algebra and of topology. In topology the limit is dimension two mainly in the latter chapters and questions of topological invariance are carefully avoided. From the technical viewpoint graphs is our only requirement. However, later, questions notably related to Kuratowski's classical theorem have demanded an easily provided treatment of 2-complexes and surfaces. January 1972 Solomon Lefschetz 4 INTRODUCTION The study of electrical networks rests upon preliminary theory of graphs. In the literature this theory has always been dealt with by special ad hoc methods. My purpose here is to show that actually this theory is nothing else than the first chapter of classical algebraic topology and may be very advantageously treated as such by the well known methods of that science. Part I of this volume covers the following ground: The first two chapters present, mainly in outline, the needed basic elements of linear algebra. In this part duality is dealt with somewhat more extensively. In Chapter III the merest elements of general topology are discussed. Graph theory proper is covered in Chapters IV and v, first structurally and then as algebra. Chapter VI discusses the applications to networks. In Chapters VII and VIII the elements of the theory of 2-dimensional complexes and surfaces are presented.
Author | : Peter Kronheimer |
Publisher | : |
Total Pages | : 796 |
Release | : 2007-12-20 |
Genre | : Mathematics |
ISBN | : 9780521880220 |
Download Monopoles and Three-Manifolds Book in PDF, ePub and Kindle
This 2007 book provides a comprehensive treatment of Floer homology, based on the Seiberg-Witten equations. Suitable for beginning graduate students and researchers in the field, this book provides a full discussion of a central part of the study of the topology of manifolds.
Author | : Benson Farb |
Publisher | : Princeton University Press |
Total Pages | : 490 |
Release | : 2012 |
Genre | : Mathematics |
ISBN | : 0691147949 |
Download A Primer on Mapping Class Groups Book in PDF, ePub and Kindle
The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. A Primer on Mapping Class Groups begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.
Author | : Michael Davis |
Publisher | : Princeton University Press |
Total Pages | : 601 |
Release | : 2008 |
Genre | : Mathematics |
ISBN | : 0691131384 |
Download The Geometry and Topology of Coxeter Groups Book in PDF, ePub and Kindle
The Geometry and Topology of Coxeter Groups is a comprehensive and authoritative treatment of Coxeter groups from the viewpoint of geometric group theory. Groups generated by reflections are ubiquitous in mathematics, and there are classical examples of reflection groups in spherical, Euclidean, and hyperbolic geometry. Any Coxeter group can be realized as a group generated by reflection on a certain contractible cell complex, and this complex is the principal subject of this book. The book explains a theorem of Moussong that demonstrates that a polyhedral metric on this cell complex is nonpositively curved, meaning that Coxeter groups are "CAT(0) groups." The book describes the reflection group trick, one of the most potent sources of examples of aspherical manifolds. And the book discusses many important topics in geometric group theory and topology, including Hopf's theory of ends; contractible manifolds and homology spheres; the Poincaré Conjecture; and Gromov's theory of CAT(0) spaces and groups. Finally, the book examines connections between Coxeter groups and some of topology's most famous open problems concerning aspherical manifolds, such as the Euler Characteristic Conjecture and the Borel and Singer conjectures.
Author | : Victor M. Buchstaber |
Publisher | : American Mathematical Soc. |
Total Pages | : 534 |
Release | : 2015-07-15 |
Genre | : Mathematics |
ISBN | : 147042214X |
Download Toric Topology Book in PDF, ePub and Kindle
This book is about toric topology, a new area of mathematics that emerged at the end of the 1990s on the border of equivariant topology, algebraic and symplectic geometry, combinatorics, and commutative algebra. It has quickly grown into a very active area with many links to other areas of mathematics, and continues to attract experts from different fields. The key players in toric topology are moment-angle manifolds, a class of manifolds with torus actions defined in combinatorial terms. Construction of moment-angle manifolds relates to combinatorial geometry and algebraic geometry of toric varieties via the notion of a quasitoric manifold. Discovery of remarkable geometric structures on moment-angle manifolds led to important connections with classical and modern areas of symplectic, Lagrangian, and non-Kaehler complex geometry. A related categorical construction of moment-angle complexes and polyhedral products provides for a universal framework for many fundamental constructions of homotopical topology. The study of polyhedral products is now evolving into a separate subject of homotopy theory. A new perspective on torus actions has also contributed to the development of classical areas of algebraic topology, such as complex cobordism. This book includes many open problems and is addressed to experts interested in new ideas linking all the subjects involved, as well as to graduate students and young researchers ready to enter this beautiful new area.