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H Principles And Flexibility In Geometry
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Book Synopsis $h$-Principles and Flexibility in Geometry by : Hansjörg Geiges
Download or read book $h$-Principles and Flexibility in Geometry written by Hansjörg Geiges and published by American Mathematical Soc.. This book was released on 2003 with total page 74 pages. Available in PDF, EPUB and Kindle. Book excerpt: The notion of homotopy principle or $h$-principle is one of the key concepts in an elegant language developed by Gromov to deal with a host of questions in geometry and topology. Roughly speaking, for a certain differential geometric problem to satisfy the $h$-principle is equivalent to saying that a solution to the problem exists whenever certain obvious topological obstructions vanish. The foundational examples for applications of Gromov's ideas include (i) Hirsch-Smale immersion theory, (ii) Nash-Kuiper $C^1$-isometric immersion theory, (iii) existence of symplectic and contact structures on open manifolds. Gromov has developed several powerful methods that allow one to prove $h$-principles. These notes, based on lectures given in the Graduiertenkolleg of Leipzig University, present two such methods which are strong enough to deal with applications (i) and (iii).
Book Synopsis Introduction to the $h$-Principle by : K. Cieliebak
Download or read book Introduction to the $h$-Principle written by K. Cieliebak and published by American Mathematical Society. This book was released on 2024-01-30 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: In differential geometry and topology one often deals with systems of partial differential equations as well as partial differential inequalities that have infinitely many solutions whatever boundary conditions are imposed. It was discovered in the 1950s that the solvability of differential relations (i.e., equations and inequalities) of this kind can often be reduced to a problem of a purely homotopy-theoretic nature. One says in this case that the corresponding differential relation satisfies the $h$-principle. Two famous examples of the $h$-principle, the Nash–Kuiper $C^1$-isometric embedding theory in Riemannian geometry and the Smale–Hirsch immersion theory in differential topology, were later transformed by Gromov into powerful general methods for establishing the $h$-principle. The authors cover two main methods for proving the $h$-principle: holonomic approximation and convex integration. The reader will find that, with a few notable exceptions, most instances of the $h$-principle can be treated by the methods considered here. A special emphasis is made on applications to symplectic and contact geometry. The present book is the first broadly accessible exposition of the theory and its applications, making it an excellent text for a graduate course on geometric methods for solving partial differential equations and inequalities. Geometers, topologists, and analysts will also find much value in this very readable exposition of an important and remarkable topic. This second edition of the book is significantly revised and expanded to almost twice of the original size. The most significant addition to the original book is the new part devoted to the method of wrinkling and its applications. Several other chapters (e.g., on multivalued holonomic approximation and foliations) are either added or completely rewritten.
Book Synopsis Partial Differential Relations by : Misha Gromov
Download or read book Partial Differential Relations written by Misha Gromov and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 372 pages. Available in PDF, EPUB and Kindle. Book excerpt: The classical theory of partial differential equations is rooted in physics, where equations (are assumed to) describe the laws of nature. Law abiding functions, which satisfy such an equation, are very rare in the space of all admissible functions (regardless of a particular topology in a function space). Moreover, some additional (like initial or boundary) conditions often insure the uniqueness of solutions. The existence of these is usually established with some apriori estimates which locate a possible solution in a given function space. We deal in this book with a completely different class of partial differential equations (and more general relations) which arise in differential geometry rather than in physics. Our equations are, for the most part, undetermined (or, at least, behave like those) and their solutions are rather dense in spaces of functions. We solve and classify solutions of these equations by means of direct (and not so direct) geometric constructions. Our exposition is elementary and the proofs of the basic results are selfcontained. However, there is a number of examples and exercises (of variable difficulty), where the treatment of a particular equation requires a certain knowledge of pertinent facts in the surrounding field. The techniques we employ, though quite general, do not cover all geometrically interesting equations. The border of the unexplored territory is marked by a number of open questions throughout the book.
Book Synopsis A Course on Holomorphic Discs by : Hansjörg Geiges
Download or read book A Course on Holomorphic Discs written by Hansjörg Geiges and published by Springer Nature. This book was released on 2023-08-07 with total page 203 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook, based on a one-semester course taught several times by the authors, provides a self-contained, comprehensive yet concise introduction to the theory of pseudoholomorphic curves. Gromov’s nonsqueezing theorem in symplectic topology is taken as a motivating example, and a complete proof using pseudoholomorphic discs is presented. A sketch of the proof is discussed in the first chapter, with succeeding chapters guiding the reader through the details of the mathematical methods required to establish compactness, regularity, and transversality results. Concrete examples illustrate many of the more complicated concepts, and well over 100 exercises are distributed throughout the text. This approach helps the reader to gain a thorough understanding of the powerful analytical tools needed for the study of more advanced topics in symplectic topology. /divThis text can be used as the basis for a graduate course, and it is also immensely suitable for independent study. Prerequisites include complex analysis, differential topology, and basic linear functional analysis; no prior knowledge of symplectic geometry is assumed. This book is also part of the Virtual Series on Symplectic Geometry.
Download or read book Modern Geometry written by Vicente Muñoz and published by American Mathematical Soc.. This book was released on 2018-09-05 with total page 426 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains a collection of survey articles of exciting new developments in geometry, written in tribute to Simon Donaldson to celebrate his 60th birthday. Reflecting the wide range of Donaldson's interests and influence, the papers range from algebraic geometry and topology through symplectic geometry and geometric analysis to mathematical physics. Their expository nature means the book acts as an invitation to the various topics described, while also giving a sense of the links between these different areas and the unity of modern geometry.
Book Synopsis An Introduction to Contact Topology by : Hansjörg Geiges
Download or read book An Introduction to Contact Topology written by Hansjörg Geiges and published by Cambridge University Press. This book was released on 2008-03-13 with total page 8 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text on contact topology is a comprehensive introduction to the subject, including recent striking applications in geometric and differential topology: Eliashberg's proof of Cerf's theorem via the classification of tight contact structures on the 3-sphere, and the Kronheimer-Mrowka proof of property P for knots via symplectic fillings of contact 3-manifolds. Starting with the basic differential topology of contact manifolds, all aspects of 3-dimensional contact manifolds are treated in this book. One notable feature is a detailed exposition of Eliashberg's classification of overtwisted contact structures. Later chapters also deal with higher-dimensional contact topology. Here the focus is on contact surgery, but other constructions of contact manifolds are described, such as open books or fibre connected sums. This book serves both as a self-contained introduction to the subject for advanced graduate students and as a reference for researchers.
Book Synopsis Symplectic, Poisson, and Noncommutative Geometry by : Tohru Eguchi
Download or read book Symplectic, Poisson, and Noncommutative Geometry written by Tohru Eguchi and published by Cambridge University Press. This book was released on 2014-08-25 with total page 303 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains seven chapters based on lectures given by invited speakers at two May 2010 workshops held at the Mathematical Sciences Research Institute.
Book Synopsis Principles of Geometry: Higher geometry by : Henry Frederick Baker
Download or read book Principles of Geometry: Higher geometry written by Henry Frederick Baker and published by . This book was released on 1925 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Principles of Geometry by : Henry Frederick Baker
Download or read book Principles of Geometry written by Henry Frederick Baker and published by . This book was released on 1923 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Principles of Projective Geometry Applied to the Straight Line and Conic by : John Leigh Smeathman Hatton
Download or read book The Principles of Projective Geometry Applied to the Straight Line and Conic written by John Leigh Smeathman Hatton and published by . This book was released on 1913 with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Introduction to the H-principle by : Y. Eliashberg
Download or read book Introduction to the H-principle written by Y. Eliashberg and published by American Mathematical Soc.. This book was released on with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: One of the most powerful modern methods of solving partial differential equations is Gromov's $h$-principle. It has also been, traditionally, one of the most difficult to explain. This book is the first broadly accessible exposition of the principle and its applications. The essence of the $h$-principle is the reduction of problems involving partial differential relations to problems of a purely homotopy-theoretic nature. Two famous examples of the $h$-principle are the Nash-Kuiper$C1$-isometric embedding theory in Riemannian geometry and the Smale-Hirsch immersion theory in differential topology. Gromov transformed these examples into a powerful general method for proving the $h$-principle. Both of these examples and their explanations in terms of the $h$-principle arecovered in detail in the book. The authors cover two main embodiments of the principle: holonomic approximation and convex integration. The first is a version of the method of continuous sheaves. The reader will find that, with a few notable exceptions, most instances of the $h$-principle can be treated by the methods considered here. There are, naturally, many connections to symplectic and contact geometry. The book would be an excellent text for a graduate course on modern methods for solvingpartial differential equations. Geometers and analysts will also find much value in this very readable exposition of an important and remarkable technique.
Book Synopsis Principles of Geometry by : Henry F. Baker
Download or read book Principles of Geometry written by Henry F. Baker and published by CUP Archive. This book was released on 1954 with total page 284 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Riemannian Geometry by : Gérard Besson
Download or read book Riemannian Geometry written by Gérard Besson and published by American Mathematical Soc.. This book was released on 1996 with total page 130 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a compendium of survey lectures presented at a conference on Riemannian Geometry sponsored by The Fields Institute for Research in Mathematical Sciences (Waterloo, Canada) in August 1993. Attended by over 80 participants, the aim of the conference was to promote research activity in Riemannian geometry. A select group of internationally established researchers in the field were invited to discuss and present current developments in a selection of contemporary topics in Riemannian geometry. This volume contains four of the five survey lectures presented at the conference. The book features basic notions of volume and entropy and the difficult and deep relations of these invariants to curvature. It also features $LP$ cohomology, in which the methods combine various areas of mathematics going beyond Riemannian geometry. It covers curvature inequalities from a general point of view, leading to the study of general spaces.
Book Synopsis Geometry, Topology and Physics, Second Edition by : Mikio Nakahara
Download or read book Geometry, Topology and Physics, Second Edition written by Mikio Nakahara and published by CRC Press. This book was released on 2003-06-04 with total page 598 pages. Available in PDF, EPUB and Kindle. Book excerpt: Differential geometry and topology have become essential tools for many theoretical physicists. In particular, they are indispensable in theoretical studies of condensed matter physics, gravity, and particle physics. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields. The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the development of the subject. The book features a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories. Chapter 2 introduces the mathematical concepts of maps, vector spaces, and topology. The following chapters focus on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber bundles, characteristic classes, and index theorems. New to this second edition is the proof of the index theorem in terms of supersymmetric quantum mechanics. The final two chapters are devoted to the most fascinating applications of geometry and topology in contemporary physics, namely the study of anomalies in gauge field theories and the analysis of Polakov's bosonic string theory from the geometrical point of view. Geometry, Topology and Physics, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics.
Book Synopsis Surgery on Contact 3-Manifolds and Stein Surfaces by : Burak Ozbagci
Download or read book Surgery on Contact 3-Manifolds and Stein Surfaces written by Burak Ozbagci and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 279 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is about an investigation of recent developments in the field of sympletic and contact structures on four- and three-dimensional manifolds from a topologist’s point of view. In it, two main issues are addressed: what kind of sympletic and contact structures we can construct via surgery theory and what kind of sympletic and contact structures are not allowed via gauge theory and the newly invented Heegaard-Floer theory.
Book Synopsis Rudimentary Principles of Geometry. The Application of Logic to Geometrical Reasoning Based on the Elements of Euclid, Books 1, 2, 3 [4-6, 11, 12]. By H. Law by :
Download or read book Rudimentary Principles of Geometry. The Application of Logic to Geometrical Reasoning Based on the Elements of Euclid, Books 1, 2, 3 [4-6, 11, 12]. By H. Law written by and published by . This book was released on 1855 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Dynamics of Topologically Generic Homeomorphisms by : Ethan Akin
Download or read book Dynamics of Topologically Generic Homeomorphisms written by Ethan Akin and published by American Mathematical Soc.. This book was released on 2003 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of this work is to describe the dynamics of generic homeomorphisms of certain compact metric spaces $X$. Here ``generic'' is used in the topological sense -- a property of homeomorphisms on $X$ is generic if the set of homeomorphisms with the property contains a residual subset (in the sense of Baire category) of the space of all homeomorphisms on $X$. The spaces $X$ we consider are those with enough local homogeneity to allow certain localized perturbations of homeomorphisms; for example, any compact manifold is such a space. We show that the dynamics of a generic homeomorphism is quite complicated, with a number of distinct dynamical behaviors coexisting (some resemble subshifts of finite type, others, which we call `generalized adding machines', appear strictly periodic when viewed to any finite precision, but are not actually periodic). Such a homeomorphism has infinitely many, intricately nested attractors and repellors, and uncountably many distinct dynamically-connected components of the chain recurrent set. We single out several types of these ``chain components'', and show that each type occurs densely (in an appropriate sense) in the chain recurrent set. We also identify one type that occurs generically in the chain recurrent set. We also show that, at least for $X$ a manifold, the chain recurrent set of a generic homeomorphism is a Cantor set, so its complement is open and dense. Somewhat surprisingly, there is a residual subset of $X$ consisting of points whose limit sets are chain components of a type other than the type of chain components that are residual in the space of all chain components. In fact, for each generic homeomorphism on $X$ there is a residual subset of points of $X$ satisfying a stability condition stronger than Lyapunov stability.
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