Virtual Fundamental Cycles In Symplectic Topology

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Virtual Fundamental Cycles in Symplectic Topology

John W. Morgan,Dusa McDuff,Mohammad Tehrani,Kenji Fukaya,Dominic Joyce
Virtual Fundamental Cycles in Symplectic Topology Book PDF

Author : John W. Morgan,Dusa McDuff,Mohammad Tehrani,Kenji Fukaya,Dominic Joyce
Publisher : American Mathematical Soc.
Page : 300 pages
File Size : 53,5 Mb
Release : 2019-04-12
Category : Geometry, Differential
ISBN : 9781470450144

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Virtual Fundamental Cycles in Symplectic Topology by John W. Morgan,Dusa McDuff,Mohammad Tehrani,Kenji Fukaya,Dominic Joyce Pdf

The method of using the moduli space of pseudo-holomorphic curves on a symplectic manifold was introduced by Mikhail Gromov in 1985. From the appearance of Gromov's original paper until today this approach has been the most important tool in global symplectic geometry. To produce numerical invariants of these manifolds using this method requires constructing a fundamental cycle associated with moduli spaces. This volume brings together three approaches to constructing the “virtual” fundamental cycle for the moduli space of pseudo-holomorphic curves. All approaches are based on the idea of local Kuranishi charts for the moduli space. Workers in the field will get a comprehensive understanding of the details of these constructions and the assumptions under which they can be made. These techniques and results will be essential in further applications of this approach to producing invariants of symplectic manifolds.

Kuranishi Structures and Virtual Fundamental Chains

Kenji Fukaya,Yong-Geun Oh,Hiroshi Ohta,Kaoru Ono
Kuranishi Structures and Virtual Fundamental Chains Book PDF

Author : Kenji Fukaya,Yong-Geun Oh,Hiroshi Ohta,Kaoru Ono
Publisher : Springer Nature
Page : 638 pages
File Size : 49,5 Mb
Release : 2020-10-16
Category : Mathematics
ISBN : 9789811555626

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Kuranishi Structures and Virtual Fundamental Chains by Kenji Fukaya,Yong-Geun Oh,Hiroshi Ohta,Kaoru Ono Pdf

The package of Gromov’s pseudo-holomorphic curves is a major tool in global symplectic geometry and its applications, including mirror symmetry and Hamiltonian dynamics. The Kuranishi structure was introduced by two of the authors of the present volume in the mid-1990s to apply this machinery on general symplectic manifolds without assuming any specific restrictions. It was further amplified by this book’s authors in their monograph Lagrangian Intersection Floer Theory and in many other publications of theirs and others. Answering popular demand, the authors now present the current book, in which they provide a detailed, self-contained explanation of the theory of Kuranishi structures. Part I discusses the theory on a single space equipped with Kuranishi structure, called a K-space, and its relevant basic package. First, the definition of a K-space and maps to the standard manifold are provided. Definitions are given for fiber products, differential forms, partitions of unity, and the notion of CF-perturbations on the K-space. Then, using CF-perturbations, the authors define the integration on K-space and the push-forward of differential forms, and generalize Stokes' formula and Fubini's theorem in this framework. Also, “virtual fundamental class” is defined, and its cobordism invariance is proved. Part II discusses the (compatible) system of K-spaces and the process of going from “geometry” to “homological algebra”. Thorough explanations of the extension of given perturbations on the boundary to the interior are presented. Also explained is the process of taking the “homotopy limit” needed to handle a system of infinitely many moduli spaces. Having in mind the future application of these chain level constructions beyond those already known, an axiomatic approach is taken by listing the properties of the system of the relevant moduli spaces and then a self-contained account of the construction of the associated algebraic structures is given. This axiomatic approach makes the exposition contained here independent of previously published construction of relevant structures.

The Adams Spectral Sequence for Topological Modular Forms

Robert R. Bruner,John Rognes
The Adams Spectral Sequence for Topological Modular Forms Book PDF

Author : Robert R. Bruner,John Rognes
Publisher : American Mathematical Society
Page : 690 pages
File Size : 51,5 Mb
Release : 2021-12-23
Category : Mathematics
ISBN : 9781470469580

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The Adams Spectral Sequence for Topological Modular Forms by Robert R. Bruner,John Rognes Pdf

The connective topological modular forms spectrum, $tmf$, is in a sense initial among elliptic spectra, and as such is an important link between the homotopy groups of spheres and modular forms. A primary goal of this volume is to give a complete account, with full proofs, of the homotopy of $tmf$ and several $tmf$-module spectra by means of the classical Adams spectral sequence, thus verifying, correcting, and extending existing approaches. In the process, folklore results are made precise and generalized. Anderson and Brown-Comenetz duality, and the corresponding dualities in homotopy groups, are carefully proved. The volume also includes an account of the homotopy groups of spheres through degree 44, with complete proofs, except that the Adams conjecture is used without proof. Also presented are modern stable proofs of classical results which are hard to extract from the literature. Tools used in this book include a multiplicative spectral sequence generalizing a construction of Davis and Mahowald, and computer software which computes the cohomology of modules over the Steenrod algebra and products therein. Techniques from commutative algebra are used to make the calculation precise and finite. The $H$-infinity ring structure of the sphere and of $tmf$ are used to determine many differentials and relations.

Algebraic Geometry over C∞-Rings

Dominic Joyce
Algebraic Geometry over C Rings Book PDF

Author : Dominic Joyce
Publisher : American Mathematical Soc.
Page : 139 pages
File Size : 52,6 Mb
Release : 2019-09-05
Category : Electronic
ISBN : 9781470436452

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Algebraic Geometry over C∞-Rings by Dominic Joyce Pdf

If X is a manifold then the R-algebra C∞(X) of smooth functions c:X→R is a C∞-ring. That is, for each smooth function f:Rn→R there is an n-fold operation Φf:C∞(X)n→C∞(X) acting by Φf:(c1,…,cn)↦f(c1,…,cn), and these operations Φf satisfy many natural identities. Thus, C∞(X) actually has a far richer structure than the obvious R-algebra structure. The author explains the foundations of a version of algebraic geometry in which rings or algebras are replaced by C∞-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are C∞-schemes, a category of geometric objects which generalize manifolds and whose morphisms generalize smooth maps. The author also studies quasicoherent sheaves on C∞-schemes, and C∞-stacks, in particular Deligne-Mumford C∞-stacks, a 2-category of geometric objects generalizing orbifolds. Many of these ideas are not new: C∞-rings and C∞ -schemes have long been part of synthetic differential geometry. But the author develops them in new directions. In earlier publications, the author used these tools to define d-manifolds and d-orbifolds, “derived” versions of manifolds and orbifolds related to Spivak's “derived manifolds”.

Sampling in Combinatorial and Geometric Set Systems

Nabil H. Mustafa
Sampling in Combinatorial and Geometric Set Systems Book PDF

Author : Nabil H. Mustafa
Publisher : American Mathematical Society
Page : 251 pages
File Size : 55,6 Mb
Release : 2022-01-14
Category : Mathematics
ISBN : 9781470461560

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Sampling in Combinatorial and Geometric Set Systems by Nabil H. Mustafa Pdf

Understanding the behavior of basic sampling techniques and intrinsic geometric attributes of data is an invaluable skill that is in high demand for both graduate students and researchers in mathematics, machine learning, and theoretical computer science. The last ten years have seen significant progress in this area, with many open problems having been resolved during this time. These include optimal lower bounds for epsilon-nets for many geometric set systems, the use of shallow-cell complexity to unify proofs, simpler and more efficient algorithms, and the use of epsilon-approximations for construction of coresets, to name a few. This book presents a thorough treatment of these probabilistic, combinatorial, and geometric methods, as well as their combinatorial and algorithmic applications. It also revisits classical results, but with new and more elegant proofs. While mathematical maturity will certainly help in appreciating the ideas presented here, only a basic familiarity with discrete mathematics, probability, and combinatorics is required to understand the material.

Amenability of Discrete Groups by Examples

Kate Juschenko
Amenability of Discrete Groups by Examples Book PDF

Author : Kate Juschenko
Publisher : American Mathematical Society
Page : 180 pages
File Size : 52,9 Mb
Release : 2022-06-30
Category : Mathematics
ISBN : 9781470470326

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Amenability of Discrete Groups by Examples by Kate Juschenko Pdf

The main topic of the book is amenable groups, i.e., groups on which there exist invariant finitely additive measures. It was discovered that the existence or non-existence of amenability is responsible for many interesting phenomena such as, e.g., the Banach-Tarski Paradox about breaking a sphere into two spheres of the same radius. Since then, amenability has been actively studied and a number of different approaches resulted in many examples of amenable and non-amenable groups. In the book, the author puts together main approaches to study amenability. A novel feature of the book is that the exposition of the material starts with examples which introduce a method rather than illustrating it. This allows the reader to quickly move on to meaningful material without learning and remembering a lot of additional definitions and preparatory results; those are presented after analyzing the main examples. The techniques that are used for proving amenability in this book are mainly a combination of analytic and probabilistic tools with geometric group theory.

Asymptotic Geometric Analysis, Part II

Shiri Artstein-Avidan,Apostolos Giannopoulos,Vitali D. Milman
Asymptotic Geometric Analysis Part II Book PDF

Author : Shiri Artstein-Avidan,Apostolos Giannopoulos,Vitali D. Milman
Publisher : American Mathematical Society
Page : 645 pages
File Size : 52,7 Mb
Release : 2021-12-13
Category : Mathematics
ISBN : 9781470463601

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Asymptotic Geometric Analysis, Part II by Shiri Artstein-Avidan,Apostolos Giannopoulos,Vitali D. Milman Pdf

This book is a continuation of Asymptotic Geometric Analysis, Part I, which was published as volume 202 in this series. Asymptotic geometric analysis studies properties of geometric objects, such as normed spaces, convex bodies, or convex functions, when the dimensions of these objects increase to infinity. The asymptotic approach reveals many very novel phenomena which influence other fields in mathematics, especially where a large data set is of main concern, or a number of parameters which becomes uncontrollably large. One of the important features of this new theory is in developing tools which allow studying high parametric families. Among the topics covered in the book are measure concentration, isoperimetric constants of log-concave measures, thin-shell estimates, stochastic localization, the geometry of Gaussian measures, volume inequalities for convex bodies, local theory of Banach spaces, type and cotype, the Banach-Mazur compactum, symmetrizations, restricted invertibility, and functional versions of geometric notions and inequalities.

Hopf Algebras and Galois Module Theory

Lindsay N. Childs,Cornelius Greither,Kevin P. Keating,Alan Koch,Timothy Kohl,Paul J. Truman,Robert G. Underwood
Hopf Algebras and Galois Module Theory Book PDF

Author : Lindsay N. Childs,Cornelius Greither,Kevin P. Keating,Alan Koch,Timothy Kohl,Paul J. Truman,Robert G. Underwood
Publisher : American Mathematical Soc.
Page : 311 pages
File Size : 47,6 Mb
Release : 2021-11-10
Category : Education
ISBN : 9781470465162

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Hopf Algebras and Galois Module Theory by Lindsay N. Childs,Cornelius Greither,Kevin P. Keating,Alan Koch,Timothy Kohl,Paul J. Truman,Robert G. Underwood Pdf

Hopf algebras have been shown to play a natural role in studying questions of integral module structure in extensions of local or global fields. This book surveys the state of the art in Hopf-Galois theory and Hopf-Galois module theory and can be viewed as a sequel to the first author's book, Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory, which was published in 2000. The book is divided into two parts. Part I is more algebraic and focuses on Hopf-Galois structures on Galois field extensions, as well as the connection between this topic and the theory of skew braces. Part II is more number theoretical and studies the application of Hopf algebras to questions of integral module structure in extensions of local or global fields. Graduate students and researchers with a general background in graduate-level algebra, algebraic number theory, and some familiarity with Hopf algebras will appreciate the overview of the current state of this exciting area and the suggestions for numerous avenues for further research and investigation.

Maximal Cohen–Macaulay Modules and Tate Cohomology

Ragnar-Olaf Buchweitz
Maximal Cohen Macaulay Modules and Tate Cohomology Book PDF

Author : Ragnar-Olaf Buchweitz
Publisher : American Mathematical Society
Page : 175 pages
File Size : 48,7 Mb
Release : 2021-12-16
Category : Mathematics
ISBN : 9781470453404

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Maximal Cohen–Macaulay Modules and Tate Cohomology by Ragnar-Olaf Buchweitz Pdf

This book is a lightly edited version of the unpublished manuscript Maximal Cohen–Macaulay modules and Tate cohomology over Gorenstein rings by Ragnar-Olaf Buchweitz. The central objects of study are maximal Cohen–Macaulay modules over (not necessarily commutative) Gorenstein rings. The main result is that the stable category of maximal Cohen–Macaulay modules over a Gorenstein ring is equivalent to the stable derived category and also to the homotopy category of acyclic complexes of projective modules. This assimilates and significantly extends earlier work of Eisenbud on hypersurface singularities. There is also an extensive discussion of duality phenomena in stable derived categories, extending Tate duality on cohomology of finite groups. Another noteworthy aspect is an extension of the classical BGG correspondence to super-algebras. There are numerous examples that illustrate these ideas. The text includes a survey of developments subsequent to, and connected with, Buchweitz's manuscript.

Completion Problems on Operator Matrices

Dragana S. Cvetković Ilić
Completion Problems on Operator Matrices Book PDF

Author : Dragana S. Cvetković Ilić
Publisher : American Mathematical Society
Page : 170 pages
File Size : 47,8 Mb
Release : 2022-06-07
Category : Mathematics
ISBN : 9781470469870

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Completion Problems on Operator Matrices by Dragana S. Cvetković Ilić Pdf

Completion problems for operator matrices are concerned with the question of whether a partially specified operator matrix can be completed to form an operator of a desired type. The research devoted to this topic provides an excellent means to investigate the structure of operators. This book provides an overview of completion problems dealing with completions to different types of operators and can be considered as a natural extension of classical results concerned with matrix completions. The book assumes some basic familiarity with functional analysis and operator theory. It will be useful for graduate students and researchers interested in operator theory and the problem of matrix completions.

Diagrammatic Algebra

J. Scott Carter,Seiichi Kamada
Diagrammatic Algebra Book PDF

Author : J. Scott Carter,Seiichi Kamada
Publisher : American Mathematical Society
Page : 365 pages
File Size : 47,9 Mb
Release : 2021-12-15
Category : Mathematics
ISBN : 9781470466718

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Diagrammatic Algebra by J. Scott Carter,Seiichi Kamada Pdf

This book is an introduction to techniques and results in diagrammatic algebra. It starts with abstract tensors and their categorifications, presents diagrammatic methods for studying Frobenius and Hopf algebras, and discusses their relations with topological quantum field theory and knot theory. The text is replete with figures, diagrams, and suggestive typography that allows the reader a glimpse into many higher dimensional processes. The penultimate chapter summarizes the previous material by demonstrating how to braid 3- and 4- dimensional manifolds into 5- and 6-dimensional spaces. The book is accessible to post-qualifier graduate students, and will also be of interest to algebraists, topologists and algebraic topologists who would like to incorporate diagrammatic techniques into their research.

Ridge Functions and Applications in Neural Networks

Vugar E. Ismailov
Ridge Functions and Applications in Neural Networks Book PDF

Author : Vugar E. Ismailov
Publisher : American Mathematical Society
Page : 186 pages
File Size : 53,8 Mb
Release : 2021-12-17
Category : Mathematics
ISBN : 9781470467654

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Ridge Functions and Applications in Neural Networks by Vugar E. Ismailov Pdf

Recent years have witnessed a growth of interest in the special functions called ridge functions. These functions appear in various fields and under various guises. They appear in partial differential equations (where they are called plane waves), in computerized tomography, and in statistics. Ridge functions are also the underpinnings of many central models in neural network theory. In this book various approximation theoretic properties of ridge functions are described. This book also describes properties of generalized ridge functions, and their relation to linear superpositions and Kolmogorov's famous superposition theorem. In the final part of the book, a single and two hidden layer neural networks are discussed. The results obtained in this part are based on properties of ordinary and generalized ridge functions. Novel aspects of the universal approximation property of feedforward neural networks are revealed. This book will be of interest to advanced graduate students and researchers working in functional analysis, approximation theory, and the theory of real functions, and will be of particular interest to those wishing to learn more about neural network theory and applications and other areas where ridge functions are used.

Attractors Under Autonomous and Non-autonomous Perturbations

Matheus C. Bortolan,Alexandre N. Carvalho,José A. Langa
Attractors Under Autonomous and Non autonomous Perturbations Book PDF

Author : Matheus C. Bortolan,Alexandre N. Carvalho,José A. Langa
Publisher : American Mathematical Soc.
Page : 246 pages
File Size : 49,6 Mb
Release : 2020-05-29
Category : Education
ISBN : 9781470453084

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Attractors Under Autonomous and Non-autonomous Perturbations by Matheus C. Bortolan,Alexandre N. Carvalho,José A. Langa Pdf

This book provides a comprehensive study of how attractors behave under perturbations for both autonomous and non-autonomous problems. Furthermore, the forward asymptotics of non-autonomous dynamical systems is presented here for the first time in a unified manner. When modelling real world phenomena imprecisions are unavoidable. On the other hand, it is paramount that mathematical models reflect the modelled phenomenon, in spite of unimportant neglectable influences discounted by simplifications, small errors introduced by empirical laws or measurements, among others. The authors deal with this issue by investigating the permanence of dynamical structures and continuity properties of the attractor. This is done in both the autonomous (time independent) and non-autonomous (time dependent) framework in four distinct levels of approximation: the upper semicontinuity, lower semicontinuity, topological structural stability and geometrical structural stability. This book is aimed at graduate students and researchers interested in dissipative dynamical systems and stability theory, and requires only a basic background in metric spaces, functional analysis and, for the applications, techniques of ordinary and partial differential equations.

Ordinary Differential Operators

Aiping Wang,Anton Zettl
Ordinary Differential Operators Book PDF

Author : Aiping Wang,Anton Zettl
Publisher : American Mathematical Soc.
Page : 250 pages
File Size : 41,9 Mb
Release : 2019-11-08
Category : Education
ISBN : 9781470453664

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Ordinary Differential Operators by Aiping Wang,Anton Zettl Pdf

In 1910 Herman Weyl published one of the most widely quoted papers of the 20th century in Analysis, which initiated the study of singular Sturm-Liouville problems. The work on the foundations of Quantum Mechanics in the 1920s and 1930s, including the proof of the spectral theorem for unbounded self-adjoint operators in Hilbert space by von Neumann and Stone, provided some of the motivation for the study of differential operators in Hilbert space with particular emphasis on self-adjoint operators and their spectrum. Since then the topic developed in several directions and many results and applications have been obtained. In this monograph the authors summarize some of these directions discussing self-adjoint, symmetric, and dissipative operators in Hilbert and Symplectic Geometry spaces. Part I of the book covers the theory of differential and quasi-differential expressions and equations, existence and uniqueness of solutions, continuous and differentiable dependence on initial data, adjoint expressions, the Lagrange Identity, minimal and maximal operators, etc. In Part II characterizations of the symmetric, self-adjoint, and dissipative boundary conditions are established. In particular, the authors prove the long standing Deficiency Index Conjecture. In Part III the symmetric and self-adjoint characterizations are extended to two-interval problems. These problems have solutions which have jump discontinuities in the interior of the underlying interval. These jumps may be infinite at singular interior points. Part IV is devoted to the construction of the regular Green's function. The construction presented differs from the usual one as found, for example, in the classical book by Coddington and Levinson.

Nonlinear Dirac Equation: Spectral Stability of Solitary Waves

Nabile Boussaïd,Andrew Comech
Nonlinear Dirac Equation Spectral Stability of Solitary Waves Book PDF

Author : Nabile Boussaïd,Andrew Comech
Publisher : American Mathematical Soc.
Page : 297 pages
File Size : 41,9 Mb
Release : 2019-11-21
Category : Education
ISBN : 9781470443955

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Nonlinear Dirac Equation: Spectral Stability of Solitary Waves by Nabile Boussaïd,Andrew Comech Pdf

This monograph gives a comprehensive treatment of spectral (linear) stability of weakly relativistic solitary waves in the nonlinear Dirac equation. It turns out that the instability is not an intrinsic property of the Dirac equation that is only resolved in the framework of the second quantization with the Dirac sea hypothesis. Whereas general results about the Dirac-Maxwell and similar equations are not yet available, we can consider the Dirac equation with scalar self-interaction, the model first introduced in 1938. In this book we show that in particular cases solitary waves in this model may be spectrally stable (no linear instability). This result is the first step towards proving asymptotic stability of solitary waves. The book presents the necessary overview of the functional analysis, spectral theory, and the existence and linear stability of solitary waves of the nonlinear Schrödinger equation. It also presents the necessary tools such as the limiting absorption principle and the Carleman estimates in the form applicable to the Dirac operator, and proves the general form of the Dirac-Pauli theorem. All of these results are used to prove the spectral stability of weakly relativistic solitary wave solutions of the nonlinear Dirac equation.