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Homotopy Theory and Arithmetic Geometry - Motivic and Diophantine Aspects
This book provides an introduction to state-of-the-art applications of homotopy theory to arithmetic geometry. The contributions to this volume are based on original lectures by leading researchers at the LMS-CMI Research School on 'Homotopy Theory and Arithmetic Geometry - Motivic and Diophantine Aspects' and the Nelder Fellow Lecturer Series, which both took place at Imperial College London in the summer of 2018. The contribution by Brazelton, based on the lectures by Wickelgren, provides an introduction to arithmetic enumerative geometry, the notes of Cisinski present motivic sheaves and new cohomological methods for intersection theory, and Schlank's contribution gives an overview of the use of 矇tale homotopy theory for obstructions to the existence of rational points on algebraic varieties. Finally, the article by Asok and ?stv疆r, based in part on the Nelder Fellow lecture series by ?stv疆r, gives a survey of the interplay between motivic homotopy theory and affine algebraic geometry, with a focus on contractible algebraic varieties. Now a major trend in arithmetic geometry, this volume offers a detailed guide to the fascinating circle of recent applications of homotopy theory to number theory. It will be invaluable to research students entering the field, as well as postdoctoral and more established researchers.
Geometric Continuum Mechanics
This contributed volume explores the applications of various topics in modern differential geometry to the foundations of continuum mechanics. In particular, the contributors use notions from areas such as global analysis, algebraic topology, and geometric measure theory. Chapter authors are experts in their respective areas, and provide important insights from the most recent research. Organized into two parts, the book first covers kinematics, forces, and stress theory, and then addresses defects, uniformity, and homogeneity. Specific topics covered include: Global stress and hyper-stress theoriesApplications of de Rham currents to singular dislocationsManifolds of mappings for continuum mechanicsKinematics of defects in solid crystalsGeometric Continuum Mechanics will appeal to graduate students and researchers in the fields of mechanics, physics, and engineering who seek a more rigorous mathematical understanding of the area. Mathematicians interested in applications of analysis and geometry will also find the topics covered here of interest.
Geometry, Mechanics, and Control in Action for the Falling Cat
The falling cat is an interesting theme to pursue, in which geometry, mechanics, and control are in action together. As is well known, cats can almost always land on their feet when tossed into the air in an upside-down attitude. If cats are not given a non-vanishing angular momentum at an initial instant, they cannot rotate during their motion, and the motion they can make in the air is vibration only. However, cats accomplish a half turn without rotation when landing on their feet. In order to solve this apparent mystery, one needs to thoroughly understand rotations and vibrations. The connection theory in differential geometry can provide rigorous definitions of rotation and vibration for many-body systems. Deformable bodies of cats are not easy to treat mechanically. A feasible way to approach the question of the falling cat is to start with many-body systems and then proceed to rigid bodies and, further, to jointed rigid bodies, which can approximate the body of a cat. In this book, the connection theory is applied first to a many-body system to show that vibrational motions of the many-body system can result in rotations without performing rotational motions and then to the cat model consisting of jointed rigid bodies. On the basis of this geometric setting, mechanics of many-body systems and of jointed rigid bodies must be set up. In order to take into account the fact that cats can deform their bodies, three torque inputs which may give a twist to the cat model are applied as control inputs under the condition of the vanishing angular momentum. Then, a control is designed according to the port-controlled Hamiltonian method for the model cat to perform a half turn and to halt the motion upon landing. The book also gives a brief review of control systems through simple examples to explain the role of control inputs.
Fiber Bundles and Homotopy
This book is an introduction to fiber bundles and fibrations. But the ultimate goal is to make the reader feel comfortable with basic ideas in homotopy theory. The author found that the classification of principal fiber bundles is an ideal motivation for this purpose. The notion of homotopy appears naturally in the classification. Basic tools in homotopy theory such as homotopy groups and their long exact sequence need to be introduced. Furthermore, the notion of fibrations, which is one of three important classes of maps in homotopy theory, can be obtained by extracting the most essential properties of fiber bundles. The book begins with elementary examples and then gradually introduces abstract definitions when necessary. The reader is assumed to be familiar with point-set topology, but it is the only requirement for this book.
The Calabi-Yau Landscape
Can artificial intelligence learn mathematics? The question is at the heart of this original monograph bringing together theoretical physics, modern geometry, and data science. The study of Calabi-Yau manifolds lies at an exciting intersection between physics and mathematics. Recently, there has been much activity in applying machine learning to solve otherwise intractable problems, to conjecture new formulae, or to understand the underlying structure of mathematics. In this book, insights from string and quantum field theory are combined with powerful techniques from complex and algebraic geometry, then translated into algorithms with the ultimate aim of deriving new information about Calabi-Yau manifolds. While the motivation comes from mathematical physics, the techniques are purely mathematical and the theme is that of explicit calculations. The reader is guided through the theory and provided with explicit computer code in standard software such as SageMath, Python and Mathematica to gain hands-on experience in applications of artificial intelligence to geometry. Driven by data and written in an informal style, The Calabi-Yau Landscape makes cutting-edge topics in mathematical physics, geometry and machine learning readily accessible to graduate students and beyond. The overriding ambition is to introduce some modern mathematics to the physicist, some modern physics to the mathematician, and machine learning to both.
Polygonal Approximation and Scale-Space Analysis of Closed Digital Curves
This book covers the most important topics in the area of pattern recognition, object recognition, computer vision, robot vision, medical computing, computational geometry, and bioinformatics systems. Students and researchers will find a comprehensive treatment of polygonal approximation and its real life applications. The book not only explains the theoretical aspects but also presents applications with detailed design parameters. The systematic development of the concept of polygonal approximation of digital curves and its scale-space analysis are useful and attractive to scholars in many fields.
Differential Geometry
Differential Geometry offers a concise introduction to some basic notions of modern differential geometry and their applications to solid mechanics and physics.Concepts such as manifolds, groups, fibre bundles and groupoids are first introduced within a purely topological framework. They are shown to be relevant to the description of space-time, configuration spaces of mechanical systems, symmetries in general, microstructure and local and distant symmetries of the constitutive response of continuous media.Once these ideas have been grasped at the topological level, the differential structure needed for the description of physical fields is introduced in terms of differentiable manifolds and principal frame bundles. These mathematical concepts are then illustrated with examples from continuum kinematics, Lagrangian and Hamiltonian mechanics, Cauchy fluxes and dislocation theory.This book will be useful for researchers and graduate students in science and engineering.
2018 Matrix Annals
MATRIX is Australia's international and residential mathematical research institute. It facilitates new collaborations and mathematical advances through intensive residential research programs, each 1-4 weeks in duration. This book is a scientific record of the eight programs held at MATRIX in 2018: - Non-Equilibrium Systems and Special Functions- Algebraic Geometry, Approximation and Optimisation- On the Frontiers of High Dimensional Computation- Month of Mathematical Biology- Dynamics, Foliations, and Geometry In Dimension 3- Recent Trends on Nonlinear PDEs of Elliptic and Parabolic Type- Functional Data Analysis and Beyond- Geometric and Categorical Representation Theory The articles are grouped into peer-reviewed contributions and other contributions. The peer-reviewed articles present original results or reviews on a topic related to the MATRIX program; the remaining contributions are predominantly lecture notes or short articles based on talks or activities at MATRIX.
Buildings and Schubert Schemes
The first part of this book introduces the Schubert Cells and varieties of the general linear group Gl (k^(r+1)) over a field k according to Ehresmann geometric way. Smooth resolutions for these varieties are constructed in terms of Flag Configurations in k^(r+1) given by linear graphs called Minimal Galleries. In the second part, Schubert Schem
Geometric Flows on Planar Lattices
This book introduces the reader to important concepts in modern applied analysis, such as homogenization, gradient flows on metric spaces, geometric evolution, Gamma-convergence tools, applications of geometric measure theory, properties of interfacial energies, etc. This is done by tackling a prototypical problem of interfacial evolution in heterogeneous media, where these concepts are introduced and elaborated in a natural and constructive way. At the same time, the analysis introduces open issues of a general and fundamental nature, at the core of important applications. The focus on two-dimensional lattices as a prototype of heterogeneous media allows visual descriptions of concepts and methods through a large amount of illustrations.
Test Configurations, Stabilities and Canonical K瓣hler Metrics
The Yau-Tian-Donaldson conjecture for anti-canonical polarization was recently solved affirmatively by Chen-Donaldson-Sun and Tian. However, this conjecture is still open for general polarizations or more generally in extremal K瓣hler cases. In this book, the unsolved cases of the conjecture will be discussed.It will be shown that the problem is closely related to the geometry of moduli spaces of test configurations for polarized algebraic manifolds. Another important tool in our approach is the Chow norm introduced by Zhang. This is closely related to Ding's functional, and plays a crucial role in our differential geometric study of stability. By discussing the Chow norm from various points of view, we shall make a systematic study of the existence problem of extremal K瓣hler metrics.
Geometric Invariant Theory, Holomorphic Vector Bundles and the Harder-Narasimhan Filtration
This book introduces key topics on Geometric Invariant Theory, a technique to obtaining quotients in algebraic geometry with a good set of properties, through various examples. It starts from the classical Hilbert classification of binary forms, advancing to the construction of the moduli space of semistable holomorphic vector bundles, and to Hitchin's theory on Higgs bundles. The relationship between the notion of stability between algebraic, differential and symplectic geometry settings is also covered.Unstable objects in moduli problems -- a result of the construction of moduli spaces -- get specific attention in this work. The notion of the Harder-Narasimhan filtration as a tool to handle them, and its relationship with GIT quotients, provide instigating new calculations in several problems. Applications include a survey of research results on correspondences between Harder-Narasimhan filtrations with the GIT picture and stratifications of the moduli space of Higgs bundles.Graduate students and researchers who want to approach Geometric Invariant Theory in moduli constructions can greatly benefit from this reading, whose key prerequisites are general courses on algebraic geometry and differential geometry.
Quantitative Tamarkin Theory
This textbook offers readers a self-contained introduction to quantitative Tamarkin category theory. Functioning as a viable alternative to the standard algebraic analysis method, the categorical approach explored in this book makes microlocal sheaf theory accessible to a wide audience of readers interested in symplectic geometry. Much of this material has, until now, been scattered throughout the existing literature; this text finally collects that information into one convenient volume. After providing an overview of symplectic geometry, ranging from its background to modern developments, the author reviews the preliminaries with precision. This refresher ensures readers are prepared for the thorough exploration of the Tamarkin category that follows. A variety of applications appear throughout, such as sheaf quantization, sheaf interleaving distance, and sheaf barcodes from projectors. An appendix offers additional perspectives by highlighting further useful topics. Quantitative Tamarkin Theory is ideal for graduate students interested in symplectic geometry who seek an accessible alternative to the algebraic analysis method. A background in algebra and differential geometry is recommended.This book is part of the "Virtual Series on Symplectic Geometry"http: //www.springer.com/series/16019
Axonometrie, Perspektive, Beleuchtung
Keine ausf羹hrliche Beschreibung f羹r "Axonometrie, Perspektive, Beleuchtung" verf羹gbar.
Lectures on Nonsmooth Differential Geometry
This book provides an introduction to some aspects of the flourishing field of nonsmooth geometric analysis. In particular, a quite detailed account of the first-order structure of general metric measure spaces is presented, and the reader is introduced to the second-order calculus on spaces - known as RCD spaces - satisfying a synthetic lower Ricci curvature bound. Examples of the main topics covered include notions of Sobolev space on abstract metric measure spaces; normed modules, which constitute a convenient technical tool for the introduction of a robust differential structure in the nonsmooth setting; first-order differential operators and the corresponding functional spaces; the theory of heat flow and its regularizing properties, within the general framework of "infinitesimally Hilbertian" metric measure spaces; the RCD condition and its effects on the behavior of heat flow; and second-order calculus on RCD spaces. The book is mainly intended for young researchers seeking acomprehensive and fairly self-contained introduction to this active research field. The only prerequisites are a basic knowledge of functional analysis, measure theory, and Riemannian geometry.
Manifolds and Local Structures
Local structures, like differentiable manifolds, fibre bundles, vector bundles and foliations, can be obtained by gluing together a family of suitable 'elementary spaces', by means of partial homeomorphisms that fix the gluing conditions and form a sort of 'intrinsic atlas', instead of the more usual system of charts living in an external framework.An 'intrinsic manifold' is defined here as such an atlas, in a suitable category of elementary spaces: open euclidean spaces, or trivial bundles, or trivial vector bundles, and so on.This uniform approach allows us to move from one basis to another: for instance, the elementary tangent bundle of an open Euclidean space is automatically extended to the tangent bundle of any differentiable manifold. The same holds for tensor calculus.Technically, the goal of this book is to treat these structures as 'symmetric enriched categories' over a suitable basis, generally an ordered category of partial mappings.This approach to gluing structures is related to Ehresmann's one, based on inductive pseudogroups and inductive categories. A second source was the theory of enriched categories and Lawvere's unusual view of interesting mathematical structures as categories enriched over a suitable basis.
Analysis and Beyond: An Introduction with Examples and Exercises
This volume aims to bridge between elementary textbooks on calculus and established books on advanced analysis. It provides elucidation of the reversible process of differentiation and integration through two featured principles: the chain rule and its inverse - the change of variable - as well as the Leibniz rule and its inverse - the integration by parts. The chain rule or differentiation of composite functions is ubiquitous since almost all (a.a.) functions are composite functions of (elementary) functions and with the change of variable method as its reverse process. The Leibniz rule or differentiation of the product of two functions is essential since it makes differentiation nonlinear and with the method of integration by parts as its reverse process.Readers will find numerous worked-out examples and exercises in this volume. Detailed solutions are provided for most of the common exercises so that readers remain enthusiastically motivated in solving and understanding the concepts better.The intention of this volume is to lead the reader into the rich fields of advanced analysis and to obtain a much better view of useful mathematics.
Klassifizieren, Sch瓣tzen, Zeichnen und Messen verschiedener Winkel. Eine Lerntheke zur F繹rderung selbstregulierten und selbstst瓣ndigen Arbeitens in einer 5. Klasse
Examensarbeit aus dem Jahr 2020 im Fachbereich Didaktik - Mathematik, Note: 1,75, Veranstaltung: Hausarbeit im Rahmen des Referendariats, Sprache: Deutsch, Abstract: Diese Arbeit stellt eine Lerntheke zum Thema Winkel f羹r eine f羹nfte Klasse vor. Die Sch羹lerinnen und Sch羹ler sollen ihre Kompetenz im Sch瓣tzen, Benennen und Messen eines Winkels vertiefen. Dabei sollen sie ihren richtigen Umgang mit dem Geodreieck weiter vertiefen. Indem die Sch羹lerinnen und Sch羹ler selbstst瓣ndig verschiedene Stationen bearbeiten und hierzu eine R羹ckmeldung anfertigen, k繹nnen sie ihre Methodenkompetenz st瓣rken. Die Fachanforderungen f羹r das Fach Mathematik sehen f羹r die f羹nfte und sechste Klassenstufe das Thema "Kreis und Winkel" vor. Eine genaue Zuordnung erfolgt in den Bereich der Leitidee Messen und der Leitidee Raum und Form. Hierzu sind bestimmte Kenntnisse sowie fachliche Anforderungen der Messung, Sch瓣tzung und Zeichnung der Winkel und der Gr繹?en an die SuS zu beachten. Die Einf羹hrung des Winkelbegriffes kann 羹ber den statischen und dynamischen Winkelbegriff erfolgen, der sachgerechte Umgang mit dem Geodreieck zum Messen und Zeichnen der Objekte ist in diesem Bereich ebenso immanent. Eine weitere Einordnung erfolgt im Bereich der Leitidee Raum und Form.
Introduction to Lipschitz Geometry of Singularities
This book presents a broad overview of the important recent progress which led to the emergence of new ideas in Lipschitz geometry and singularities, and started to build bridges to several major areas of singularity theory. Providing all the necessary background in a series of introductory lectures, it also contains Pham and Teissier's previously unpublished pioneering work on the Lipschitz classification of germs of plane complex algebraic curves. While a real or complex algebraic variety is topologically locally conical, it is in general not metrically conical; there are parts of its link with non-trivial topology which shrink faster than linearly when approaching the special point. The essence of the Lipschitz geometry of singularities is captured by the problem of building classifications of the germs up to local bi-Lipschitz homeomorphism. The Lipschitz geometry of a singular space germ is then its equivalence class in this category. The book is aimedat graduate students and researchers from other fields of geometry who are interested in studying the multiple open questions offered by this new subject.
Constrained Graph Layouts
Constraining graph layouts - that is, restricting the placement of vertices and the routing of edges to obey certain constraints - is common practice in graph drawing. In this book, we discuss algorithmic results on two different restriction types: placing vertices on the outer face and on the integer grid. For the first type, we look into the outer k-planar and outer k-quasi-planar graphs, as well as giving a linear-time algorithm to recognize full and closed outer k-planar graphs Monadic Second-order Logic. For the second type, we consider the problem of transferring a given planar drawing onto the integer grid while perserving the original drawings topology; we also generalize a variant of Cauchy's rigidity theorem for orthogonal polyhedra of genus 0 to those of arbitrary genus.
The Wonder Book of Geometry
How can we be sure that Pythagoras's theorem is really true? Why is the 'angle in a semicircle' always 90 degrees? And how can tangents help determine the speed of a bullet? David Acheson takes the reader on a highly illustrated tour through the history of geometry, from ancient Greece to the present day. He emphasizes throughout elegant deduction and practical applications, and argues that geometry can offer the quickest route to the whole spirit of mathematics at its best. Along the way, we encounter the quirky and the unexpected, meet the great personalities involved, and uncover some of the loveliest surprises in mathematics.
Vedic Mathematics: A Mathematics Tale from the Ancient Veda to Modern Times
This is a book about Mathematics but not a book of Mathematics. It is an attempt, between the serious and facetious, of conveying the idea that a mathematical thought is the result of different experiences, geographical and social factors. Even though it is not clear when Mathematics had started, it is evident that it had been used at an early stage of human history and by ancient Babylonians and Egyptians who have already developed a sophisticated corpus of mathematical items, which were the workhorse tools in engineering, navigation, trades and astronomy. The book sweeps across the mathematical minds of the Greek and Arab traditions, concepts by Assyro-Babylonians, and ancient Indian Vedic culture. The mathematical mind has modeled the evolution of societies and has been modeled by it. It is now in the midst of a great revolution and it is not clear where it will bring us. The current new epoch needs new mathematical tools and, above this, a new way of looking at Mathematics. This book tells the tale of what went on and what might go on.
Rules and Examples of Perspective Proper for Painters and Architects
Rules and Examples of Perspective Proper for Painters and Architects - in English and Latin: containing a most easie and expeditious method to delineate in perspective all designs relating to architecture is an unchanged, high-quality reprint of the original edition of 1693. Hansebooks is editor of the literature on different topic areas such as research and science, travel and expeditions, cooking and nutrition, medicine, and other genres. As a publisher we focus on the preservation of historical literature. Many works of historical writers and scientists are available today as antiques only. Hansebooks newly publishes these books and contributes to the preservation of literature which has become rare and historical knowledge for the future.
In the Tradition of Thurston
This book consists of 16 surveys on Thurston's work and its later development. The authors are mathematicians who were strongly influenced by Thurston's publications and ideas. The subjects discussed include, among others, knot theory, the topology of 3-manifolds, circle packings, complex projective structures, hyperbolic geometry, Kleinian groups, foliations, mapping class groups, Teichm羹ller theory, anti-de Sitter geometry, and co-Minkowski geometry. The book is addressed to researchers and students who want to learn about Thurston's wide-ranging mathematical ideas and their impact. At the same time, it is a tribute to Thurston, one of the greatest geometers of all time, whose work extended over many fields in mathematics and who had a unique way of perceiving forms and patterns, and of communicating and writing mathematics.
Knot Theory
Over the last fifteen years, the face of knot theory has changed due to various new theories and invariants coming from physics, topology, combinatorics and algebra. It suffices to mention the great progress in knot homology theory (Khovanov homology and Ozsvath-Szabo Heegaard-Floer homology), the A-polynomial which give rise to strong invariant
Einf羹hrung in Die Algebraische Geometrie
Die algebraische Geometrie ist eines der gro?en aktuellen Forschungsgebiete der Mathematik und hat sich in verschiedene Richtungen und in die Anwendungen hinein verzweigt. Ihre grundlegenden Ideen sind aber bereits im Anschluss an die Algebra-Vorlesung gut zug瓣nglich und stellen f羹r viele weitere Vertiefungsrichtungen eine Bereicherung dar. Diese Einf羹hrung baut deshalb auf der Algebra auf und richtet sich an Bachelor- und Master-Studierende etwa ab dem f羹nften Semester. Die geometrischen Begriffe werden erst nah an der Algebra eingef羹hrt - illustriert durch viele Beispiele. Anschlie?end werden sie auf die projektive Geometrie 羹bertragen und weiterentwickelt. Auch weiterf羹hrende Konzepte aus der kommutativen Algebra und die Grundlagen der Computer-Algebra kommen dabei zum Tragen, ohne die technischen Anforderungen zu hoch zu schrauben.Der AutorDaniel Plaumann ist seit 2016 Professor f羹r Algebra und ihre Anwendungen an der TU Dortmund. Sein Forschungsgebiet ist die reelle algebraische Geometrie.
Metacyclic Groups and the D(2) Problem
The D(2) problem is a fundamental problem in low dimensional topology. In broad terms, it asks when a three-dimensional space can be continuously deformed into a two-dimensional space without changing the essential algebraic properties of the spaces involved.The problem is parametrized by the fundamental group of the spaces involved; that is, each group G has its own D(2) problem whose difficulty varies considerably with the individual nature of G.This book solves the D(2) problem for a large, possibly infinite, number of finite metacyclic groups G(p, q). Prior to this the author had solved the D(2) problem for the groups G(p, 2). However, for q > 2, the only previously known solutions were for the groups G(7, 3), G(5, 4) and G(7, 6), all done by difficult direct calculation by two of the author's students, Jonathan Remez (2011) and Jason Vittis (2019).The method employed is heavily algebraic and involves precise analysis of the integral representation theory of G(p, q). Some noteworthy features are a new cancellation theory of modules (Chapters 10 and 11) and a simplified treatment (Chapters 5 and 12) of the author's theory of Swan homomorphisms.
Greek Geometry From Thales To Euclid
Where did proof first take shape? A landmark study of geometry. George Johnston Allman's Greek Geometry From Thales to Euclid maps the emergence of rigorous demonstration in ancient Greece, guiding readers from the problem-solving practices of pre-euclidean geometry to the sweeping order of Euclid's system. Drawing on the works of Thales and Euclid, Allman reconstructs the origins of geometric proofs and situates them within the broader history of Greek mathematics. Equal parts classical geometry collection and ancient mathematics anthology, the book balances clear exposition with historical insight: mathematical thought in antiquity is treated as both intellectual history and technical achievement. Its lucid account of the foundations of Euclidean geometry explains how simple constructions, definitions and axioms coalesced into a durable logical framework. Valued as a textbook for math students and used as a steady reference for educators, the volume also serves as a companion to Greek mathematicians for anyone exploring ancient Greek mathematics. Far from dry antiquarianism, Allman's prose captures the tension between practical measurement and abstract deduction that shaped the Western geometric tradition, making a neglected past audible and relevant to modern readers. First published in the nineteenth century, Allman's study helped to shape later surveys of the history of Greek mathematics through its orderly narration and judicious scholarship. Its restraint and clarity make it an engaging bridge between specialist history and popular explanation. Republished by Alpha Editions in a careful modern edition, this volume preserves the spirit of the original while making it effortless to enjoy today - a heritage title prepared for readers and collectors alike. A welcome choice for casual readers and classic-literature collectors, it sits equally well beside novels and scholarly shelves. Its lucid explanations and historical perspective make it ideal as a companion for students and a dependable reference for educators, while collectors will prize the intellectual precision and clear prose that mark a cultural treasure.
Global Nonlinear Stability of Schwarzschild Spacetime Under Polarized Perturbations
Essential mathematical insights into one of the most important and challenging open problems in general relativity-the stability of black holes One of the major outstanding questions about black holes is whether they remain stable when subject to small perturbations. An affirmative answer to this question would provide strong theoretical support for the physical reality of black holes. In this book, Sergiu Klainerman and J矇r矇mie Szeftel take a first important step toward solving the fundamental black hole stability problem in general relativity by establishing the stability of nonrotating black holes-or Schwarzschild spacetimes-under so-called polarized perturbations. This restriction ensures that the final state of evolution is itself a Schwarzschild space. Building on the remarkable advances made in the past fifteen years in establishing quantitative linear stability, Klainerman and Szeftel introduce a series of new ideas to deal with the strongly nonlinear, covariant features of the Einstein equations. Most preeminent among them is the general covariant modulation (GCM) procedure that allows them to determine the center of mass frame and the mass of the final black hole state. Essential reading for mathematicians and physicists alike, this book introduces a rich theoretical framework relevant to situations such as the full setting of the Kerr stability conjecture.
An Introduction to Mathematical Relativity
This concise textbook introduces the reader to advanced mathematical aspects of general relativity, covering topics like Penrose diagrams, causality theory, singularity theorems, the Cauchy problem for the Einstein equations, the positive mass theorem, and the laws of black hole thermodynamics. It emerged from lecture notes originally conceived for a one-semester course in Mathematical Relativity which has been taught at the Instituto Superior T矇cnico (University of Lisbon, Portugal) since 2010 to Masters and Doctorate students in Mathematics and Physics. Mostly self-contained, and mathematically rigorous, this book can be appealing to graduate students in Mathematics or Physics seeking specialization in general relativity, geometry or partial differential equations. Prerequisites include proficiency in differential geometry and the basic principles of relativity. Readers who are familiar with special relativity and have taken a course either inRiemannian geometry (for students of Mathematics) or in general relativity (for those in Physics) can benefit from this book.
3d-Fraktale
Fraktale Geometrie und Chaostheorie gelten seit den 1970er Jahren als zentrale Forschungsgebiete der Mathematik und haben bereits zahlreiche Anwendungsbereiche in Naturwissenschaften und Technik gefunden. Die zumeist auf relativ einfachen mathematischen Gesetzm瓣?igkeiten beruhenden fraktalen Gebilde lassen sich sowohl in der zweidimensionalen Ebene als auch im dreidimensionalen Raum entwickeln, wodurch sie f羹r die stereoskopische Visualisierung zu interessanten Untersuchungsobjekten geraten. Das vorliegende Buch gibt einen kurzen ?berblick 羹ber das allgemeine Wesen der Fraktalgeometrie und lenkt sein Augenmerk in weiterer Folge auf die 3D-Darstellung fraktaler Strukturen aller Art, wobei hier zahlreiche Bildbeispiele zur Pr瓣sentation gelangen.
Arithmetic Geometry of Logarithmic Pairs and Hyperbolicity of Moduli Spaces
This textbook introduces exciting new developments and cutting-edge results on the theme of hyperbolicity. Written by leading experts in their respective fields, the chapters stem from mini-courses given alongside three workshops that took place in Montr矇al between 2018 and 2019. Each chapter is self-contained, including an overview of preliminaries for each respective topic. This approach captures the spirit of the original lectures, which prepared graduate students and those new to the field for the technical talks in the program. The four chapters turn the spotlight on the following pivotal themes: The basic notions of o-minimal geometry, which build to the proof of the Ax-Schanuel conjecture for variations of Hodge structures;A broad introduction to the theory of orbifold pairs and Campana's conjectures, with a special emphasis on the arithmetic perspective;A systematic presentation and comparison between different notions of hyperbolicity, as an introduction to the Lang-Vojta conjectures in the projective case;An exploration of hyperbolicity and the Lang-Vojta conjectures in the general case of quasi-projective varieties.Arithmetic Geometry of Logarithmic Pairs and Hyperbolicity of Moduli Spaces is an ideal resource for graduate students and researchers in number theory, complex algebraic geometry, and arithmetic geometry. A basic course in algebraic geometry is assumed, along with some familiarity with the vocabulary of algebraic number theory.
A Perspective on Canonical Riemannian Metrics
This book focuses on a selection of special topics, with emphasis on past and present research of the authors on "canonical" Riemannian metrics on smooth manifolds. On the backdrop of the fundamental contributions given by many experts in the field, the volume offers a self-contained view of the wide class of "Curvature Conditions" and "Critical Metrics" of suitable Riemannian functionals. The authors describe the classical examples and the relevant generalizations. This monograph is the winner of the 2020 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics.
Elementare Koordinatengeometrie
Dieses strukturell und didaktisch gut durchdachte Lehrbuch f羹r die Ausbildung von Lehrerinnen und Lehrern im Fach Mathematik m繹chte den Studierenden die klassische Geometrie, die in der Schule leider ein Schattendasein fristet, unter einem etwas ver瓣nderten, neuartigen Blickwinkel nahe bringen. Von besonderem Reiz ist in diesem Zusammenhang die alte, urspr羹ngliche Descartes'sche Idee einer algebraischen L繹sung geometrischer Probleme - ohne dabei die Geometrie durch den Formalismus der linearen Algebra und durch das Jonglieren mit Matrizen zu verdecken. F羹r die rechnerische L繹sung der geometrischen Probleme sind nur einfache algebraische Verfahren n繹tig, wobei der gezielte Einsatz eines Computer-Algebra-Systems langwierige Berechnungen vermeidet und gleichzeitig die erworbenen Kenntnisse vertieft. Das Buch gibt hierf羹r eine praxis- und anwendungsbezogene Anleitung in den sinnvollen Gebrauch des Computer-Algebra-Systems Maxima. Eine vielf瓣ltige Aufgabensammlung rundet das Buch ab, L繹sungen findet man kostenlos auf der Internetseite des Verlages.
Geometry and Analysis of Metric Spaces Via Weighted Partitions
The aim of these lecture notes is to propose a systematic framework for geometry and analysis on metric spaces. The central notion is a partition (an iterated decomposition) of a compact metric space. Via a partition, a compact metric space is associated with an infinite graph whose boundary is the original space. Metrics and measures on the space are then studied from an integrated point of view as weights of the partition. In the course of the text: It is shown that a weight corresponds to a metric if and only if the associated weighted graph is Gromov hyperbolic.Various relations between metrics and measures such as bilipschitz equivalence, quasisymmetry, Ahlfors regularity, and the volume doubling property are translated to relations between weights. In particular, it is shown that the volume doubling property between a metric and a measure corresponds to a quasisymmetry between two metrics in the language of weights.The Ahlfors regular conformal dimension of a compact metric space is characterized as the critical index of p-energies associated with the partition and the weight function corresponding to the metric. These notes should interest researchers and PhD students working in conformal geometry, analysis on metric spaces, and related areas.
The Theory of Quantum Torus Knots
The mathematical building block presented in the four-volume set is called the theory of quantum torus knots (QTK), a theory that is anchored in the principles of differential geometry and 2D Riemannian manifolds for 3D curved surfaces. The reader is given a mathematical setting from which they will be able to witness the derivations, solutions, and interrelationships between theories and equations taken from classical and modern physics. Included are the equations of Ginzburg-Landau, Gross-Pitaevskii, Kortewig-de Vries, Landau-Lifshitz, nonlinear Schr繹dinger, Schr繹dinger-Ginzburg-Landau, Maxwell, Navier-Stokes, and Sine-Gordon. They are applied to the fields of aerodynamics, electromagnetics, hydrodynamics, quantum mechanics, and superfluidity. These will be utilized to elucidate discussions and examples involving longitudinal and transverse waves, convected waves, solitons, special relativity, torus knots, and vortices.
An Introduction to Tensor Analysis
The subject of Tensor Analysis deals with the problem of the formulation of the relation between various entities in forms which remain invariant when we pass from one system of coordinates to another. The invariant form of equation is necessarily related to the possible system of coordinates with reference to which the equation remains invariant. The primary purpose of this book is the study of the invariance form of equation relative to the totally of the rectangular co-ordinate system in the three-dimensional Euclidean space. We start with the consideration of the way the sets representing various entities are transformed when we pass from one system of rectangular co-ordinates to another. A Tensor may be a physical entity that can be described as a Tensor only with respect to the manner of its representation by means of multi-sux sets associated with different system of axes such that the sets associated with different system of co-ordinate obey the transformation law for Tensor. We have employed sux notation for tensors of any order, we could also employ single letter such A, B to denote Tensors.
Planar Maps, Random Walks and Circle Packing
This open access book focuses on the interplay between random walks on planar maps and Koebe's circle packing theorem. Further topics covered include electric networks, the He-Schramm theorem on infinite circle packings, uniform spanning trees of planar maps, local limits of finite planar maps and the almost sure recurrence of simple random walks on these limits. One of its main goals is to present a self-contained proof that the uniform infinite planar triangulation (UIPT) is almost surely recurrent. Full proofs of all statements are provided. A planar map is a graph that can be drawn in the plane without crossing edges, together with a specification of the cyclic ordering of the edges incident to each vertex. One widely applicable method of drawing planar graphs is given by Koebe's circle packing theorem (1936). Various geometric properties of these drawings, such as existence of accumulation points and bounds on the radii, encode important probabilistic information, such as the recurrence/transience of simple random walks and connectivity of the uniform spanning forest. This deep connection is especially fruitful to the study of random planar maps. The book is aimed at researchers and graduate students in mathematics and is suitable for a single-semester course; only a basic knowledge of graduate level probability theory is assumed. This work was published by Saint Philip Street Press pursuant to a Creative Commons license permitting commercial use. All rights not granted by the work's license are retained by the author or authors.
Planar Maps, Random Walks and Circle Packing
This open access book focuses on the interplay between random walks on planar maps and Koebe's circle packing theorem. Further topics covered include electric networks, the He-Schramm theorem on infinite circle packings, uniform spanning trees of planar maps, local limits of finite planar maps and the almost sure recurrence of simple random walks on these limits. One of its main goals is to present a self-contained proof that the uniform infinite planar triangulation (UIPT) is almost surely recurrent. Full proofs of all statements are provided. A planar map is a graph that can be drawn in the plane without crossing edges, together with a specification of the cyclic ordering of the edges incident to each vertex. One widely applicable method of drawing planar graphs is given by Koebe's circle packing theorem (1936). Various geometric properties of these drawings, such as existence of accumulation points and bounds on the radii, encode important probabilistic information, such as the recurrence/transience of simple random walks and connectivity of the uniform spanning forest. This deep connection is especially fruitful to the study of random planar maps. The book is aimed at researchers and graduate students in mathematics and is suitable for a single-semester course; only a basic knowledge of graduate level probability theory is assumed. This work was published by Saint Philip Street Press pursuant to a Creative Commons license permitting commercial use. All rights not granted by the work's license are retained by the author or authors.
Mathematical problems on the first and second divisions of the schedule of subjects for the Cambridge mathematical tripos examination Devised and Arranged
A gateway to the intellectual rigour of Victorian Cambridge, Joseph Wolstenholme's *Mathematical Problems On The First And Second Divisions Of The Schedule Of Subjects For The Cambridge Mathematical Tripos Examination Devised And Arranged* stands as both a formidable challenge and a scholarly touchstone. Each page brims with advanced mathematics problems, meticulously curated for the legendary Cambridge Mathematical Tripos-an examination that shaped generations of mathematical talent and defined British academic traditions. This volume is more than a mathematical examination guide; it is a window into the problem-solving techniques that propelled the brightest minds of nineteenth-century mathematics. Wolstenholme's careful arrangement offers clarity and progression, making it an invaluable mathematics students reference for competitive exam preparation or for anyone wishing to experience the intellectual atmosphere of Cambridge University mathematics at its historical height. The problems, devised during a period when mathematical rigour and creativity flourished, continue to inspire and challenge, inviting readers to test their skill and deepen their understanding. Republished by Alpha Editions in a careful modern edition, this volume preserves the spirit of the original while making it effortless to enjoy today - a heritage title prepared for readers and collectors alike. Whether you are a dedicated mathematician, a lover of historical mathematics collections, or a collector seeking a cultural treasure from the era of Joseph Wolstenholme's works, this book offers a rare glimpse into the foundations of competitive mathematics and the enduring legacy of Victorian era mathematics.
Mathematical problems on the first and second divisions of the schedule of subjects for the Cambridge mathematical tripos examination Devised and Arranged
Step into the intellectual crucible of 19th century academia, where the Cambridge Mathematical Tripos shaped generations of mathematical minds. Joseph Wolstenholme's Mathematical Problems on the First and Second Divisions of the Schedule of Subjects for the Cambridge Mathematical Tripos Examination offers a rare window into the rigour and creativity of Victorian era mathematics. Each problem is a challenge drawn from the very heart of higher education mathematics, meticulously devised for those preparing for the most demanding competitive exams of the age. This classic mathematical problem collection stands as both a formidable exam preparation guide and a fascinating resource for math students and enthusiasts today. Through advanced math exercises and historical math challenges, Wolstenholme's work invites readers to test their skill and deepen their understanding, echoing the intensity that once defined Cambridge's legendary examinations. The carefully arranged problems reflect not only the academic standards of their time but also the enduring appeal of classic math problems that continue to inspire and instruct. Republished by Alpha Editions in a careful modern edition, this volume preserves the spirit of the original while making it effortless to enjoy today - a heritage title prepared for readers and collectors alike. Whether you are a student seeking authentic practice, a lover of mathematical history, or a collector of Joseph Wolstenholme works, this book is more than a study aid; it is a cultural artefact from the golden age of mathematical thought.
Kuranishi Structures and Virtual Fundamental Chains
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.
Theory of Algebraic Surfaces
This is an English translation of the book in Japanese, published as the volume 20 in the series of Seminar Notes from The University of Tokyo that grew out of a course of lectures by Professor Kunihiko Kodaira in 1967. It serves as an almost self-contained introduction to the theory of complex algebraic surfaces, including concise proofs of Gorenstein's theorem for curves on a surface and Noether's formula for the arithmetic genus. It also discusses the behavior of the pluri-canonical maps of surfaces of general type as a practical application of the general theory. The book is aimed at graduate students and also at anyone interested in algebraic surfaces, and readers are expected to have only a basic knowledge of complex manifolds as a prerequisite.
Introduction to Algebraic Geometry
The goal of this book is to provide an introduction to algebraic geometry accessible to students. Starting from solutions of polynomial equations, modern tools of the subject soon appear, motivated by how they improve our understanding of geometrical concepts. In many places, analogies and differences with related mathematical areas are explained. The text approaches foundations of algebraic geometry in a complete and self-contained way, also covering the underlying algebra. The last two chapters include a comprehensive treatment of cohomology and discuss some of its applications in algebraic geometry.
Differential Calculus
The textbook "DIFFERENTIAL CALCULUS. A Mathematical Analysis for Applied Sciences" is dedicated to all university students in distance or face-to-face programs that require learning Differential Calculus in order to use its contents in Applied Sciences. The relevant contributions of this textbook focus on Pedagogy for the Development of Autonomous Learning together with the logical and complete selection of topics. These topics offer the student the necessary knowledge to achieve the conceptual and applied management of Differential Calculus with command in applied sciences, without including demonstrative processes or theoretical constructions typical of pure mathematics. The methodological design offers the student alternatives in learning to learn and learning autonomously in a logical-analytical-constructive and application process of concepts. The aim of the textbook "DIFFERENTIAL CALCULUS. A Mathematical Analysis for Applied Sciences" is to make the university student learn for himself the fundamental theory of differential calculus as part of a teaching-learning process. This process mainly includes theoretical management and conceptual application oriented towards the socioeconomic and administrative sciences.
Algebraic Geometry I: Schemes
This book introduces the reader to modern algebraic geometry. It presents Grothendieck's technically demanding language of schemes that is the basis of the most important developments in the last fifty years within this area. A systematic treatment and motivation of the theory is emphasized, using concrete examples to illustrate its usefulness. Several examples from the realm of Hilbert modular surfaces and of determinantal varieties are used methodically to discuss the covered techniques. Thus the reader experiences that the further development of the theory yields an ever better understanding of these fascinating objects. The text is complemented by many exercises that serve to check the comprehension of the text, treat further examples, or give an outlook on further results. The volume at hand is an introduction to schemes. To get started, it requires only basic knowledge in abstract algebra and topology. Essential facts from commutative algebra are assembled in an appendix. It will be complemented by a second volume on the cohomology of schemes. For the second edition, several mistakes and many smaller errors and misprints have been corrected.
Lectures on Convex Geometry
This book provides a self-contained introduction to convex geometry in Euclidean space. After covering the basic concepts and results, it develops Brunn-Minkowski theory, with an exposition of mixed volumes, the Brunn-Minkowski inequality, and some of its consequences, including the isoperimetric inequality. Further central topics are then treated, such as surface area measures, projection functions, zonoids, and geometric valuations. Finally, an introduction to integral-geometric formulas in Euclidean space is provided. The numerous exercises and the supplementary material at the end of each section form an essential part of the book. Convexity is an elementary and natural concept. It plays a key role in many mathematical fields, including functional analysis, optimization, probability theory, and stochastic geometry. Paving the way to the more advanced and specialized literature, the material will be accessible to students in the third year and can be covered in one semester.
