Dual Jet Geometrization for Time-Dependent Hamiltonians and Applications
This book studies a category of mathematical objects called Hamiltonians, which are dependent on both time and momenta. The authors address the development of the distinguished geometrization on dual 1-jet spaces for time-dependent Hamiltonians, in contrast with the time-independent variant on cotangent bundles. Two parts are presented to include both geometrical theory and the applicative models: Part One: Time-dependent Hamilton Geometry and Part Two: Applications to Dynamical Systems, Economy and Theoretical Physics. The authors present 1-jet spaces and their duals as appropriate fundamental ambient mathematical spaces used to model classical and quantum field theories. In addition, the authors present dual jet Hamilton geometry as a distinct metrical approach to various interdisciplinary problems.
Mathematical Principles of Natural Philosophy
"I have presented principles of philosophy that are not, however, philosophical but strictly mathematicalthat is, those on which the study of philosophy can be based."Translated from the classic text Philosophi疆 Naturalis Principia Mathematica, which was originally published in Latin in 1687. Such is its influence, the book is often referred to simply as the Principia.The text alongside its numerous, historically important diagrams formed the foundation of classical mechanics, Newton's law of universal gravitation and a derivation of Kepler's laws of planetary motion.The French mathematical physicist Alexis Clairaut summed up the influence of the book in 1747 by saying that Principia was: "The epoch of a great revolution in physics. The method followed by its illustrious author... spread the light of mathematics on a science which up to then had remained in the darkness of conjectures and hypotheses."
Parametric Geometry of Curves and Surfaces
This textbook provides a thorough introduction to the differential geometry of parametrized curves and surfaces, along with a wealth of applications to specific architectural elements. Geometric elements in architecture respond to practical, physical and aesthetic needs. Proper understanding of the mathematics underlying the geometry provides control over the construction. This book relates the classical mathematical theory of parametrized curves and surfaces to multiple applications in architecture. The presentation is mathematically complete with numerous figures and animations illustrating the theory, and special attention is given to some of the recent trends in the field. Solved exercises are provided to see the theory in practice.Intended as a textbook for lecture courses, Parametric Geometry of Curves and Surfaces is suitable for mathematically-inclined students in engineering, architecture and related fields, and can also serve as a textbook for traditional differential geometry courses to mathematics students. Researchers interested in the mathematics of architecture or computer-aided design will also value its combination of precise mathematics and architectural examples.
A Ludic Journey Into Geometric Topology
Preface.- Mathematical Models.- The Big Bang Theory of Ancient Greece.- Geometry: From disorder to order.- Topology.- Fourth dimension.- Non-orientable surfaces.- Hypersurfaces.
Decomposition of Jacobians by Prym Varieties
This monograph studies decompositions of the Jacobian of a smooth projective curve, induced by the action of a finite group, into a product of abelian subvarieties. The authors give a general theorem on how to decompose the Jacobian which works in many cases and apply it for several groups, as for groups of small order and some series of groups. In many cases, these components are given by Prym varieties of pairs of subcovers. As a consequence, new proofs are obtained for the classical bigonal and trigonal constructions which have the advantage to generalize to more general situations. Several isogenies between Prym varieties also result.
Convex Cones
This book provides the foundations for geometric applications of convex cones and presents selected examples from a wide range of topics, including polytope theory, stochastic geometry, and Brunn-Minkowski theory. Giving an introduction to convex cones, it describes their most important geometric functionals, such as conic intrinsic volumes and Grassmann angles, and develops general versions of the relevant formulas, namely the Steiner formula and kinematic formula. In recent years questions related to convex cones have arisen in applied mathematics, involving, for example, properties of random cones and their non-trivial intersections. The prerequisites for this work, such as integral geometric formulas and results on conic intrinsic volumes, were previously scattered throughout the literature, but no coherent presentation was available. The present book closes this gap. It includes several pearls from the theory of convex cones, which should be better known.
An Essay on the Foundations of Geometry
An Essay on the Foundations of Geometry was first published in 1897 and marks Bertrand Russell's first foray into analytic philosophy, a movement in which Russell is one of the founding members and figurehead. This Routledge Classics edition includes a new Foreword by Michael Potter.
Field Engineering
Field Engineering - A Handbook of the Theory and Practice of Railway Surveying, Location, and Construction is an unchanged, high-quality reprint of the original edition of 1882. 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.
Kontsevich's Deformation Quantization and Quantum Field Theory
- 1. Introduction. - 2. Foundations of Differential Geometry. - 3. Symplectic Geometry. - 4. Poisson Geometry. - 5. Deformation Quantization. - 6. Quantum Field Theoretic Approach to Deformation Quantization.
Sacred Geometry
Is there a secret visual language all around us? What's so special about the shape of the Great Pyramid? Why is there something so sexy about circles? How many ways can you tile the plane? Lavishly illustrated by the author, this enchanting small introduction to one of the oldest and most widely-used ancient traditions on Earth will forever change the way you look at a triangle, arch, window, fabric repeat, ceramic pattern, graphic design, painting, spiral, or flower.
Pentagons and Pentagrams
A fascinating exploration of the pentagon and its role in various cultures The pentagon and its close cousin, the pentagram, have inspired individuals for the last two and half millennia, from mathematicians and philosophers to artists and naturalists. Despite the pentagon's wide-ranging history, no single book has explored the important role of this shape in various cultures, until now. Richly illustrated, Pentagons and Pentagrams offers a sweeping view of the five-sided polygon, revealing its intriguing geometric properties and its essential influence on a variety of fields. Traversing time, Eli Maor narrates vivid stories, both celebrated and unknown, about the pentagon and pentagram. He discusses the early Pythagoreans, who ascribed to the pentagon mythical attributes, adopted it as their emblem, and figured out its construction with a straightedge and compass. Maor looks at how a San Diego housewife uncovered four previously unknown types of pentagonal tilings, and how in 1982 a scientist's discovery of fivefold symmetries in certain alloys caused an uproar in crystallography and led to a Nobel Prize. Maor also discusses the pentagon's impact on many buildings, from medieval fortresses to the Pentagon in Washington, D.C. Eugen Jost's superb illustrations provide sumptuous visual context, and the book's puzzles and mazes offer fun challenges for readers, with solutions given in an appendix.
Simultaneous Tracking and Shape Estimation of Extended Objects
This work is concerned with the simultaneous tracking and shape estimation of a mobile extended object based on noisy sensor measurements. Novel methods are developed for coping with the following two main challenges: i) The computational complexity due to the nonlinearity and high-dimensionality of the problem, and ii) the lack of statistical knowledge about possible measurement sources on the extended object.
Veech Groups and Translation Coverings
A translation surface is obtained by taking plane polygons and gluing their edges by translations. We ask which subgroups of the Veech group of a primitive translation surface can be realised via a translation covering. For many primitive surfaces we prove that partition stabilising congruence subgroups are the Veech group of a covering surface. We also address the coverings via their monodromy groups and present examples of cyclic coverings in short orbits, i.e. with large Veech groups.
Geometry and Discrete Mathematics
In the two-volume set 'A Selection of Highlights' we present basics of mathematics in an exciting and pedagogically sound way. This volume examines many fundamental results in Geometry and Discrete Mathematics along with their proofs and their history. In the second edition we include a new chapter on Topological Data Analysis and enhanced the chapter on Graph Theory for solving further classical problems such as the Traveling Salesman Problem.
Polynomials, Dynamics, and Choice
This book is organized in two parts, the first of which develops an account of polynomial symmetry that relies on considerations of algebra and geometry. The second explores beyond polynomials to spaces consisting of choices ranging from mundane decisions to evolutionary algorithms that search for optimal outcomes.
Plane and Solid Geometry Essentials
In one textbook, Plane and Solid Geometry Essentials presents all the foundational materials of a semester and a half of high school geometry! It incorporates the impressive teaching methods of Professors H. Fawcett and George Polya plus suggestions from the fifth and thirteenth yearbooks of the National Council of Teachers of Mathematics and the 1950's yearbook of Central Association of Science and Math Teachers (now School Science and Mathematics Association). This text prepares students with the Plane and Solid Geeometry needed for the for the first year calculus as well the trades and decision making. James Elander's career of high school and college teaching provides insight into what students and teachers need to make learning enjoyable, gratifying, and learn the skills of decision-making!
Plane and Solid Geometry Essentials
In one textbook, Plane and Solid Geometry Essentials presents all the foundational materials of a semester and a half of high school geometry! It incorporates the impressive teaching methods of Professors H. Fawcett and George Polya plus suggestions from the fifth and thirteenth yearbooks of the National Council of Teachers of Mathematics and the 1950's yearbook of Central Association of Science and Math Teachers (now School Science and Mathematics Association). This text prepares students with the Plane and Solid Geeometry needed for the for the first year calculus as well the trades and decision making. James Elander's career of high school and college teaching provides insight into what students and teachers need to make learning enjoyable, gratifying, and learn the skills of decision-making!
Classical and Discrete Differential Geometry
This book introduces differential geometry and cutting-edge findings from the discipline by incorporating both classical approaches and modern discrete differential geometry across all facets and applications, including graphics and imaging, physics and networks.
Pythagorean Theorem for Babies
The bestselling scientific series continues to expand! Fans of Chris Ferrie's Organic Chemistry for Babies, Rocket Science for Babies, and 8 Little Planets will love this introduction to the Pythagorean Theorem for babies and toddlers!It only takes a small spark to ignite a child's mind.Written by an expert, with mathematical information from an expert, this is the perfect book for enlightening the next generation of geniuses.From the #1 science author for kids comes this next installment in the bestselling Baby University series! Pythagorean Theorem for Babies gives babies (and grownups!) the answers to the common question: what is the Pythagorean Theorem and how can I prove it?With a tongue-in-cheek approach that adults will love, this installment of the Baby University board book series is the perfect way to introduce basic concepts to even the youngest mathematician. After all, it's never too early to start loving Math!If you're looking for the perfect math or science gift, or more Baby University books for your little one, look no further! Pythagorean Theorem for Babies offers fun early learning for your little mathematician!
Differential Geometry of Curves and Surfaces
The book explains the reasons for various definitions. The interactive applets offer motivation for definitions, allowing students to explore examples, and give a visual explanation of complicated theorems. More elementary exercises are added and some challenging problems are moved later in exercise sets to assure more graduated progress.
Multiplicative Differential Geometry
The author introduces the main conceptions for multiplicative surfaces: multiplicative first fundamental form, the main multiplicative rules for differentiations on multiplicative surfaces, and the main multiplicative regularity conditions for multiplicative surfaces. Many examples and problems are included.
Problems and Solutions in Mathematical Olympiad (Secondary 3)
The series is edited by the head coaches of China's IMO National Team. Each volume, catering to different grades, is contributed by the senior coaches of the IMO National Team. The Chinese edition has won the award of Top 50 Most Influential Educational Brands in China.The series is created in line with the mathematics cognition and intellectual development levels of the students in the corresponding grades. All hot mathematics topics of the competition are included in the volumes and are organized into chapters where concepts and methods are gradually introduced to equip the students with necessary knowledge until they can finally reach the competition level.In each chapter, well-designed problems including those collected from real competitions are provided so that the students can apply the skills and strategies they have learned to solve these problems. Detailed solutions are provided selectively. As a feature of the series, we also include some solutions generously offered by the members of Chinese national team and national training team.
Problems and Solutions in Mathematical Olympiad (Secondary 3)
The series is edited by the head coaches of China's IMO National Team. Each volume, catering to different grades, is contributed by the senior coaches of the IMO National Team. The Chinese edition has won the award of Top 50 Most Influential Educational Brands in China.The series is created in line with the mathematics cognition and intellectual development levels of the students in the corresponding grades. All hot mathematics topics of the competition are included in the volumes and are organized into chapters where concepts and methods are gradually introduced to equip the students with necessary knowledge until they can finally reach the competition level.In each chapter, well-designed problems including those collected from real competitions are provided so that the students can apply the skills and strategies they have learned to solve these problems. Detailed solutions are provided selectively. As a feature of the series, we also include some solutions generously offered by the members of Chinese national team and national training team.
Topics in Global Real Analytic Geometry
In the first two chapters we review the theory developped by Cartan, Whitney and Tognoli. Then Nullstellensatz is proved both for Stein algebras and for the algebra of real analytic functions on a C-analytic space. Here we find a relation between real Nullstellensatz and seventeenth Hilbert's problem for positive semidefinite analytic functions. Namely, a positive answer to Hilbert's problem implies a solution for the real Nullstellensatz more similar to the one for real polinomials. A chapter is devoted to the state of the art on this problem that is far from a complete answer. In the last chapter we deal with inequalities. We describe a class of semianalytic sets defined by countably many global real analytic functions that is stable under topological properties and under proper holomorphic maps between Stein spaces, that is, verifies a direct image theorem. A smaller class admits also a decomposition into irreducible components as it happens for semialgebraic sets. Duringthe redaction some proofs have been simplified with respect to the original ones.
Geometry and Its Applications
This unique textbook combines traditional geometry with current ideas to present a contemporary approach that is grounded in real-world applications. It balances introduces axiomatic, Euclidean geometry, non-Euclidean geometry, and transformational geometry. The text integrates applications and examples throughout and includes historical notes.
Developments in Lorentzian Geometry
This proceedings volume gathers selected, revised papers presented at the X International Meeting on Lorentzian Geometry (GeLoCor 2021), virtually held at the University of C籀rdoba, Spain, on February 1-5, 2021. It includes surveys describing the state-of-the-art in specific areas, and a selection of the most relevant results presented at the conference. Taken together, the papers offer an invaluable introduction to key topics discussed at the conference and an overview of the main techniques in use today.This volume also gathers extended revisions of key studies in this field. Bringing new results and examples, these unique contributions offer new perspectives to the original problems and, in most cases, extend and reinforce the robustness of previous findings.Hosted every two years since 2001, the International Meeting on Lorentzian Geometry has become one of the main events bringing together the leading experts on Lorentzian geometry. Inthis volume, the reader will find studies on spatial and null hypersurfaces, low regularity in general relativity, conformal structures, Lorentz-Finsler spacetimes, and more.Given its scope, the book will be of interest to both young and experienced mathematicians and physicists whose research involves general relativity and semi-Riemannian geometry.
A Gyrovector Space Approach to Hyperbolic Geometry
The mere mention of hyperbolic geometry is enough to strike fear in the heart of the undergraduate mathematics and physics student. Some regard themselves as excluded from the profound insights of hyperbolic geometry so that this enormous portion of human achievement is a closed door to them. The mission of this book is to open that door by making the hyperbolic geometry of Bolyai and Lobachevsky, as well as the special relativity theory of Einstein that it regulates, accessible to a wider audience in terms of novel analogies that the modern and unknown share with the classical and familiar. These novel analogies that this book captures stem from Thomas gyration, which is the mathematical abstraction of the relativistic effect known as Thomas precession. Remarkably, the mere introduction of Thomas gyration turns Euclidean geometry into hyperbolic geometry, and reveals mystique analogies that the two geometries share. Accordingly, Thomas gyration gives rise to the prefix "gyro" that isextensively used in the gyrolanguage of this book, giving rise to terms like gyrocommutative and gyroassociative binary operations in gyrogroups, and gyrovectors in gyrovector spaces. Of particular importance is the introduction of gyrovectors into hyperbolic geometry, where they are equivalence classes that add according to the gyroparallelogram law in full analogy with vectors, which are equivalence classes that add according to the parallelogram law. A gyroparallelogram, in turn, is a gyroquadrilateral the two gyrodiagonals of which intersect at their gyromidpoints in full analogy with a parallelogram, which is a quadrilateral the two diagonals of which intersect at their midpoints. Table of Contents: Gyrogroups / Gyrocommutative Gyrogroups / Gyrovector Spaces / Gyrotrigonometry
Extrinsic Geometry of Foliations
This book is devoted to geometric problems of foliation theory, in particular those related to extrinsic geometry, modern branch of Riemannian Geometry. The concept of mixed curvature is central to the discussion, and a version of the deep problem of the Ricci curvature for the case of mixed curvature of foliations is examined. The book is divided into five chapters that deal with integral and variation formulas and curvature and dynamics of foliations. Different approaches and methods (local and global, regular and singular) in solving the problems are described using integral and variation formulas, extrinsic geometric flows, generalizations of the Ricci and scalar curvatures, pseudo-Riemannian and metric-affine geometries, and 'computable' Finsler metrics.The book presents the state of the art in geometric and analytical theory of foliations as a continuation of the authors' life-long work in extrinsic geometry. It is designed for newcomers to the field as well as experienced geometers working in Riemannian geometry, foliation theory, differential topology, and a wide range of researchers in differential equations and their applications. It may also be a useful supplement to postgraduate level work and can inspire new interesting topics to explore.
Introduction to Differential Geometry
This textbook is suitable for a one semester lecture course on differential geometry for students of mathematics or STEM disciplines with a working knowledge of analysis, linear algebra, complex analysis, and point set topology. The book treats the subject both from an extrinsic and an intrinsic view point.The first chapters give a historical overview of the field and contain an introduction to basic concepts such as manifolds and smooth maps, vector fields and flows, and Lie groups, leading up to the theorem of Frobenius. Subsequent chapters deal with the Levi-Civita connection, geodesics, the Riemann curvature tensor, a proof of the Cartan-Ambrose-Hicks theorem, as well as applications to flat spaces, symmetric spaces, and constant curvature manifolds. Also included are sections about manifolds with nonpositive sectional curvature, the Ricci tensor, the scalar curvature, and the Weyl tensor. An additional chapter goes beyond the scope of a one semester lecture course and deals with subjects such as conjugate points and the Morse index, the injectivity radius, the group of isometries and the Myers-Steenrod theorem, and Donaldson's differential geometric approach to Lie algebra theory.
Oliver Byrne`s Elements of Euclid
In one of the most stunning expositions of mathematical publishing, Oliver Byrne combines Euclid's geometric theories with vibrant colour proofs, turning what was already a cornerstone academic text into a pedagogical work of art. First published in 1847, Oliver Byrne's Elements of Euclid is an innovative educational masterpiece. Uniquely beautiful in its presentation, Byrne's edition was the first attempt to illustrate the classic books of mathematical theorems written by the ancient Greek mathematician, Euclid of Alexandria, in 300BC. Scattered across each page in brilliant reds, blues and yellows are triangles, squares and circles combined in a myriad of combinations with intersecting lines and numbers. These intricate figures express the proofs of many of the iconic geometric equations that form the bedrock of mathematical study. This stunning example of numerical visual study greatly influenced the history of mathematics, with Euclid's text being used in classrooms until the late nineteenth century. It has also proven to be an indispensable inspiration for following twentieth-century art movements, with avant-garde groups like De Stijl and The Bauhaus using Byrne's iconic colour, line work and form in many of their works. A facsimile edition of this legacy work has been painstakingly restored for a new generation to enjoy. Taking special care to conserve the colours, shapes and text as they were printed on publication in the hope to recapture the magic of this beautiful volume for future readers, both inside and outside of the classroom.
Elements of Surveying
This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work.This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.
New Horizons in Differential Geometry and Its Related Fields
This volume presents recent developments in geometric structures on Riemannian manifolds and their discretizations. With chapters written by recognized experts, these discussions focus on contact structures, K瓣hler structures, fiber bundle structures and Einstein metrics. It also contains works on the geometric approach on coding theory.For researchers and students, this volume forms an invaluable source to learn about these subjects that are not only in the field of differential geometry but also in other wide related areas. It promotes and deepens the study of geometric structures.
A First Course in Algebraic Geometry and Algebraic Varieties
This book provides a gentle introduction to the foundations of Algebraic Geometry, starting from computational topics (ideals and homogeneous ideals, zero loci of ideals) up to increasingly intrinsic and abstract arguments, such as 'Algebraic Varieties', whose natural continuation is a more advanced course on the theory of schemes, vector bundles, and sheaf-cohomology.Valuable to students studying Algebraic Geometry and Geometry, this title contains around 60 exercises (with solutions) to help students thoroughly understand the theories introduced in the book. Proofs of the results are carried out in full detail. Many examples are discussed in order to reinforce the understanding of both the theoretical elements and their consequences, as well as the possible applications of the material.
A First Course in Algebraic Geometry and Algebraic Varieties
This book provides a gentle introduction to the foundations of Algebraic Geometry, starting from computational topics (ideals and homogeneous ideals, zero loci of ideals) up to increasingly intrinsic and abstract arguments, such as 'Algebraic Varieties', whose natural continuation is a more advanced course on the theory of schemes, vector bundles, and sheaf-cohomology.Valuable to students studying Algebraic Geometry and Geometry, this title contains around 60 exercises (with solutions) to help students thoroughly understand the theories introduced in the book. Proofs of the results are carried out in full detail. Many examples are discussed in order to reinforce the understanding of both the theoretical elements and their consequences, as well as the possible applications of the material.
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.
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.
Plane Geometry
The contents of the book are1 GEOMETRY.2 PLANE GEOMETRY.BOOK I. RECTILINEAR FIGURESBOOK II. THE CIRCLE.BOOK III. PROPORTION. SIMILAR POLYGONSBOOK IV. AREAS OF POLYGONS.BOOK V. REGULAR POLYGONS AND CIRCLES
Periodic Monopoles and Difference Modules
This book studies a class of monopoles defined by certain mild conditions, called periodic monopoles of generalized Cherkis-Kapustin (GCK) type. It presents a classification of the latter in terms of difference modules with parabolic structure, revealing a kind of Kobayashi-Hitchin correspondence between differential geometric objects and algebraic objects. It also clarifies the asymptotic behaviour of these monopoles around infinity.The theory of periodic monopoles of GCK type has applications to Yang-Mills theory in differential geometry and to the study of difference modules in dynamical algebraic geometry. A complete account of the theory is given, including major generalizations of results due to Charbonneau, Cherkis, Hurtubise, Kapustin, and others, and a new and original generalization of the nonabelian Hodge correspondence first studied by Corlette, Donaldson, Hitchin and Simpson.This work will be of interest to graduate students and researchers in differential and algebraic geometry, as well as in mathematical physics.
Surveys in Geometry I
The volume consists of a set of surveys on geometry in the broad sense. The goal is to present a certain number of research topics in a non-technical and appealing manner.The topics surveyed include spherical geometry, the geometry of finite-dimensional normed spaces, metric geometry (Bishop--Gromov type inequalities in Gromov-hyperbolic spaces), convexity theory and inequalities involving volumes and mixed volumes of convex bodies, 4-dimensional topology, Teichm羹ller spaces and mapping class groups actions, translation surfaces and their dynamics, and complex higher-dimensional geometry.Several chapters are based on lectures given by their authors to middle-advanced level students and young researchers. The whole book is intended to be an introduction to current research trends in geometry.
Geometric Transformations
This textbook teaches the transformations of plane Euclidean geometry through problems, offering a transformation-based perspective on problems that have appeared in recent years at mathematics competitions around the globe, as well as on some classical examples and theorems. It is based on the combined teaching experience of the authors (coaches of several Mathematical Olympiad teams in Brazil, Romania and the USA) and presents comprehensive theoretical discussions of isometries, homotheties and spiral similarities, and inversions, all illustrated by examples and followed by myriad problems left for the reader to solve. These problems were carefully selected and arranged to introduce students to the topics by gradually moving from basic to expert level. Most of them have appeared in competitions such as Mathematical Olympiads or in mathematical journals aimed at an audience interested in mathematics competitions, while some are fundamental facts of mathematics discussed in the framework of geometric transformations. The book offers a global view of the geometric content of today's mathematics competitions, bringing many new methods and ideas to the attention of the public.Talented high school and middle school students seeking to improve their problem-solving skills can benefit from this book, as well as high school and college instructors who want to add nonstandard questions to their courses. People who enjoy solving elementary math problems as a hobby will also enjoy this work.
Directed Quantities in Electrodynamics
This monograph explores classical electrodynamics from a geometrical perspective with a clear visual presentation throughout. Featuring over 200 figures, readers will delve into the definitions, properties, and uses of directed quantities in classical field theory. With an emphasis on both mathematical and electrodynamic concepts, the author's illustrative approach will help readers understand the critical role directed quantities play in physics and mathematics. Chapters are organized so that they gradually scale in complexity, and carefully guide readers through important topics. The first three chapters introduce directed quantities in three dimensions with and without the metric, as well as the development of the algebra and analysis of directed quantities. Chapters four through seven then focus on electrodynamics without the metric, such as the premetric case, waves, and fully covariant four-dimensional electrodynamics. Complementing the book's careful structure, exercises are included throughout for readers seeking further opportunities to practice the material. Directed Quantities in Electrodynamics will appeal to students, lecturers, and researchers of electromagnetism. It is particularly suitable as a supplement to standard textbooks on electrodynamics.
Differential Geometry
This book combines the classical and contemporary approaches to differential geometry. An introduction to the Riemannian geometry of manifolds is preceded by a detailed discussion of properties of curves and surfaces.The chapter on the differential geometry of plane curves considers local and global properties of curves, evolutes and involutes, and affine and projective differential geometry. Various approaches to Gaussian curvature for surfaces are discussed. The curvature tensor, conjugate points, and the Laplace-Beltrami operator are first considered in detail for two-dimensional surfaces, which facilitates studying them in the many-dimensional case. A separate chapter is devoted to the differential geometry of Lie groups.
Research Directions in Symplectic and Contact Geometry and Topology
This book highlights a number of recent research advances in the field of symplectic and contact geometry and topology, and related areas in low-dimensional topology. This field has experienced significant and exciting growth in the past few decades, and this volume provides an accessible introduction into many active research problems in this area. The papers were written with a broad audience in mind so as to reach a wide range of mathematicians at various levels. Aside from teaching readers about developing research areas, this book will inspire researchers to ask further questions to continue to advance the field.The volume contains both original results and survey articles, presenting the results of collaborative research on a wide range of topics. These projects began at the Research Collaboration Conference for Women in Symplectic and Contact Geometry and Topology (WiSCon) in July 2019 at ICERM, Brown University. Each group of authors includedfemale and nonbinary mathematicians at different career levels in mathematics and with varying areas of expertise. This paved the way for new connections between mathematicians at all career levels, spanning multiple continents, and resulted in the new collaborations and directions that are featured in this work.
Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry
For more than thirty years the senior author has been trying to learn algebraic geometry. In the process he discovered that many of the classic textbooks in algebraic geometry require substantial knowledge of cohomology, homological algebra, and sheaf theory. In an attempt to demystify these abstract concepts and facilitate understanding for a new generation of mathematicians, he along with co-author wrote this book for an audience who is familiar with basic concepts of linear and abstract algebra, but who never has had any exposure to the algebraic geometry or homological algebra. As such this book consists of two parts. The first part gives a crash-course on the homological and cohomological aspects of algebraic topology, with a bias in favor of cohomology. The second part is devoted to presheaves, sheaves, Cech cohomology, derived functors, sheaf cohomology, and spectral sequences. All important concepts are intuitively motivated and the associated proofs of the quintessential theorems are presented in detail rarely found in the standard texts.
Math Foundation +
The Math Labs are designed using the "Chunking Method" of learning math concepts. Besides focused labs involving Math Foundation, an introduction to Geometry, Algebra, and Functions are also included. This book is perfect for 6th and 7th-grade students who are about to learn Pre-Algebra and Algebra 1. Others interested in this book may be those studying for standardized test exams, homeschool, and GED students.
