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Easily accessible Includes recent developments Assumes very little knowledge of differentiable manifolds and functional analysis Particular emphasis on topics related to mirror symmetry (SUSY, Kaehler-Einstein metrics, Tian-Todorov lemma)
An account for graduate students of this new technique in diophantine geometry; includes account of higher dimensional theory.
In recent years geometry seems to have lost large parts of its former central position in mathematics teaching in most countries. However, new trends have begun to counteract this tendency. There is an increasing awareness that geometry plays a key role in mathematics and learning mathematics. Although geometry has been eclipsed in the mathematics curriculum, research in geometry has blossomed as new ideas have arisen from inside mathematics and other disciplines, including computer science. Due to reassessment of the role of geometry, mathematics educators and mathematicians face new challenges. In the present ICMI study, the whole spectrum of teaching and learning of geometry is analysed. Experts from all over the world took part in this study, which was conducted on the basis of recent international research, case studies, and reports on actual school practice. This book will be of particular interest to mathematics educators and mathematicians who are involved in the teaching of geometry at all educational levels, as well as to researchers in mathematics education.
Learn geometry at your own pace What are congruent circles? How do you find the hypotenuse of a triangle? What is the sum of the angles in a decagon? How can you apply geometric equations to your daily life? With the unbeatable study companion Geometry: A Self-Teaching Guide, you'll discover the answers to these questions and many more. This thorough primer presents an easy-to-follow, proven method for grasping the key concepts of geometry. You'll progress step by step through plane, solid, and analytic geometry and then move on to geometric applications for calculus. You'll build your problem-solving skills along the way through detailed examples, reviews, exercises, and answer explanations. The clearly structured format of Geometry makes it fully accessible, providing an easily understood, comprehensive overview for everyone from high school students to adult learners to math mavens. Like all Self-Teaching Guides, Geometry allows you to build gradually on what you have learned-at your own pace. Questions and self-tests reinforce the information in each chapter and allow you to skip ahead or focus on specific areas of concern. Packed with useful, up-to-date information, this clear, concise volume is a valuable learning tool and reference source for anyone who wants to improve his or her understanding of basic geometry.
IMPACT (Interweaving Mathematics Pedagogy and Content for Teaching) is an exciting new series of texts for teacher education which aims to advance the learning and teaching of mathematics by integrating mathematics content with the broader research and theoretical base of mathematics education. The Learning and Teaching of Geometry in Secondary Schools reviews past and present research on the teaching and learning of geometry in secondary schools and proposes an approach for design research on secondary geometry instruction. Areas covered include: teaching and learning secondary geometry through history; the representations of geometric figures; students’ cognition in geometry; teacher knowledge, practice and, beliefs; teaching strategies, instructional improvement, and classroom interventions; research designs and problems for secondary geometry. Drawing on a team of international authors, this new text will be essential reading for experienced teachers of mathematics, graduate students, curriculum developers, researchers, and all those interested in exploring students’ study of geometry in secondary schools.
An interactive guide to shapes for 5 to 8 year olds, this bright and bold lift-the-flap activity book helps children understand the properties of 2-D and 3-D shapes. Shapes are an important topic for early learners, and this visually appealing book will make it a lot of fun, too! Geometry Genius features fun geometric characters, like Fox and Lion, and lift-the-flap activities that help kids relate shapes to everyday life. Characters pose key questions, such as "What's special about a sphere?," "What is an equilateral triangle?," and "How many lines of symmetry does a hexagon have?" Children can then lift the flaps and find the answers. An interactive pop-up will also bring learning to life by encouraging kids to spot different shapes within the scene. Geometry Genius helps kids identify and describe 2-D and 3-D shapes, compare and contrast features of regular and irregular shapes, discuss the size and orientation of shapes, understand nets, identify and count lines of symmetry, and more! It gets kids thinking about shapes in their world and not just on the pages of a math book. Quiz questions and fun activities are found sprinkled throughout the book, encouraging kids to lift the flaps and find out more. Learning shapes is a highly visual topic, and this book tackles the subject in a visually appealing, fully interactive, and playful way.
Several years ago our statistical friends and relations introduced us to the work of Amari and Barndorff-Nielsen on applications of differential geometry to statistics. This book has arisen because we believe that there is a deep relationship between statistics and differential geometry and moreoever that this relationship uses parts of differential geometry, particularly its 'higher-order' aspects not readily accessible to a statistical audience from the existing literature. It is, in part, a long reply to the frequent requests we have had for references on differential geometry! While we have not gone beyond the path-breaking work of Amari and Barndorff- Nielsen in the realm of applications, our book gives some new explanations of their ideas from a first principles point of view as far as geometry is concerned. In particular it seeks to explain why geometry should enter into parametric statistics, and how the theory of asymptotic expansions involves a form of higher-order differential geometry. The first chapter of the book explores exponential families as flat geometries. Indeed the whole notion of using log-likelihoods amounts to exploiting a particular form of flat space known as an affine geometry, in which straight lines and planes make sense, but lengths and angles are absent. We use these geometric ideas to introduce the notion of the second fundamental form of a family whose vanishing characterises precisely the exponential families.
Sure to be influential, Watanabe's book lays the foundations for the use of algebraic geometry in statistical learning theory. Many models/machines are singular: mixture models, neural networks, HMMs, Bayesian networks, stochastic context-free grammars are major examples. The theory achieved here underpins accurate estimation techniques in the presence of singularities.
This is the first comprehensive book on information geometry, written by the founder of the field. It begins with an elementary introduction to dualistic geometry and proceeds to a wide range of applications, covering information science, engineering, and neuroscience. It consists of four parts, which on the whole can be read independently. A manifold with a divergence function is first introduced, leading directly to dualistic structure, the heart of information geometry. This part (Part I) can be apprehended without any knowledge of differential geometry. An intuitive explanation of modern differential geometry then follows in Part II, although the book is for the most part understandable without modern differential geometry. Information geometry of statistical inference, including time series analysis and semiparametric estimation (the Neyman–Scott problem), is demonstrated concisely in Part III. Applications addressed in Part IV include hot current topics in machine learning, signal processing, optimization, and neural networks. The book is interdisciplinary, connecting mathematics, information sciences, physics, and neurosciences, inviting readers to a new world of information and geometry. This book is highly recommended to graduate students and researchers who seek new mathematical methods and tools useful in their own fields.