Metric Algebraic Geometry

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Metric Algebraic Geometry

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Author : Paul Breiding
language : en
Publisher: Springer Nature
Release Date : 2024-02-27

Metric Algebraic Geometry written by Paul Breiding and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-02-27 with Mathematics categories.

Metric algebraic geometry combines concepts from algebraic geometry and differential geometry. Building on classical foundations, it offers practical tools for the 21st century. Many applied problems center around metric questions, such as optimization with respect to distances. After a short dive into 19th-century geometry of plane curves, we turn to problems expressed by polynomial equations over the real numbers. The solution sets are real algebraic varieties. Many of our metric problems arise in data science, optimization and statistics. These include minimizing Wasserstein distances in machine learning, maximum likelihood estimation, computing curvature, or minimizing the Euclidean distance to a variety. This book addresses a wide audience of researchers and students and can be used for a one-semester course at the graduate level. The key prerequisite is a solid foundation in undergraduate mathematics, especially in algebra and geometry. This is an openaccess book.

Lipschitz Algebras

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Author : Nik Weaver
language : en
Publisher: World Scientific
Release Date : 1999

Lipschitz Algebras written by Nik Weaver and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999 with Mathematics categories.

The Lipschitz algebras Lp(M), for M a complete metric space, are quite analogous to the spaces C(omega) and Linfinity(X), for omega a compact Hausdorff space and X a sigma-finite measure space. Although the Lipschitz algebras have not been studied as thoroughly as these better-known cousins, it is becoming increasingly clear that they play a fundamental role in functional analysis, and are also useful in many applications, especially in the direction of metric geometry. This book gives a comprehensive treatment of (what is currently known about) the beautiful theory of these algebras.

Geometry

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Author : Richard S. Millman
language : en
Publisher: Springer Science & Business Media
Release Date : 1993-05-07

Geometry written by Richard S. Millman and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993-05-07 with Mathematics categories.

Geometry: A Metric Approach with Models, imparts a real feeling for Euclidean and non-Euclidean (in particular, hyperbolic) geometry. Intended as a rigorous first course, the book introduces and develops the various axioms slowly, and then, in a departure from other texts, continually illustrates the major definitions and axioms with two or three models, enabling the reader to picture the idea more clearly. The second edition has been expanded to include a selection of expository exercises. Additionally, the authors have designed software with computational problems to accompany the text. This software may be obtained from George Parker.

Metric Structures In Differential Geometry

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Author : Gerard Walschap
language : en
Publisher: Springer Science & Business Media
Release Date : 2004-03-18

Metric Structures In Differential Geometry written by Gerard Walschap and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004-03-18 with Mathematics categories.

This book offers an introduction to the theory of differentiable manifolds and fiber bundles. It examines bundles from the point of view of metric differential geometry: Euclidean bundles, Riemannian connections, curvature, and Chern-Weil theory are discussed, including the Pontrjagin, Euler, and Chern characteristic classes of a vector bundle. These concepts are illustrated in detail for bundles over spheres.

Algebraic Geometry Iii

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Author : A.N. Parshin
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-17

Algebraic Geometry Iii written by A.N. Parshin and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-17 with Mathematics categories.

Starting with the end of the seventeenth century, one of the most interesting directions in mathematics (attracting the attention as J. Bernoulli, Euler, Jacobi, Legendre, Abel, among others) has been the study of integrals of the form r dz l Aw(T) = -, TO W where w is an algebraic function of z. Such integrals are now called abelian. Let us examine the simplest instance of an abelian integral, one where w is defined by the polynomial equation (1) where the polynomial on the right hand side has no multiple roots. In this case the function Aw is called an elliptic integral. The value of Aw is determined up to mv + nv , where v and v are complex numbers, and m and n are 1 2 1 2 integers. The set of linear combinations mv+ nv forms a lattice H C C, and 1 2 so to each elliptic integral Aw we can associate the torus C/ H. 2 On the other hand, equation (1) defines a curve in the affine plane C = 2 2 {(z,w)}. Let us complete C2 to the projective plane lP' = lP' (C) by the addition of the "line at infinity", and let us also complete the curve defined 2 by equation (1). The result will be a nonsingular closed curve E C lP' (which can also be viewed as a Riemann surface). Such a curve is called an elliptic curve.

Differential Geometry

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Author : Clifford Taubes
language : en
Publisher: Oxford University Press
Release Date : 2011-10-13

Differential Geometry written by Clifford Taubes and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-10-13 with Mathematics categories.

Bundles, connections, metrics and curvature are the lingua franca of modern differential geometry and theoretical physics. Supplying graduate students in mathematics or theoretical physics with the fundamentals of these objects, this book would suit a one-semester course on the subject of bundles and the associated geometry.

Complex Geometry

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Author : Daniel Huybrechts
language : en
Publisher: Springer Science & Business Media
Release Date : 2005

Complex Geometry written by Daniel Huybrechts and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with Computers categories.

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)

Computational Algebraic Geometry

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Author : Frederic Eyssette
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Computational Algebraic Geometry written by Frederic Eyssette and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

The theory and practice of computation in algebraic geometry and related domains, from a mathematical point of view, has generated an increasing interest both for its rich theoretical possibilities and its usefulness in applications in science and engineering. In fact, it is one of the master keys for future significant improvement of the computer algebra systems (e.g., Reduce, Macsyma, Maple, Mathematica, Axiom, Macaulay, etc.) that have become such useful tools for many scientists in a variety of disciplines. The major themes covered in this volume, arising from papers p- sented at the conference MEGA-92 were: - Effective methods and complexity issues in commutative algebra, projective geometry, real geometry, and algebraic number theory - Algebra-geometric methods in algebraic computing and applica tions. MEGA-92 was the second of a new series of European conferences on the general theme of Effective Methods in Algebraic Geometry. It was held in Nice, France, on April 21-25, 1992 and built on the themes presented at MEGA-90 (Livomo, Italy, April 17-21, 1990). The next conference - MEGA-94 - will be held in Santander, Spain in the spring of 1994. The Organizing committee that initiatiod and supervises this bi enniel conference consists of A. Conte (Torino), J.H. Davenport (Bath), A. Galligo (Nice), D. Yu. Grigoriev (Petersburg), J. Heintz (Buenos Aires), W. Lassner (Leipzig), D. Lazard (paris), H.M. MOller (Hagen), T. Mora (Genova), M. Pohst (DUsseldort), T. Recio (Santander), J.J.

Introduction To Tropical Geometry

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Author : Diane Maclagan
language : en
Publisher: American Mathematical Society
Release Date : 2021-12-13

Introduction To Tropical Geometry written by Diane Maclagan and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-12-13 with Mathematics categories.

Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of information about their classical counterparts. Tropical geometry is a young subject that has undergone a rapid development since the beginning of the 21st century. While establishing itself as an area in its own right, deep connections have been made to many branches of pure and applied mathematics. This book offers a self-contained introduction to tropical geometry, suitable as a course text for beginning graduate students. Proofs are provided for the main results, such as the Fundamental Theorem and the Structure Theorem. Numerous examples and explicit computations illustrate the main concepts. Each of the six chapters concludes with problems that will help the readers to practice their tropical skills, and to gain access to the research literature. This wonderful book will appeal to students and researchers of all stripes: it begins at an undergraduate level and ends with deep connections to toric varieties, compactifications, and degenerations. In between, the authors provide the first complete proofs in book form of many fundamental results in the subject. The pages are sprinkled with illuminating examples, applications, and exercises, and the writing is lucid and meticulous throughout. It is that rare kind of book which will be used equally as an introductory text by students and as a reference for experts. —Matt Baker, Georgia Institute of Technology Tropical geometry is an exciting new field, which requires tools from various parts of mathematics and has connections with many areas. A short definition is given by Maclagan and Sturmfels: “Tropical geometry is a marriage between algebraic and polyhedral geometry”. This wonderful book is a pleasant and rewarding journey through different landscapes, inviting the readers from a day at a beach to the hills of modern algebraic geometry. The authors present building blocks, examples and exercises as well as recent results in tropical geometry, with ingredients from algebra, combinatorics, symbolic computation, polyhedral geometry and algebraic geometry. The volume will appeal both to beginning graduate students willing to enter the field and to researchers, including experts. —Alicia Dickenstein, University of Buenos Aires, Argentina

Real Analytic And Algebraic Geometry

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Author : Fabrizio Broglia
language : en
Publisher: Walter de Gruyter
Release Date : 2011-07-11

Real Analytic And Algebraic Geometry written by Fabrizio Broglia and has been published by Walter de Gruyter this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-07-11 with Mathematics categories.

The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.