Introduction to Tropical Geometry

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Author: Diane Maclagan,Bernd Sturmfels

Publisher: American Mathematical Soc.

ISBN: 0821851985

Category: Algebraic geometry -- Special varieties -- Toric varieties, Newton polyhedra

Page: 363

View: 6372

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.

Combinatorial Algebraic Geometry

Selected Papers From the 2016 Apprenticeship Program

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Author: Gregory G. Smith,Bernd Sturmfels

Publisher: Springer

ISBN: 1493974866

Category: Mathematics

Page: 390

View: 5726

This volume consolidates selected articles from the 2016 Apprenticeship Program at the Fields Institute, part of the larger program on Combinatorial Algebraic Geometry that ran from July through December of 2016. Written primarily by junior mathematicians, the articles cover a range of topics in combinatorial algebraic geometry including curves, surfaces, Grassmannians, convexity, abelian varieties, and moduli spaces. This book bridges the gap between graduate courses and cutting-edge research by connecting historical sources, computation, explicit examples, and new results.

Algebraic Geometry

Salt Lake City 2015 : 2015 Summer Research Institute, July 13-31, 2015, University of Utah, Salt Lake City, Utah

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Author: Richard Thomas

Publisher: American Mathematical Soc.

ISBN: 1470435780

Category: Geometry, Algebraic

Page: 635

View: 3562

This is Part 2 of a two-volume set. Since Oscar Zariski organized a meeting in 1954, there has been a major algebraic geometry meeting every decade: Woods Hole (1964), Arcata (1974), Bowdoin (1985), Santa Cruz (1995), and Seattle (2005). The American Mathematical Society has supported these summer institutes for over 50 years. Their proceedings volumes have been extremely influential, summarizing the state of algebraic geometry at the time and pointing to future developments. The most recent Summer Institute in Algebraic Geometry was held July 2015 at the University of Utah in Salt Lake City, sponsored by the AMS with the collaboration of the Clay Mathematics Institute. This volume includes surveys growing out of plenary lectures and seminar talks during the meeting. Some present a broad overview of their topics, while others develop a distinctive perspective on an emerging topic. Topics span both complex algebraic geometry and arithmetic questions, specifically, analytic techniques, enumerative geometry, moduli theory, derived categories, birational geometry, tropical geometry, Diophantine questions, geometric representation theory, characteristic and -adic tools, etc. The resulting articles will be important references in these areas for years to come.

Introduction to Algebraic Geometry

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Author: Steven Dale Cutkosky

Publisher: American Mathematical Soc.

ISBN: 1470435187

Category: Geometry, Algebraic

Page: 484

View: 9501

This book presents a readable and accessible introductory course in algebraic geometry, with most of the fundamental classical results presented with complete proofs. An emphasis is placed on developing connections between geometric and algebraic aspects of the theory. Differences between the theory in characteristic and positive characteristic are emphasized. The basic tools of classical and modern algebraic geometry are introduced, including varieties, schemes, singularities, sheaves, sheaf cohomology, and intersection theory. Basic classical results on curves and surfaces are proved. More advanced topics such as ramification theory, Zariski's main theorem, and Bertini's theorems for general linear systems are presented, with proofs, in the final chapters. With more than 200 exercises, the book is an excellent resource for teaching and learning introductory algebraic geometry.

Tropical Geometry and Integrable Systems

A Conference on Tropical Geometry and Integrable Systems, July 3-8, 2011, School of Mathematics and Statistics, University of Glasgow, United Kingdom

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Author: Chris Athorne,Diane Maclagan,Ian Strachan

Publisher: American Mathematical Soc.

ISBN: 0821875531

Category: Mathematics

Page: 155

View: 3779

This volume contains the proceedings of the conference on tropical geometry and integrable systems, held July 3-8, 2011, at the University of Glasgow, United Kingdom. One of the aims of this conference was to bring together researchers in the field of tropical geometry and its applications, from apparently disparate ends of the spectrum, to foster a mutual understanding and establish a common language which will encourage further developments of the area. This aim is reflected in these articles, which cover areas from automata, through cluster algebras, to enumerative geometry. In addition, two survey articles are included which introduce ideas from researchers on one end of this spectrum to researchers on the other. This book is intended for graduate students and researchers interested in tropical geometry and integrable systems and the developing links between these two areas.

Introduction to Algebraic Geometry

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Author: Brendan Hassett

Publisher: Cambridge University Press

ISBN: 1139464590

Category: Mathematics

Page: N.A

View: 6089

Algebraic geometry, central to pure mathematics, has important applications in such fields as engineering, computer science, statistics and computational biology, which exploit the computational algorithms that the theory provides. Users get the full benefit, however, when they know something of the underlying theory, as well as basic procedures and facts. This book is a systematic introduction to the central concepts of algebraic geometry most useful for computation. Written for advanced undergraduate and graduate students in mathematics and researchers in application areas, it focuses on specific examples and restricts development of formalism to what is needed to address these examples. In particular, it introduces the notion of Gröbner bases early on and develops algorithms for almost everything covered. It is based on courses given over the past five years in a large interdisciplinary programme in computational algebraic geometry at Rice University, spanning mathematics, computer science, biomathematics and bioinformatics.

Algebraic and Combinatorial Aspects of Tropical Geometry

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Author: Erwan Brugalle,Maria Angelica Cueto,Alicia Dickenstein,Eva-Maria Feichtner,Ilia Itenberg

Publisher: American Mathematical Soc.

ISBN: 0821891464

Category: Mathematics

Page: 350

View: 8056

This volume contains the proceedings of the CIEM workshop on Tropical Geometry, held December 12-16, 2011, at the International Centre for Mathematical Meetings (CIEM), Castro Urdiales, Spain. Tropical geometry is a new and rapidly developing field of mat

Introduction to Intersection Theory in Algebraic Geometry

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Author: William Fulton

Publisher: American Mathematical Soc.

ISBN: 0821807048

Category: Mathematics

Page: 82

View: 8694

This book introduces some of the main ideas of modern intersection theory, traces their origins in classical geometry and sketches a few typical applications. It requires little technical background: much of the material is accessible to graduate students in mathematics. A broad survey, the book touches on many topics, most importantly introducing a powerful new approach developed by the author and R. MacPherson. It was written from the expository lectures delivered at the NSF-supported CBMS conference at George Mason University, held June 27-July 1, 1983.The author describes the construction and computation of intersection products by means of the geometry of normal cones. In the case of properly intersecting varieties, this yields Samuel's intersection multiplicity; at the other extreme it gives the self-intersection formula in terms of a Chern class of the normal bundle; in general it produces the excess intersection formula of the author and R. MacPherson. Among the applications presented are formulas for degeneracy loci, residual intersections, and multiple point loci; dynamic interpretations of intersection products; Schubert calculus and solutions to enumerative geometry problems; and Riemann-Roch theorems.

An Algebraic Introduction to Complex Projective Geometry

Commutative Algebra

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Author: Christian Peskine,Peskine Christian

Publisher: Cambridge University Press

ISBN: 9780521480727

Category: Mathematics

Page: 244

View: 5583

In this introduction to commutative algebra, the author choses a route that leads the reader through the essential ideas, without getting embroiled in technicalities. He takes the reader quickly to the fundamentals of complex projective geometry, requiring only a basic knowledge of linear and multilinear algebra and some elementary group theory. The author divides the book into three parts. In the first, he develops the general theory of noetherian rings and modules. He includes a certain amount of homological algebra, and he emphasizes rings and modules of fractions as preparation for working with sheaves. In the second part, he discusses polynomial rings in several variables with coefficients in the field of complex numbers. After Noether's normalization lemma and Hilbert's Nullstellensatz, the author introduces affine complex schemes and their morphisms; he then proves Zariski's main theorem and Chevalley's semi-continuity theorem. Finally, the author's detailed study of Weil and Cartier divisors provides a solid background for modern intersection theory. This is an excellent textbook for those who seek an efficient and rapid introduction to the geometric applications of commutative algebra.

An Invitation to Arithmetic Geometry

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Author: Dino Lorenzini

Publisher: American Mathematical Soc.

ISBN: 0821802674

Category: Arithmetical algebraic geometry

Page: 397

View: 8948

Extremely carefully written, masterfully thought out, and skillfully arranged introduction ... to the arithmetic of algebraic curves, on the one hand, and to the algebro-geometric aspects of number theory, on the other hand. ... an excellent guide for beginners in arithmetic geometry, just as an interesting reference and methodical inspiration for teachers of the subject ... a highly welcome addition to the existing literature. --Zentralblatt MATH The interaction between number theory and algebraic geometry has been especially fruitful. In this volume, the author gives a unified presentation of some of the basic tools and concepts in number theory, commutative algebra, and algebraic geometry, and for the first time in a book at this level, brings out the deep analogies between them. The geometric viewpoint is stressed throughout the book. Extensive examples are given to illustrate each new concept, and many interesting exercises are given at the end of each chapter. Most of the important results in the one-dimensional case are proved, including Bombieri's proof of the Riemann Hypothesis for curves over a finite field. While the book is not intended to be an introduction to schemes, the author indicates how many of the geometric notions introduced in the book relate to schemes, which will aid the reader who goes to the next level of this rich subject.

An Introduction to Commutative Algebra

From the Viewpoint of Normalization

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Author: Huishi Li

Publisher: World Scientific

ISBN: 9789812389510

Category: Mathematics

Page: 175

View: 1798

- Contains many examples and problems (with hints) - Provides a good introduction for beginners in algebraic number theory and algebraic geometry

Introduction to Algebraic Curves

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Author: Phillip Griffiths,Sigurdur Helgason

Publisher: American Mathematical Soc.

ISBN: 0821845306

Category: Courbes algébriques

Page: 221

View: 7399

This book offers a comprehensive treatment of geometric analysis on symmetric spaces, with applications to representation theory. The author's thorough, accurate approach brings the reader up to date on current research in analysis on symmetric spaces and the analytic approach to the representation theory of semisimple Lie groups. The author is a 1988 Steele Prize recipient for his earlier books in this area.

Algebraic Geometry

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Author: Masayoshi Miyanishi

Publisher: American Mathematical Soc.

ISBN: 9780821887707

Category: Mathematics

Page: 246

View: 6281

Students often find, in setting out to study algebraic geometry, that most of the serious textbooks on the subject require knowledge of ring theory, field theory, local rings, and transcendental field extensions, and even sheaf theory. Often the expected background goes well beyond college mathematics. This book, aimed at senior undergraduates and graduate students, grew out of Miyanishi's attempt to lead students to an understanding of algebraic surfaces while presenting thenecessary background along the way. Originally published in Japanese in 1990, it presents a self-contained introduction to the fundamentals of algebraic geometry. This book begins with background on commutative algebras, sheaf theory, and related cohomology theory. The next part introduces schemes andalgebraic varieties, the basic language of algebraic geometry. The last section brings readers to a point at which they can start to learn about the classification of algebraic surfaces.

Introduction to Topological Manifolds

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Author: John Lee

Publisher: Springer Science & Business Media

ISBN: 1441979409

Category: Mathematics

Page: 433

View: 9712

This book is an introduction to manifolds at the beginning graduate level, and accessible to any student who has completed a solid undergraduate degree in mathematics. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of differential geometry, algebraic topology, and related fields. Although this second edition has the same basic structure as the first edition, it has been extensively revised and clarified; not a single page has been left untouched. The major changes include a new introduction to CW complexes (replacing most of the material on simplicial complexes in Chapter 5); expanded treatments of manifolds with boundary, local compactness, group actions, and proper maps; and a new section on paracompactness.

A Guide to Plane Algebraic Curves

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Author: Keith Kendig

Publisher: MAA

ISBN: 0883853531

Category: Mathematics

Page: 193

View: 4149

This is an informal and accessible introduction to plane algebraic curves that also serves as a natural entry point to algebraic geometry. There is a unifying theme to the book: give curves enough living space and beautiful theorems will follow. This book provides the reader with a solid intuition for the subject, while at the same time keeping the exposition simple and understandable, by introducing abstract concepts with concrete examples and pictures. It can be used as the text for an undergraduate course on plane algebraic curves, or as a companion to algebraic geometry at graduate level. This book is accessible to those with a limited mathematical background. This is because for those outside mathematics there is a growing need for an entre to algebraic geometry, a need created by the ever-expanding role algebraic geometry is playing in areas ranging from biology to chemistry and robotics to cryptology.

Introduction to the Geometry of Foliations, Part A

Foliations on Compact Surfaces, Fundamentals for Arbitrary Codimension, and Holonomy

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Author: Gilbert Hector,Ulrich Hirsch

Publisher: Springer-Verlag

ISBN: 3322984826

Category: Juvenile Nonfiction

Page: 236

View: 1563

Semidefinite Optimization and Convex Algebraic Geometry

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Author: Grigoriy Blekherman,Pablo A. Parrilo,Rekha R. Thomas

Publisher: SIAM

ISBN: 1611972280

Category: Mathematics

Page: 476

View: 8452

An accessible introduction to convex algebraic geometry and semidefinite optimization. For graduate students and researchers in mathematics and computer science.

An Invitation to Noncommutative Geometry

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Author: Matilde Marcolli

Publisher: World Scientific

ISBN: 9812814337

Category: Mathematics

Page: 506

View: 335

This is the first existing volume that collects lectures on this important and fast developing subject in mathematics. The lectures are given by leading experts in the field and the range of topics is kept as broad as possible by including both the algebraic and the differential aspects of noncommutative geometry as well as recent applications to theoretical physics and number theory. Sample Chapter(s). A Walk in the Noncommutative Garden (1,639 KB). Contents: A Walk in the Noncommutative Garden (A Connes & M Marcolli); Renormalization of Noncommutative Quantum Field Theory (H Grosse & R Wulkenhaar); Lectures on Noncommutative Geometry (M Khalkhali); Noncommutative Bundles and Instantons in Tehran (G Landi & W D van Suijlekom); Lecture Notes on Noncommutative Algebraic Geometry and Noncommutative Tori (S Mahanta); Lectures on Derived and Triangulated Categories (B Noohi); Examples of Noncommutative Manifolds: Complex Tori and Spherical Manifolds (J Plazas); D-Branes in Noncommutative Field Theory (R J Szabo). Readership: Researchers in mathematical and theoretical physics, geometry and topology, algebra and number theory.