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Algebraic Topology
Algebraic Topology is an introductory textbook based on a class for advanced high-school students at the Stanford University Mathematics Camp (SUMaC) that the authors have taught for many years.Each chapter, or lecture, corresponds to one day of class at SUMaC.The book begins with the preliminaries needed for the formal definition of a surface.Other topics covered in the book include the classification of surfaces, group theory, the fundamental group, and homology. This book assumes no background in abstract algebra or real analysis, and the material from those subjects is presented as needed in the text.This makes the book readable to undergraduates or high-school students who do not have the background typically assumed in an algebraic topology book or class.The book contains many examples and exercises, allowing it to be used for both self-study and for an introductory undergraduate topology course.
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Algebraic Geometry
Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P.Serre and A. Grothendieck in Paris. After receiving his Ph.D. from Princeton in 1963, Hartshorne became a Junior Fellow at Harvard, then taught there for several years.In 1972 he moved to California where he is now Professor at the University of California at Berkeley.He is the author of "Residues and Duality" (1966), "Foundations of Projective Geometry (1968), "Ample Subvarieties of Algebraic Varieties" (1970), and numerous research titles.His current research interest is the geometry of projective varieties and vector bundles.He has been a visiting professor at the College de France and at Kyoto University, where he gave lectures in French and in Japanese, respectively.Professor Hartshorne is married to Edie Churchill, educator and psychotherapist, and has two sons.He has travelled widely, speaks several foreign languages, and is an experienced mountain climber.He is also an accomplished amateur musician: he has played the flute for many years, and during his last visit to Kyoto he began studying the shakuhachi.
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Algebraic Number Theory
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First Course In Algebraic Geometry And Algebraic Varieties, A
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.
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Complex Algebraic Threefolds
The first book on the explicit birational geometry of complex algebraic threefolds arising from the minimal model program, this text is sure to become an essential reference in the field of birational geometry.Threefolds remain the interface between low and high-dimensional settings and a good understanding of them is necessary in this actively evolving area.Intended for advanced graduate students as well as researchers working in birational geometry, the book is as self-contained as possible.Detailed proofs are given throughout and more than 100 examples help to deepen understanding of birational geometry.The first part of the book deals with threefold singularities, divisorial contractions and flips.After a thorough explanation of the Sarkisov program, the second part is devoted to the analysis of outputs, specifically minimal models and Mori fibre spaces.The latter are divided into conical fibrations, del Pezzo fibrations and Fano threefolds according to the relative dimension.
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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.
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Elements Of Algebraic Topology
Elements of Algebraic Topology provides the most concrete approach to the subject.With coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in manifolds, and applications to classical theorems of point-set topology, this book is perfect for comunicating complex topics and the fun nature of algebraic topology for beginners.
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Lectures On Algebraic Topology
Algebraic Topology and basic homotopy theory form a fundamental building block for much of modern mathematics.These lecture notes represent a culmination of many years of leading a two-semester course in this subject at MIT.The style is engaging and student-friendly, but precise.Every lecture is accompanied by exercises. It begins slowly in order to gather up students with a variety of backgrounds, but gains pace as the course progresses, and by the end the student has a command of all the basic techniques of classical homotopy theory.
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