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Lectures on Riemann Surfaces
Author: Otto Forster
Publisher: Springer Science & Business Media
ISBN: 1461259614
Pages: 256
Year: 2012-12-06
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This book grew out of lectures on Riemann surfaces given by Otto Forster at the universities of Munich, Regensburg, and Münster. It provides a concise modern introduction to this rewarding subject, as well as presenting methods used in the study of complex manifolds in the special case of complex dimension one. From the reviews: "This book deserves very serious consideration as a text for anyone contemplating giving a course on Riemann surfaces."—-MATHEMATICAL REVIEWS
Riemann Surfaces
Author: Simon Donaldson
Publisher: Oxford University Press
ISBN: 0198526393
Pages: 286
Year: 2011-03-24
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An authoritative but accessible text on one dimensional complex manifolds or Riemann surfaces. Dealing with the main results on Riemann surfaces from a variety of points of view; it pulls together material from global analysis, topology, and algebraic geometry, and covers the essential mathematical methods and tools.
Handbook of Teichmüller Theory
Author: Athanase Papadopoulos
Publisher: European Mathematical Society
ISBN: 3037190558
Pages: 874
Year: 2012
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Lectures on K3 Surfaces
Author: Daniel Huybrechts
Publisher: Cambridge University Press
ISBN: 1316797252
Year: 2016-09-26
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K3 surfaces are central objects in modern algebraic geometry. This book examines this important class of Calabi–Yau manifolds from various perspectives in eighteen self-contained chapters. It starts with the basics and guides the reader to recent breakthroughs, such as the proof of the Tate conjecture for K3 surfaces and structural results on Chow groups. Powerful general techniques are introduced to study the many facets of K3 surfaces, including arithmetic, homological, and differential geometric aspects. In this context, the book covers Hodge structures, moduli spaces, periods, derived categories, birational techniques, Chow rings, and deformation theory. Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular, and each chapter ends with questions and open problems. Based on lectures at the advanced graduate level, this book is suitable for courses and as a reference for researchers.
Counting Surfaces
Author: Bertrand Eynard
Publisher: Springer Science & Business Media
ISBN: 3764387971
Pages: 414
Year: 2016-03-21
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The problem of enumerating maps (a map is a set of polygonal "countries" on a world of a certain topology, not necessarily the plane or the sphere) is an important problem in mathematics and physics, and it has many applications ranging from statistical physics, geometry, particle physics, telecommunications, biology, ... etc. This problem has been studied by many communities of researchers, mostly combinatorists, probabilists, and physicists. Since 1978, physicists have invented a method called "matrix models" to address that problem, and many results have been obtained. Besides, another important problem in mathematics and physics (in particular string theory), is to count Riemann surfaces. Riemann surfaces of a given topology are parametrized by a finite number of real parameters (called moduli), and the moduli space is a finite dimensional compact manifold or orbifold of complicated topology. The number of Riemann surfaces is the volume of that moduli space. More generally, an important problem in algebraic geometry is to characterize the moduli spaces, by computing not only their volumes, but also other characteristic numbers called intersection numbers. Witten's conjecture (which was first proved by Kontsevich), was the assertion that Riemann surfaces can be obtained as limits of polygonal surfaces (maps), made of a very large number of very small polygons. In other words, the number of maps in a certain limit, should give the intersection numbers of moduli spaces. In this book, we show how that limit takes place. The goal of this book is to explain the "matrix model" method, to show the main results obtained with it, and to compare it with methods used in combinatorics (bijective proofs, Tutte's equations), or algebraic geometry (Mirzakhani's recursions). The book intends to be self-contained and accessible to graduate students, and provides comprehensive proofs, several examples, and gives the general formula for the enumeration of maps on surfaces of any topology. In the end, the link with more general topics such as algebraic geometry, string theory, is discussed, and in particular a proof of the Witten-Kontsevich conjecture is provided.
Against the Day
Author: Thomas Pynchon
Publisher: Penguin
ISBN: 1101594667
Pages: 1584
Year: 2012-06-13
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A New York Times Notable Book of the Year, a Washington Post Best Book of the Year Spanning the era between the Chicago World’s Fair of 1893 and the years just after World War I, and constantly moving between locations across the globe (and to a few places not strictly speaking on the map at all), Against the Day unfolds with a phantasmagoria of characters that includes anarchists, balloonists, gamblers, drug enthusiasts, mathematicians, mad scientists, shamans, spies, and hired guns. As an era of uncertainty comes crashing down around their ears and an unpredictable future commences, these folks are mostly just trying to pursue their lives. Sometimes they manage to catch up; sometimes it’s their lives that pursue them.
Characteristic Classes. (AM-76)
Author: John Milnor, James D. Stasheff
Publisher: Princeton University Press
ISBN: 140088182X
Pages: 340
Year: 2016-03-02
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The theory of characteristic classes provides a meeting ground for the various disciplines of differential topology, differential and algebraic geometry, cohomology, and fiber bundle theory. As such, it is a fundamental and an essential tool in the study of differentiable manifolds. In this volume, the authors provide a thorough introduction to characteristic classes, with detailed studies of Stiefel-Whitney classes, Chern classes, Pontrjagin classes, and the Euler class. Three appendices cover the basics of cohomology theory and the differential forms approach to characteristic classes, and provide an account of Bernoulli numbers. Based on lecture notes of John Milnor, which first appeared at Princeton University in 1957 and have been widely studied by graduate students of topology ever since, this published version has been completely revised and corrected.
Riemannian Geometry and Geometric Analysis
Author: Jürgen Jost
Publisher: Springer
ISBN: 3319618601
Pages: 697
Year: 2017-10-13
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This established reference work continues to provide its readers with a gateway to some of the most interesting developments in contemporary geometry. It offers insight into a wide range of topics, including fundamental concepts of Riemannian geometry, such as geodesics, connections and curvature; the basic models and tools of geometric analysis, such as harmonic functions, forms, mappings, eigenvalues, the Dirac operator and the heat flow method; as well as the most important variational principles of theoretical physics, such as Yang-Mills, Ginzburg-Landau or the nonlinear sigma model of quantum field theory. The present volume connects all these topics in a systematic geometric framework. At the same time, it equips the reader with the working tools of the field and enables her or him to delve into geometric research. The 7th edition has been systematically reorganized and updated. Almost no page has been left unchanged. It also includes new material, for instance on symplectic geometry, as well as the Bishop-Gromov volume growth theorem which elucidates the geometric role of Ricci curvature. From the reviews:“This book provides a very readable introduction to Riemannian geometry and geometric analysis... With the vast development of the mathematical subject of geometric analysis, the present textbook is most welcome.” Mathematical Reviews “For readers familiar with the basics of differential geometry and some acquaintance with modern analysis, the book is reasonably self-contained. The book succeeds very well in laying out the foundations of modern Riemannian geometry and geometric analysis. It introduces a number of key techniques and provides a representative overview of the field.” Monatshefte für Mathematik
Supersymmetry for Mathematicians
Author: V. S. Varadarajan
Publisher: American Mathematical Soc.
ISBN: 0821835742
Pages: 300
Year: 2004
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Supersymmetry has been the object of study by theoretical physicists since the early 1970's. In recent years it has attracted the interest of mathematicians because of its novelty, and because of significance, both in mathematics and physics, of the main issues it raises. This book presents the foundations of supersymmetry to the mathematically minded reader in a cogent and self-contained manner. It begins with a brief introduction to the physical foundations of the theory, especially the classification of relativistic particles and their wave equations, such as the equations of Dirac and Weyl. It then continues the development of the theory of supermanifolds stressing the analogy with the Grothendieck theory of schemes. All the super linear algebra needed for the book is developed here and the basic theorems are established: differential and integral calculus in supermanifolds, Frobenius theorem, foundations of the theory of super Lie groups, and so on. A special feature of the book is the treatment in depth of the theory of spinors in all dimensions and signatures, which is the basis of all developments of supergeometry both in physics and mathematics, especially in quantum field theory and supergravity.
The Survival of a Mathematician
Author: Steven George Krantz
Publisher: American Mathematical Soc.
ISBN: 0821846299
Pages: 310
Year: 2009-01
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"One of the themes of the book is how to have a fulfilling professional life. In order to achieve this goal, Krantz discusses keeping a vigorous scholarly program going and finding new challenges, as well as dealing with the everyday tasks of research, teaching, and administration." "In short, this is a survival manual for the professional mathematician - both in academics and in industry and government agencies. It is a sequel to the author's A Mathematician's Survival Guide."--BOOK JACKET.
A Panoramic View of Riemannian Geometry
Author: Marcel Berger
Publisher: Springer Science & Business Media
ISBN: 3642182453
Pages: 824
Year: 2012-12-06
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This book introduces readers to the living topics of Riemannian Geometry and details the main results known to date. The results are stated without detailed proofs but the main ideas involved are described, affording the reader a sweeping panoramic view of almost the entirety of the field. From the reviews "The book has intrinsic value for a student as well as for an experienced geometer. Additionally, it is really a compendium in Riemannian Geometry." --MATHEMATICAL REVIEWS
Complex Analysis
Author: Eberhard Freitag, Rolf Busam
Publisher: Springer Science & Business Media
ISBN: 3540939830
Pages: 532
Year: 2009-04-28
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All needed notions are developed within the book: with the exception of fundamentals which are presented in introductory lectures, no other knowledge is assumed Provides a more in-depth introduction to the subject than other existing books in this area Over 400 exercises including hints for solutions are included
Complex Variables and Applications
Author: Brown
Publisher: McGraw-Hill Higher Education
ISBN: 0073530859
Year: 2013-08-30
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Mathematical Masterpieces
Author: Art Knoebel, Reinhard Laubenbacher, Jerry Lodder, David Pengelley
Publisher: Springer Science & Business Media
ISBN: 0387330623
Pages: 340
Year: 2007-10-16
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Intended for juniors and seniors majoring in mathematics, as well as anyone pursuing independent study, this book traces the historical development of four different mathematical concepts by presenting readers with the original sources. Each chapter showcases a masterpiece of mathematical achievement, anchored to a sequence of selected primary sources. The authors examine the interplay between the discrete and continuous, with a focus on sums of powers. They then delineate the development of algorithms by Newton, Simpson and Smale. Next they explore our modern understanding of curvature, and finally they look at the properties of prime numbers. The book includes exercises, numerous photographs, and an annotated bibliography.
Differential Analysis on Complex Manifolds
Author: R. O. Wells
Publisher: Springer Science & Business Media
ISBN: 147573946X
Pages: 262
Year: 2013-04-17
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