Geometry and Quantum Field Theory

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Author: Daniel S. Freed,Karen K. Uhlenbeck

Publisher: American Mathematical Soc.

ISBN: 0821804006

Category: Science

Page: 459

View: 4144

Exploring topics from classical and quantum mechanics and field theory, this book is based on lectures presented in the Graduate Summer School at the Regional Geometry Institute in Park City, Utah, in 1991. The chapter by Bryant treats Lie groups and symplectic geometry, examining not only the connection with mechanics but also the application to differential equations and the recent work of the Gromov school. Rabin's discussion of quantum mechanics and field theory is specifically aimed at mathematicians. Alvarez describes the application of supersymmetry to prove the Atiyah-Singer index theorem, touching on ideas that also underlie more complicated applications of supersymmetry. Quinn's account of the topological quantum field theory captures the formal aspects of the path integral and shows how these ideas can influence branches of mathematics which at first glance may not seem connected. Presenting material at a level between that of textbooks and research papers, much of the book would provide excellent material for graduate courses. The book provides an entree into a field that promises to remain exciting and important for years to come.

Quantum Field Theory, Supersymmetry, and Enumerative Geometry

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Author: Daniel S. Freed,David R. Morrison,Isadore Manuel Singer

Publisher: American Mathematical Soc.

ISBN: 0821834312

Category: Mathematics

Page: 285

View: 8622

Each summer the IAS/Park City Mathematics Institute Graduate Summer School gathers some of the best researchers and educators in a particular field to present diverse sets of lectures. This volume presents three weeks of lectures given at the Summer School on Quantum Field Theory, Super symmetry, and Enumerative Geometry, three very active research areas in mathematics and theoretical physics. With this volume, the Park City Mathematics Institute returns to the general topic of the first institute: the interplay between quantum field theory and mathematics.Two major themes at this institute were super symmetry and algebraic geometry, particularly enumerative geometry. The volume contains two lecture series on methods of enumerative geometry that have their roots in QFT. The first series covers the Schubert calculus and quantum cohomology. The second discusses methods from algebraic geometry for computing Gromov-Witten invariants. There are also three sets of lectures of a more introductory nature: an overview of classical field theory and super symmetry, an introduction to supermanifolds, and an introduction to general relativity. This volume is recommended for independent study and is suitable for graduate students and researchers interested in geometry and physics.

Mirror Symmetry and Algebraic Geometry

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Author: David A. Cox,Sheldon Katz

Publisher: American Mathematical Soc.

ISBN: 082182127X

Category: Mathematics

Page: 469

View: 7095

Mathematicians wanting to get into the field ... will find a very well written and encyclopaedic account of the mathematics which was needed in, and was developed from, what now might be termed classical mirror symmetry. --Bulletin of the LMS The book is highly recommended for everyone who wants to learn about the fascinating recent interplay between physics and mathematics. --Mathematical Reviews Mirror symmetry began when theoretical physicists made some astonishing predictions about rational curves on quintic hypersurfaces in four-dimensional projective space. Understanding the mathematics behind these predictions has been a substantial challenge. This book is a completely comprehensive monograph on mirror symmetry, covering the original observations by the physicists through the most recent progress made to date. Subjects discussed include toric varieties, Hodge theory, Kahler geometry, moduli of stable maps, Calabi-Yau manifolds, quantum cohomology, Gromov-Witten invariants, and the mirror theorem.

Symplectic Geometry and Topology

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Author: Yakov Eliashberg,Lisa M. Traynor

Publisher: American Mathematical Soc.

ISBN: 9780821886892

Category: Mathematics

Page: 430

View: 9920

Symplectic geometry has its origins as a geometric language for classical mechanics. But it has recently exploded into an independent field interconnected with many other areas of mathematics and physics. The goal of the IAS/Park City Mathematics Institute Graduate Summer School on Symplectic Geometry and Topology was to give an intensive introduction to these exciting areas of current research. Included in this proceedings are lecture notes from the following courses: Introductionto Symplectic Topology by D. McDuff; Holomorphic Curves and Dynamics in Dimension Three by H. Hofer; An Introduction to the Seiberg-Witten Equations on Symplectic Manifolds by C. Taubes; Lectures on Floer Homology by D. Salamon; A Tutorial on Quantum Cohomology by A. Givental; Euler Characteristicsand Lagrangian Intersections by R. MacPherson; Hamiltonian Group Actions and Symplectic Reduction by L. Jeffrey; and Mechanics: Symmetry and Dynamics by J. Marsden. Information for our distributors: Titles in this series are copublished with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

Encyclopedia of mathematical physics

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Author: Sheung Tsun Tsou

Publisher: Academic Pr

ISBN: 9780125126601

Category: Science

Page: 3500

View: 5757

The Encyclopedia of Mathematical Physics provides a complete resource for researchers, students and lecturers with an interest in mathematical physics. It enables readers to access basic information on topics peripheral to their own areas, to provide a repository of the core information in the area that can be used to refresh the researcher's own memory banks, and aid teachers in directing students to entries relevant to their course-work. The Encyclopedia does contain information that has been distilled, organised and presented as a complete reference tool to the user and a landmark to the body of knowledge that has accumulated in this domain. It also is a stimulus for new researchers working in mathematical physics or in areas using the methods originated from work in mathematical physics by providing them with focused high quality background information. * First comprehensive interdisciplinary coverage * Mathematical Physics explained to stimulate new developments and foster new applications of its methods to other fields * Written by an international group of experts * Contains several undergraduate-level introductory articles to facilitate acquisition of new expertise * Thematic index and extensive cross-referencing to provide easy access and quick search functionality * Also available online with active linking.

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Numerical mathematics

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Author: N.A

Publisher: N.A

ISBN: N.A

Category: Electronic journals

Page: N.A

View: 6119

Nonlinear Partial Differential Equations in Differential Geometry

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Author: Robert Hardt

Publisher: American Mathematical Soc.

ISBN: 9780821804315

Category: Mathematics

Page: 339

View: 1969

What distinguishes differential geometry in the last half of the twentieth century from its earlier history is the use of nonlinear partial differential equations in the study of curved manifolds, submanifolds, mapping problems, and function theory on manifolds, among other topics. The differential equations appear as tools and as objects of study, with analytic and geometric advances fueling each other in the current explosion of progress in this area of geometry in the last twenty years. This book contains lecture notes of minicourses at the Regional Geometry Institute at Park City, Utah, in July 1992. Presented here are surveys of breaking developments in a number of areas of nonlinear partial differential equations in differential geometry. The authors of the articles are not only excellent expositors, but are also leaders in this field of research. All of the articles provide in-depth treatment of the topics and require few prerequisites and less background than current research articles.

Quantum Symmetries/Symetries Quantiques

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Author: Alain Connes,K. Gawędzki,Jean Zinn-Justin

Publisher: North-Holland

ISBN: 9780444828675

Category: Science

Page: 990

View: 9059

One of the greatest challenges that theoretical physics faces at the end of the century is to blend together the two revolutions of the beginning of the century, namely general relativity and quantum mechanics. One message that has become clear from the start and is common to both revolutions is that there is no limit to the level of sophistication of mathematics that will become essential to physics. This book comprises the lecture notes of the 1995 Les Houches Summer School. The aim of the school was to cover a wide range of areas, from theoretical physics to abstract mathematics, that are relevant in the search of a quantum theory of gravity. The lectures provide a systematic introduction to topological and conformal field theories, supersymmetry and super Yang-Mills theories, string theory and superstring dualities, integrable lattice models and quantum groups, non commutative geometry and the theory of diffeomorphism groups. It is directed at graduate students and researchers in theoretical physics and mathematics.

Enumerative Geometry and String Theory

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Author: Sheldon Katz

Publisher: American Mathematical Soc.

ISBN: 0821836870

Category: Mathematics

Page: 206

View: 6850

Perhaps the most famous example of how ideas from modern physics have revolutionized mathematics is the way string theory has led to an overhaul of enumerative geometry, an area of mathematics that started in the eighteen hundreds. Century-old problems of enumerating geometric configurations have now been solved using new and deep mathematical techniques inspired by physics!The book begins with an insightful introduction to enumerative geometry. From there, the goal becomes explaining the more advanced elements of enumerative algebraic geometry. Along the way, there are some crash courses on intermediate topics which are essential tools for the student of modern mathematics, such as cohomology and other topics in geometry. The physics content assumes nothing beyond a first undergraduate course. The focus is on explaining the action principle in physics, the idea of string theory, and how these directly lead to questions in geometry. Once these topics are in place, the connection between physics and enumerative geometry is made with the introduction of topological quantum field theory and quantum cohomology.

Differentialgeometrie, Topologie und Physik

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Author: Mikio Nakahara

Publisher: Springer-Verlag

ISBN: 3662453002

Category: Science

Page: 597

View: 652

Differentialgeometrie und Topologie sind wichtige Werkzeuge für die Theoretische Physik. Insbesondere finden sie Anwendung in den Gebieten der Astrophysik, der Teilchen- und Festkörperphysik. Das vorliegende beliebte Buch, das nun erstmals ins Deutsche übersetzt wurde, ist eine ideale Einführung für Masterstudenten und Forscher im Bereich der theoretischen und mathematischen Physik. - Im ersten Kapitel bietet das Buch einen Überblick über die Pfadintegralmethode und Eichtheorien. - Kapitel 2 beschäftigt sich mit den mathematischen Grundlagen von Abbildungen, Vektorräumen und der Topologie. - Die folgenden Kapitel beschäftigen sich mit fortgeschritteneren Konzepten der Geometrie und Topologie und diskutieren auch deren Anwendungen im Bereich der Flüssigkristalle, bei suprafluidem Helium, in der ART und der bosonischen Stringtheorie. - Daran anschließend findet eine Zusammenführung von Geometrie und Topologie statt: es geht um Faserbündel, characteristische Klassen und Indextheoreme (u.a. in Anwendung auf die supersymmetrische Quantenmechanik). - Die letzten beiden Kapitel widmen sich der spannendsten Anwendung von Geometrie und Topologie in der modernen Physik, nämlich den Eichfeldtheorien und der Analyse der Polakov'schen bosonischen Stringtheorie aus einer gemetrischen Perspektive. Mikio Nakahara studierte an der Universität Kyoto und am King’s in London Physik sowie klassische und Quantengravitationstheorie. Heute ist er Physikprofessor an der Kinki-Universität in Osaka (Japan), wo er u. a. über topologische Quantencomputer forscht. Diese Buch entstand aus einer Vorlesung, die er während Forschungsaufenthalten an der University of Sussex und an der Helsinki University of Sussex gehalten hat.

Geometry of Moduli Spaces and Representation Theory

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Author: Roman Bezrukavnikov,Alexander Braverman,Zhiwei Yun

Publisher: American Mathematical Soc.

ISBN: 1470435748

Category: Algebraic varieties

Page: 436

View: 6203

This book is based on lectures given at the Graduate Summer School of the 2015 Park City Mathematics Institute program “Geometry of moduli spaces and representation theory”, and is devoted to several interrelated topics in algebraic geometry, topology of algebraic varieties, and representation theory. Geometric representation theory is a young but fast developing research area at the intersection of these subjects. An early profound achievement was the famous conjecture by Kazhdan–Lusztig about characters of highest weight modules over a complex semi-simple Lie algebra, and its subsequent proof by Beilinson-Bernstein and Brylinski-Kashiwara. Two remarkable features of this proof have inspired much of subsequent development: intricate algebraic data turned out to be encoded in topological invariants of singular geometric spaces, while proving this fact required deep general theorems from algebraic geometry. Another focus of the program was enumerative algebraic geometry. Recent progress showed the role of Lie theoretic structures in problems such as calculation of quantum cohomology, K-theory, etc. Although the motivation and technical background of these constructions is quite different from that of geometric Langlands duality, both theories deal with topological invariants of moduli spaces of maps from a target of complex dimension one. Thus they are at least heuristically related, while several recent works indicate possible strong technical connections. The main goal of this collection of notes is to provide young researchers and experts alike with an introduction to these areas of active research and promote interaction between the two related directions.