[PDF] A Gentle Introduction To Homological Mirror Symmetry eBook

Author : Rafaël Bocklandt
Publisher :
Page : pages
File Size : 15,79 MB
Release : 2021-09
Category :
ISBN : 9781108692458

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"This book grew out of an advanced masters course that I teach biannually at the University of Amsterdam. The course is aimed at students who are doing a masters in algebra and geometry or mathematical physics. In this course I try to give them the feeling of what homological mirror symmetry is and how it ties together many different areas of mathematics. The focus of the course is to explain the main concepts and results and to illustrate them with examples, without getting too technical. In this way the students will be better prepared to delve into the primary literature if they want to understand the theory at a deeper and more detailed level"--

Author : Raf Bocklandt
Publisher : Cambridge University Press
Page : 403 pages
File Size : 21,86 MB
Release : 2021-08-19
Category : Mathematics
ISBN : 110848350X

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Introduction to homological mirror symmetry from the point of view of representation theory, suitable for graduate students.

Author : Anton Kapustin
Publisher : Springer Science & Business Media
Page : 281 pages
File Size : 10,94 MB
Release : 2009
Category : Mathematics
ISBN : 3540680292

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An ideal reference on the mathematical aspects of quantum field theory, this volume provides a set of lectures and reviews that both introduce and representatively review the state-of-the art in the field from different perspectives.

Author : Paul Seidel
Publisher : American Mathematical Soc.
Page : 142 pages
File Size : 26,29 MB
Release : 2015-06-26
Category : Mathematics
ISBN : 1470410974

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The author proves Kontsevich's form of the mirror symmetry conjecture for (on the symplectic geometry side) a quartic surface in C .

Author : Raf Bocklandt
Publisher : Cambridge University Press
Page : 404 pages
File Size : 47,63 MB
Release : 2021-08-19
Category : Mathematics
ISBN : 1108644112

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Homological mirror symmetry has its origins in theoretical physics but is now of great interest in mathematics due to the deep connections it reveals between different areas of geometry and algebra. This book offers a self-contained and accessible introduction to the subject via the representation theory of algebras and quivers. It is suitable for graduate students and others without a great deal of background in homological algebra and modern geometry. Each part offers a different perspective on homological mirror symmetry. Part I introduces the A-infinity formalism and offers a glimpse of mirror symmetry using representations of quivers. Part II discusses various A- and B-models in mirror symmetry and their connections through toric and tropical geometry. Part III deals with mirror symmetry for Riemann surfaces. The main mathematical ideas are illustrated by means of simple examples coming mainly from the theory of surfaces, helping the reader connect theory with intuition.

Author : Ricardo Castano-Bernard
Publisher : Springer
Page : 445 pages
File Size : 46,91 MB
Release : 2014-10-07
Category : Mathematics
ISBN : 3319065149

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The relationship between Tropical Geometry and Mirror Symmetry goes back to the work of Kontsevich and Y. Soibelman (2000), who applied methods of non-archimedean geometry (in particular, tropical curves) to Homological Mirror Symmetry. In combination with the subsequent work of Mikhalkin on the “tropical” approach to Gromov-Witten theory and the work of Gross and Siebert, Tropical Geometry has now become a powerful tool. Homological Mirror Symmetry is the area of mathematics concentrated around several categorical equivalences connecting symplectic and holomorphic (or algebraic) geometry. The central ideas first appeared in the work of Maxim Kontsevich (1993). Roughly speaking, the subject can be approached in two ways: either one uses Lagrangian torus fibrations of Calabi-Yau manifolds (the so-called Strominger-Yau-Zaslow picture, further developed by Kontsevich and Soibelman) or one uses Lefschetz fibrations of symplectic manifolds (suggested by Kontsevich and further developed by Seidel). Tropical Geometry studies piecewise-linear objects which appear as “degenerations” of the corresponding algebro-geometric objects.

Author : Anton Kapustin
Publisher : Springer
Page : 272 pages
File Size : 46,41 MB
Release : 2009-08-29
Category : Science
ISBN : 9783540863748

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Homological Mirror Symmetry, the study of dualities of certain quantum field theories in a mathematically rigorous form, has developed into a flourishing subject on its own over the past years. The present volume bridges a gap in the literature by providing a set of lectures and reviews that both introduce and representatively review the state-of-the art in the field from different perspectives. With contributions by K. Fukaya, M. Herbst, K. Hori, M. Huang, A. Kapustin, L. Katzarkov, A. Klemm, M. Kontsevich, D. Page, S. Quackenbush, E. Sharpe, P. Seidel, I. Smith and Y. Soibelman, this volume will be a reference on the topic for everyone starting to work or actively working on mathematical aspects of quantum field theory.

Author : Andrew Allan Port
Publisher :
Page : pages
File Size : 24,63 MB
Release : 2012
Category :
ISBN : 9781267969439

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Here we carefully construct an equivalence between the derived category of coherent sheaves on an elliptic curve and a version of the Fukaya category on its mirror. This is the most accessible case of homological mirror symmetry. We also provide introductory background on the general Calabi-Yau case of The Homological Mirror Symmetry Conjecture.

Author : Stephanie Frank Singer
Publisher : Springer Science & Business Media
Page : 201 pages
File Size : 11,19 MB
Release : 2012-12-06
Category : Mathematics
ISBN : 1461201896

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"And what is the use," thought Alice, "of a book without pictures or conversations in it?" -Lewis Carroll This book is written for modem undergraduate students - not the ideal stu dents that mathematics professors wish for (and who occasionally grace our campuses), but the students like many the author has taught: talented but ap preciating review and reinforcement of past course work; willing to work hard, but demanding context and motivation for the mathematics they are learning. To suit this audience, the author eschews density of topics and efficiency of presentation in favor of a gentler tone, a coherent story, digressions on mathe maticians, physicists and their notations, simple examples worked out in detail, and reinforcement of the basics. Dense and efficient texts play a crucial role in the education of budding (and budded) mathematicians and physicists. This book does not presume to improve on the classics in that genre. Rather, it aims to provide those classics with a large new generation of appreciative readers. This text introduces some basic constructs of modern symplectic geometry in the context of an old celestial mechanics problem, the two-body problem. We present the derivation of Kepler's laws of planetary motion from Newton's laws of gravitation, first in the style of an undergraduate physics course, and x Preface then again in the language of symplectic geometry. No previous exposure to symplectic geometry is required: we introduce and illustrate all necessary con structs.

Author : Shing-Tung Yau
Publisher : American Mathematical Soc.
Page : 460 pages
File Size : 15,99 MB
Release : 1998
Category : Mathematics
ISBN : 082182743X

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Vol. 1 represents a new ed. of papers which were originally published in Essays on mirror manifolds (1992); supplemented by the additional volume: Mirror symmetry 2 which presents papers by both physicists and mathematicians. Mirror symmetry 1 (the 1st volume) constitutes the proceedings of the Mathematical Sciences Research Institute Workshop of 1991.