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Homological Mirror Symmetry
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Book Synopsis A Gentle Introduction to Homological Mirror Symmetry by : Raf Bocklandt
Download or read book A Gentle Introduction to Homological Mirror Symmetry written by Raf Bocklandt and published by Cambridge University Press. This book was released on 2021-08-19 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Homological Mirror Symmetry by : Anton Kapustin
Download or read book Homological Mirror Symmetry written by Anton Kapustin and published by Springer Science & Business Media. This book was released on 2009 with total page 281 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Dirichlet Branes and Mirror Symmetry by : Paul Aspinwall
Download or read book Dirichlet Branes and Mirror Symmetry written by Paul Aspinwall and published by American Mathematical Soc.. This book was released on 2009 with total page 698 pages. Available in PDF, EPUB and Kindle. Book excerpt: Research in string theory has generated a rich interaction with algebraic geometry, with exciting work that includes the Strominger-Yau-Zaslow conjecture. This monograph builds on lectures at the 2002 Clay School on Geometry and String Theory that sought to bridge the gap between the languages of string theory and algebraic geometry.
Book Synopsis Homological Mirror Symmetry and Tropical Geometry by : Ricardo Castano-Bernard
Download or read book Homological Mirror Symmetry and Tropical Geometry written by Ricardo Castano-Bernard and published by Springer. This book was released on 2014-10-07 with total page 445 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Mirror Symmetry written by Kentaro Hori and published by American Mathematical Soc.. This book was released on 2003 with total page 954 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thorough and detailed exposition is the result of an intensive month-long course on mirror symmetry sponsored by the Clay Mathematics Institute. It develops mirror symmetry from both mathematical and physical perspectives with the aim of furthering interaction between the two fields. The material will be particularly useful for mathematicians and physicists who wish to advance their understanding across both disciplines. Mirror symmetry is a phenomenon arising in string theory in which two very different manifolds give rise to equivalent physics. Such a correspondence has significant mathematical consequences, the most familiar of which involves the enumeration of holomorphic curves inside complex manifolds by solving differential equations obtained from a ``mirror'' geometry. The inclusion of D-brane states in the equivalence has led to further conjectures involving calibrated submanifolds of the mirror pairs and new (conjectural) invariants of complex manifolds: the Gopakumar-Vafa invariants. This book gives a single, cohesive treatment of mirror symmetry. Parts 1 and 2 develop the necessary mathematical and physical background from ``scratch''. The treatment is focused, developing only the material most necessary for the task. In Parts 3 and 4 the physical and mathematical proofs of mirror symmetry are given. From the physics side, this means demonstrating that two different physical theories give isomorphic physics. Each physical theory can be described geometrically, and thus mirror symmetry gives rise to a ``pairing'' of geometries. The proof involves applying $R\leftrightarrow 1/R$ circle duality to the phases of the fields in the gauged linear sigma model. The mathematics proof develops Gromov-Witten theory in the algebraic setting, beginning with the moduli spaces of curves and maps, and uses localization techniques to show that certain hypergeometric functions encode the Gromov-Witten invariants in genus zero, as is predicted by mirror symmetry. Part 5 is devoted to advanced topi This one-of-a-kind book is suitable for graduate students and research mathematicians interested in mathematics and mathematical and theoretical physics.
Book Synopsis A Gentle Introduction to Homological Mirror Symmetry by : Raf Bocklandt
Download or read book A Gentle Introduction to Homological Mirror Symmetry written by Raf Bocklandt and published by Cambridge University Press. This book was released on 2021-08-19 with total page 403 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduction to homological mirror symmetry from the point of view of representation theory, suitable for graduate students.
Book Synopsis Homological Mirror Symmetry for the Quartic Surface by : Paul Seidel
Download or read book Homological Mirror Symmetry for the Quartic Surface written by Paul Seidel and published by American Mathematical Soc.. This book was released on 2015-06-26 with total page 236 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author proves Kontsevich's form of the mirror symmetry conjecture for (on the symplectic geometry side) a quartic surface in C .
Book Synopsis Symplectic Geometry and Mirror Symmetry by : Kodŭng Kwahagwŏn (Korea). International Conference
Download or read book Symplectic Geometry and Mirror Symmetry written by Kodŭng Kwahagwŏn (Korea). International Conference and published by World Scientific. This book was released on 2001 with total page 940 pages. Available in PDF, EPUB and Kindle. Book excerpt: In 1993, M. Kontsevich proposed a conceptual framework for explaining the phenomenon of mirror symmetry. Mirror symmetry had been discovered by physicists in string theory as a duality between families of three-dimensional Calabi–Yau manifolds. Kontsevich's proposal uses Fukaya's construction of the A∞-category of Lagrangian submanifolds on the symplectic side and the derived category of coherent sheaves on the complex side. The theory of mirror symmetry was further enhanced by physicists in the language of D-branes and also by Strominger–Yau–Zaslow in the geometric set-up of (special) Lagrangian torus fibrations. It rapidly expanded its scope across from geometry, topology, algebra to physics. In this volume, leading experts in the field explore recent developments in relation to homological mirror symmetry, Floer theory, D-branes and Gromov–Witten invariants. Kontsevich-Soibelman describe their solution to the mirror conjecture on the abelian variety based on the deformation theory of A∞-categories, and Ohta describes recent work on the Lagrangian intersection Floer theory by Fukaya–Oh–Ohta–Ono which takes an important step towards a rigorous construction of the A∞-category. There follow a number of contributions on the homological mirror symmetry, D-branes and the Gromov–Witten invariants, e.g. Getzler shows how the Toda conjecture follows from recent work of Givental, Okounkov and Pandharipande. This volume provides a timely presentation of the important developments of recent years in this rapidly growing field.
Book Synopsis Tropical Geometry and Mirror Symmetry by : Mark Gross
Download or read book Tropical Geometry and Mirror Symmetry written by Mark Gross and published by American Mathematical Soc.. This book was released on 2011-01-20 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt: Tropical geometry provides an explanation for the remarkable power of mirror symmetry to connect complex and symplectic geometry. The main theme of this book is the interplay between tropical geometry and mirror symmetry, culminating in a description of the recent work of Gross and Siebert using log geometry to understand how the tropical world relates the A- and B-models in mirror symmetry. The text starts with a detailed introduction to the notions of tropical curves and manifolds, and then gives a thorough description of both sides of mirror symmetry for projective space, bringing together material which so far can only be found scattered throughout the literature. Next follows an introduction to the log geometry of Fontaine-Illusie and Kato, as needed for Nishinou and Siebert's proof of Mikhalkin's tropical curve counting formulas. This latter proof is given in the fourth chapter. The fifth chapter considers the mirror, B-model side, giving recent results of the author showing how tropical geometry can be used to evaluate the oscillatory integrals appearing. The final chapter surveys reconstruction results of the author and Siebert for ``integral tropical manifolds.'' A complete version of the argument is given in two dimensions.
Book Synopsis Proceedings of the International Congress of Mathematicians by : S.D. Chatterji
Download or read book Proceedings of the International Congress of Mathematicians written by S.D. Chatterji and published by Birkhäuser. This book was released on 2012-12-06 with total page 1669 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since the first ICM was held in Zürich in 1897, it has become the pinnacle of mathematical gatherings. It aims at giving an overview of the current state of different branches of mathematics and its applications as well as an insight into the treatment of special problems of exceptional importance. The proceedings of the ICMs have provided a rich chronology of mathematical development in all its branches and a unique documentation of contemporary research. They form an indispensable part of every mathematical library. The Proceedings of the International Congress of Mathematicians 1994, held in Zürich from August 3rd to 11th, 1994, are published in two volumes. Volume I contains an account of the organization of the Congress, the list of ordinary members, the reports on the work of the Fields Medalists and the Nevanlinna Prize Winner, the plenary one-hour addresses, and the invited addresses presented at Section Meetings 1 - 6. Volume II contains the invited address for Section Meetings 7 - 19. A complete author index is included in both volumes. '...the content of these impressive two volumes sheds a certain light on the present state of mathematical sciences and anybody doing research in mathematics should look carefully at these Proceedings. For young people beginning research, this is even more important, so these are a must for any serious mathematics library. The graphical presentation is, as always with Birkhäuser, excellent....' (Revue Roumaine de Mathematiques pures et Appliquées)
Book Synopsis Mirror Symmetry and Algebraic Geometry by : David A. Cox
Download or read book Mirror Symmetry and Algebraic Geometry written by David A. Cox and published by American Mathematical Soc.. This book was released on 1999 with total page 469 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Fukaya Categories and Picard-Lefschetz Theory by : Paul Seidel
Download or read book Fukaya Categories and Picard-Lefschetz Theory written by Paul Seidel and published by European Mathematical Society. This book was released on 2008 with total page 340 pages. Available in PDF, EPUB and Kindle. Book excerpt: The central objects in the book are Lagrangian submanifolds and their invariants, such as Floer homology and its multiplicative structures, which together constitute the Fukaya category. The relevant aspects of pseudo-holomorphic curve theory are covered in some detail, and there is also a self-contained account of the necessary homological algebra. Generally, the emphasis is on simplicity rather than generality. The last part discusses applications to Lefschetz fibrations and contains many previously unpublished results. The book will be of interest to graduate students and researchers in symplectic geometry and mirror symmetry.
Book Synopsis Handbook for Mirror Symmetry of Calabi-yau and Fano Manifolds by : Shing-tung Shing-tung Yau
Download or read book Handbook for Mirror Symmetry of Calabi-yau and Fano Manifolds written by Shing-tung Shing-tung Yau and published by . This book was released on with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Mirror Symmetry I by : Shing-Tung Yau
Download or read book Mirror Symmetry I written by Shing-Tung Yau and published by American Mathematical Soc.. This book was released on 1998 with total page 460 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Homological Mirror Symmetry by : Anton Kapustin
Download or read book Homological Mirror Symmetry written by Anton Kapustin and published by Springer. This book was released on 2009-08-29 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Riemannian Holonomy Groups and Calibrated Geometry by : Dominic D. Joyce
Download or read book Riemannian Holonomy Groups and Calibrated Geometry written by Dominic D. Joyce and published by Oxford University Press. This book was released on 2007 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: Covering an exciting and active area of research at the crossroads of several different fields in mathematics and physics, and drawing on the author's previous work, this text has been written to explain the advanced mathematics involved simply and clearly to graduate students in both disciplines.
Book Synopsis Homological Theory of Representations by : Henning Krause
Download or read book Homological Theory of Representations written by Henning Krause and published by Cambridge University Press. This book was released on 2021-11-18 with total page 518 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modern developments in representation theory rely heavily on homological methods. This book for advanced graduate students and researchers introduces these methods from their foundations up and discusses several landmark results that illustrate their power and beauty. Categorical foundations include abelian and derived categories, with an emphasis on localisation, spectra, and purity. The representation theoretic focus is on module categories of Artin algebras, with discussions of the representation theory of finite groups and finite quivers. Also covered are Gorenstein and quasi-hereditary algebras, including Schur algebras, which model polynomial representations of general linear groups, and the Morita theory of derived categories via tilting objects. The final part is devoted to a systematic introduction to the theory of purity for locally finitely presented categories, covering pure-injectives, definable subcategories, and Ziegler spectra. With its clear, detailed exposition of important topics in modern representation theory, many of which were unavailable in one volume until now, it deserves a place in every representation theorist's library.
