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Homological Mirror Symmetry and Tropical Geometry

  • Book
  • © 2014

Overview

Editors:
  1. Ricardo Castano-Bernard

    1. Mathematics Department, Kansas State University, Manhattan, USA

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  2. Fabrizio Catanese

    1. Mathematisches Institut, Universität Bayreuth, Bayreuth, Germany

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  3. Maxim Kontsevich

    1. Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France

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  4. Tony Pantev

    1. Mathematics Department, University of Pennsylvania, Philadelphia, USA

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  5. Yan Soibelman

    1. Department of Mathematics, Kansas State University, Manhattan, USA

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  6. Ilia Zharkov

    1. Department of Mathematics, Kansas State University, Manhattan, USA

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Part of the book series: Lecture Notes of the Unione Matematica Italiana (UMILN, volume 15)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xi
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  2. Moduli Stacks of Bundles on Local Surfaces

    • Oren Ben-Bassat, Elizabeth Gasparim
    Pages 1-32

  3. An Orbit Construction of Phantoms, Orlov Spectra, and Knörrer Periodicity

    • David Favero, Fabian Haiden, Ludmil Katzarkov
    Pages 33-42

  4. Microlocal Theory of Sheaves and Tamarkin’s Non Displaceability Theorem

    • Stéphane Guillermou, Pierre Schapira
    Pages 43-85

  5. A-Polynomial, B-Model, and Quantization

    • Sergei Gukov, Piotr Sułkowski
    Pages 87-151

  6. Spherical Hall Algebra of \(\overline{\text{Spec }(\mathbb{Z})}\)

    • M. Kapranov, O. Schiffmann, E. Vasserot
    Pages 153-196

  7. Wall-Crossing Structures in Donaldson–Thomas Invariants, Integrable Systems and Mirror Symmetry

    • Maxim Kontsevich, Yan Soibelman
    Pages 197-308

  8. Tropical Eigenwave and Intermediate Jacobians

    • Grigory Mikhalkin, Ilia Zharkov
    Pages 309-349

  9. Notes on a New Construction of Hyperkahler Metrics

    • Andrew Neitzke
    Pages 351-375

  10. Mirror Duality of Landau–Ginzburg Models via Discrete Legendre Transforms

    • Helge Ruddat
    Pages 377-406

  11. Mirror Symmetry in Dimension 1 and Fourier–Mukai Transforms

    • Nicolò Sibilla
    Pages 407-428

  12. The Very Good Property for Moduli of Parabolic Bundles and the Additive Deligne–Simpson Problem

    • Alexander Soibelman
    Pages 429-436

  13. Back Matter

    Pages 437-437
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Keywords

About this book

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.

Editors and Affiliations

  • Mathematics Department, Kansas State University, Manhattan, USA

    Ricardo Castano-Bernard

  • Mathematisches Institut, Universität Bayreuth, Bayreuth, Germany

    Fabrizio Catanese

  • Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France

    Maxim Kontsevich

  • Mathematics Department, University of Pennsylvania, Philadelphia, USA

    Tony Pantev

  • Department of Mathematics, Kansas State University, Manhattan, USA

    Yan Soibelman, Ilia Zharkov

Bibliographic Information

  • Book Title: Homological Mirror Symmetry and Tropical Geometry

  • Editors: Ricardo Castano-Bernard, Fabrizio Catanese, Maxim Kontsevich, Tony Pantev, Yan Soibelman, Ilia Zharkov

  • Series Title: Lecture Notes of the Unione Matematica Italiana

  • DOI: https://doi.org/10.1007/978-3-319-06514-4

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing Switzerland 2014

  • Softcover ISBN: 978-3-319-06513-7Published: 16 October 2014

  • eBook ISBN: 978-3-319-06514-4Published: 07 October 2014

  • Series ISSN: 1862-9113

  • Series E-ISSN: 1862-9121

  • Edition Number: 1

  • Number of Pages: XI, 436

  • Number of Illustrations: 25 b/w illustrations, 18 illustrations in colour

  • Topics: Algebraic Geometry, Differential Geometry

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