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Book cover
  • Textbook
  • © 2007

Tropical Algebraic Geometry

Birkhäuser

Authors:

  1. Ilia Itenberg

    1. IRMA, Université Louis Pasteur, Strasbourg Cedex, France

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  2. Grigory Mikhalkin

    1. Department of Mathematics, University of Toronto, Toronto, Canada

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  3. Eugenii Shustin

    1. School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel

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  • Contains polished notes of three introductory courses to tropical geometry
  • Based on a seminar at the Mathematical Research Center in Oberwolfach in October 2004
  • Includes supplementary material: sn.pub/extras

Part of the book series: Oberwolfach Seminars (OWS, volume 35)

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Table of contents (3 chapters)

  1. Front Matter

    Pages i-viii
    PDF

  5. Back Matter

    Pages 97-103
    PDF
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About this book

This book is based on the lectures given at the Oberwolfach Seminar on Tropical Algebraic Geometry in October 2004. Tropical Geometry ?rst appeared as a subject of its own in 2002, while its roots can be traced back at least to Bergman’s work [1] on logarithmic limit sets. Tropical Geometry is now a rapidly developing area of mathematics. It is int- twined with algebraic and symplectic geometry, geometric combinatorics, in- grablesystems, and statistical physics. Tropical Geometry can be viewed as a sort of algebraic geometry with the underlying algebra based on the so-called tropical numbers. The tropicalnumbers (the term “tropical” comesfrom computer science and commemorates Brazil, in particular a contribution of the Brazilian school to the language recognition problem) are the real numbers enhanced with negative in?nity and equipped with two arithmetic operations called tropical addition and tropical multiplication. The tropical addition is the operation of taking the m- imum. The tropical multiplication is the conventional addition. These operations are commutative, associative and satisfy the distribution law. It turns out that such tropical algebra describes some meaningful geometric objects, namely, the Tropical Varieties. From the topological point of view the tropical varieties are piecewise-linearpolyhedral complexes equipped with a particular geometric str- ture coming from tropical algebra. From the point of view of complex geometry this geometric structure is the worst possible degeneration of complex structure on a manifold.
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Keywords

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Authors and Affiliations

  • IRMA, Université Louis Pasteur, Strasbourg Cedex, France

    Ilia Itenberg

  • Department of Mathematics, University of Toronto, Toronto, Canada

    Grigory Mikhalkin

  • School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel

    Eugenii Shustin

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Bibliographic Information

  • Book Title: Tropical Algebraic Geometry

  • Authors: Ilia Itenberg, Grigory Mikhalkin, Eugenii Shustin

  • Series Title: Oberwolfach Seminars

  • DOI: https://doi.org/10.1007/978-3-7643-8310-7

  • Publisher: Birkhäuser Basel

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

  • Copyright Information: Birkhäuser Basel 2007

  • eBook ISBN: 978-3-7643-8310-7Published: 08 January 2008

  • Series ISSN: 1661-237X

  • Series E-ISSN: 2296-5041

  • Edition Number: 1

  • Number of Pages: VIII, 104

  • Number of Illustrations: 30 b/w illustrations

  • Topics: Algebraic Geometry

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