Tropical Geometry And Mirror Symmetry

Author : Ricardo Castano-Bernard,Fabrizio Catanese,Maxim Kontsevich,Tony Pantev,Yan Soibelman,Ilia Zharkov
Publisher : Springer
Page : 445 pages
File Size : 47,7 Mb
Release : 2014-10-07
Category : Mathematics
ISBN : 9783319065144

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Homological Mirror Symmetry and Tropical Geometry by Ricardo Castano-Bernard,Fabrizio Catanese,Maxim Kontsevich,Tony Pantev,Yan Soibelman,Ilia Zharkov Pdf

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 : Mark Gross
Publisher : American Mathematical Soc.
Page : 338 pages
File Size : 42,9 Mb
Release : 2011-01-20
Category : Mathematics
ISBN : 9780821852323

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Tropical Geometry and Mirror Symmetry by Mark Gross Pdf

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.

Author : Ricardo Castaño-Bernard,Yan S. Soibelman,Ilia Zharkov
Publisher : American Mathematical Soc.
Page : 184 pages
File Size : 52,5 Mb
Release : 2010
Category : Science
ISBN : 9780821858516

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Mirror Symmetry and Tropical Geometry by Ricardo Castaño-Bernard,Yan S. Soibelman,Ilia Zharkov Pdf

This volume contains contributions from the NSF-CBMS Conference on Tropical Geometry and Mirror Symmetry, which was held from December 13-17, 2008 at Kansas State University in Manhattan, Kansas. It gives an excellent picture of numerous connections of mirror symmetry with other areas of mathematics (especially with algebraic and symplectic geometry) as well as with other areas of mathematical physics. The techniques and methods used by the authors of the volume are at the frontier of this very active area of research.

Author : Ricardo Castaño-Bernard,Yan S. Soibelman,Ilia Zharkov
Publisher : American Mathematical Soc.
Page : 184 pages
File Size : 49,8 Mb
Release : 2010
Category : Algebraic varieties
ISBN : 9780821848845

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Mirror Symmetry and Tropical Geometry by Ricardo Castaño-Bernard,Yan S. Soibelman,Ilia Zharkov Pdf

This volume contains contributions from the NSF-CBMS Conference on Tropical Geometry and Mirror Symmetry, which was held from December 13-17, 2008 at Kansas State University in Manhattan, Kansas. --

Author : David A. Cox,Sheldon Katz
Publisher : American Mathematical Soc.
Page : 469 pages
File Size : 55,7 Mb
Release : 1999
Category : Mathematics
ISBN : 9780821821275

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Mirror Symmetry and Algebraic Geometry by David A. Cox,Sheldon Katz Pdf

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.

Author : David A. Cox,Sheldon Katz
Publisher : American Mathematical Society(RI)
Page : 0 pages
File Size : 45,8 Mb
Release : 1999
Category : Geometry, Algebraic
ISBN : 0821810596

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Mirror Symmetry and Algebraic Geometry by David A. Cox,Sheldon Katz Pdf

This text presents a comprehensive monograph on mirror symmetry, covering the original observations by the physicists through the most recent progress made up to date. Subjects discussed include toric varieties, Hodge theory, Kahler geometry, moduli of stable maps and Calabi-Yau manifolds.

Author : Ricardo Castaño-Bernard,Yan S. Soibelman,Ilia Zharkov
Publisher : Unknown
Page : 168 pages
File Size : 50,8 Mb
Release : 2010
Category : Electronic
ISBN : OCLC:804925824

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Mirror Symmetry and Tropical Geometry by Ricardo Castaño-Bernard,Yan S. Soibelman,Ilia Zharkov Pdf

Author : Diane Maclagan,Bernd Sturmfels
Publisher : American Mathematical Society
Page : 363 pages
File Size : 43,6 Mb
Release : 2021-12-13
Category : Mathematics
ISBN : 9781470468569

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Introduction to Tropical Geometry by Diane Maclagan,Bernd Sturmfels Pdf

Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of information about their classical counterparts. Tropical geometry is a young subject that has undergone a rapid development since the beginning of the 21st century. While establishing itself as an area in its own right, deep connections have been made to many branches of pure and applied mathematics. This book offers a self-contained introduction to tropical geometry, suitable as a course text for beginning graduate students. Proofs are provided for the main results, such as the Fundamental Theorem and the Structure Theorem. Numerous examples and explicit computations illustrate the main concepts. Each of the six chapters concludes with problems that will help the readers to practice their tropical skills, and to gain access to the research literature. This wonderful book will appeal to students and researchers of all stripes: it begins at an undergraduate level and ends with deep connections to toric varieties, compactifications, and degenerations. In between, the authors provide the first complete proofs in book form of many fundamental results in the subject. The pages are sprinkled with illuminating examples, applications, and exercises, and the writing is lucid and meticulous throughout. It is that rare kind of book which will be used equally as an introductory text by students and as a reference for experts. —Matt Baker, Georgia Institute of Technology Tropical geometry is an exciting new field, which requires tools from various parts of mathematics and has connections with many areas. A short definition is given by Maclagan and Sturmfels: “Tropical geometry is a marriage between algebraic and polyhedral geometry”. This wonderful book is a pleasant and rewarding journey through different landscapes, inviting the readers from a day at a beach to the hills of modern algebraic geometry. The authors present building blocks, examples and exercises as well as recent results in tropical geometry, with ingredients from algebra, combinatorics, symbolic computation, polyhedral geometry and algebraic geometry. The volume will appeal both to beginning graduate students willing to enter the field and to researchers, including experts. —Alicia Dickenstein, University of Buenos Aires, Argentina

Author : Renzo Cavalieri,Hannah Markwig,Dhruv Ranganathan
Publisher : Springer Nature
Page : 163 pages
File Size : 47,7 Mb
Release : 2023-11-01
Category : Mathematics
ISBN : 9783031394010

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Tropical and Logarithmic Methods in Enumerative Geometry by Renzo Cavalieri,Hannah Markwig,Dhruv Ranganathan Pdf

This book is based on the lectures given at the Oberwolfach Seminar held in Fall 2021. Logarithmic Gromov-Witten theory lies at the heart of modern approaches to mirror symmetry, but also opens up a number of new directions in enumerative geometry of a more classical flavour. Tropical geometry forms the calculus through which calculations in this subject are carried out. These notes cover the foundational aspects of this tropical calculus, geometric aspects of the degeneration formula for Gromov-Witten invariants, and the practical nuances of working with and enumerating tropical curves. Readers will get an assisted entry route to the subject, focusing on examples and explicit calculations.

Author : Raf Bocklandt
Publisher : Cambridge University Press
Page : 403 pages
File Size : 50,8 Mb
Release : 2021-08-19
Category : Mathematics
ISBN : 9781108483506

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A Gentle Introduction to Homological Mirror Symmetry by Raf Bocklandt Pdf

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

Author : Diane Maclagan,Bernd Sturmfels
Publisher : American Mathematical Soc.
Page : 363 pages
File Size : 40,8 Mb
Release : 2015-04-15
Category : Algebraic geometry -- Special varieties -- Toric varieties, Newton polyhedra
ISBN : 9780821851982

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Introduction to Tropical Geometry by Diane Maclagan,Bernd Sturmfels Pdf

Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of information about their classical counterparts. Tropical geometry is a young subject that has undergone a rapid development since the beginning of the 21st century. While establishing itself as an area in its own right, deep connections have been made to many branches of pure and applied mathematics. This book offers a self-contained introduction to tropical geometry, suitable as a course text for beginning graduate students. Proofs are provided for the main results, such as the Fundamental Theorem and the Structure Theorem. Numerous examples and explicit computations illustrate the main concepts. Each of the six chapters concludes with problems that will help the readers to practice their tropical skills, and to gain access to the research literature.

Author : Anonim
Publisher : American Mathematical Soc.
Page : 698 pages
File Size : 40,6 Mb
Release : 2009
Category : Mathematics
ISBN : 9780821838488

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Dirichlet Branes and Mirror Symmetry by Anonim Pdf

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.

Author : Richard Thomas
Publisher : American Mathematical Soc.
Page : 635 pages
File Size : 40,7 Mb
Release : 2018-06-01
Category : Geometry, Algebraic
ISBN : 9781470435783

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Algebraic Geometry by Richard Thomas Pdf

This is Part 2 of a two-volume set. Since Oscar Zariski organized a meeting in 1954, there has been a major algebraic geometry meeting every decade: Woods Hole (1964), Arcata (1974), Bowdoin (1985), Santa Cruz (1995), and Seattle (2005). The American Mathematical Society has supported these summer institutes for over 50 years. Their proceedings volumes have been extremely influential, summarizing the state of algebraic geometry at the time and pointing to future developments. The most recent Summer Institute in Algebraic Geometry was held July 2015 at the University of Utah in Salt Lake City, sponsored by the AMS with the collaboration of the Clay Mathematics Institute. This volume includes surveys growing out of plenary lectures and seminar talks during the meeting. Some present a broad overview of their topics, while others develop a distinctive perspective on an emerging topic. Topics span both complex algebraic geometry and arithmetic questions, specifically, analytic techniques, enumerative geometry, moduli theory, derived categories, birational geometry, tropical geometry, Diophantine questions, geometric representation theory, characteristic and -adic tools, etc. The resulting articles will be important references in these areas for years to come.

Author : Ilia Itenberg,Grigory Mikhalkin,Eugenii I. Shustin
Publisher : Springer Science & Business Media
Page : 113 pages
File Size : 46,5 Mb
Release : 2009-05-30
Category : Mathematics
ISBN : 9783034600484

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Tropical Algebraic Geometry by Ilia Itenberg,Grigory Mikhalkin,Eugenii I. Shustin Pdf

These notes present a polished introduction to tropical geometry and contain some applications of this rapidly developing and attractive subject. It consists of three chapters which complete each other and give a possibility for non-specialists to make the first steps in the subject which is not yet well represented in the literature. The notes are based on a seminar at the Mathematical Research Center in Oberwolfach in October 2004. The intended audience is graduate, post-graduate, and Ph.D. students as well as established researchers in mathematics.

Author : Matthew Baker,Sam Payne
Publisher : Springer
Page : 526 pages
File Size : 44,6 Mb
Release : 2016-08-18
Category : Mathematics
ISBN : 9783319309453

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Nonarchimedean and Tropical Geometry by Matthew Baker,Sam Payne Pdf

This volume grew out of two Simons Symposia on "Nonarchimedean and tropical geometry" which took place on the island of St. John in April 2013 and in Puerto Rico in February 2015. Each meeting gathered a small group of experts working near the interface between tropical geometry and nonarchimedean analytic spaces for a series of inspiring and provocative lectures on cutting edge research, interspersed with lively discussions and collaborative work in small groups. The articles collected here, which include high-level surveys as well as original research, mirror the main themes of the two Symposia. Topics covered in this volume include: Differential forms and currents, and solutions of Monge-Ampere type differential equations on Berkovich spaces and their skeletons; The homotopy types of nonarchimedean analytifications; The existence of "faithful tropicalizations" which encode the topology and geometry of analytifications; Relations between nonarchimedean analytic spaces and algebraic geometry, including logarithmic schemes, birational geometry, and the geometry of algebraic curves; Extended notions of tropical varieties which relate to Huber's theory of adic spaces analogously to the way that usual tropical varieties relate to Berkovich spaces; and Relations between nonarchimedean geometry and combinatorics, including deep and fascinating connections between matroid theory, tropical geometry, and Hodge theory.