Nonarchimedean And Tropical Geometry

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Nonarchimedean and Tropical Geometry

Matthew Baker,Sam Payne
Nonarchimedean and Tropical Geometry Book PDF

Author : Matthew Baker,Sam Payne
Publisher : Springer
Page : 526 pages
File Size : 53,8 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.

Tropical and Non-Archimedean Geometry

Omid Amini,Matthew Baker,Xander Faber
Tropical and Non Archimedean Geometry Book PDF

Author : Omid Amini,Matthew Baker,Xander Faber
Publisher : American Mathematical Soc.
Page : 274 pages
File Size : 53,5 Mb
Release : 2014-12-26
Category : Mathematics
ISBN : 9781470410216

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Tropical and Non-Archimedean Geometry by Omid Amini,Matthew Baker,Xander Faber Pdf

Over the past decade, it has become apparent that tropical geometry and non-Archimedean geometry should be studied in tandem; each subject has a great deal to say about the other. This volume is a collection of articles dedicated to one or both of these disciplines. Some of the articles are based, at least in part, on the authors' lectures at the 2011 Bellairs Workshop in Number Theory, held from May 6-13, 2011, at the Bellairs Research Institute, Holetown, Barbados. Lecture topics covered in this volume include polyhedral structures on tropical varieties, the structure theory of non-Archimedean curves (algebraic, analytic, tropical, and formal), uniformisation theory for non-Archimedean curves and abelian varieties, and applications to Diophantine geometry. Additional articles selected for inclusion in this volume represent other facets of current research and illuminate connections between tropical geometry, non-Archimedean geometry, toric geometry, algebraic graph theory, and algorithmic aspects of systems of polynomial equations.

Homological Mirror Symmetry and Tropical Geometry

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

Author : Ricardo Castano-Bernard,Fabrizio Catanese,Maxim Kontsevich,Tony Pantev,Yan Soibelman,Ilia Zharkov
Publisher : Springer
Page : 445 pages
File Size : 53,8 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.

Tropical Algebraic Geometry

Ilia Itenberg,Grigory Mikhalkin,Eugenii I. Shustin
Tropical Algebraic Geometry Book PDF

Author : Ilia Itenberg,Grigory Mikhalkin,Eugenii I. Shustin
Publisher : Springer Science & Business Media
Page : 113 pages
File Size : 45,7 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.

Introduction to Tropical Geometry

Diane Maclagan,Bernd Sturmfels
Introduction to Tropical Geometry Book PDF

Author : Diane Maclagan,Bernd Sturmfels
Publisher : American Mathematical Society
Page : 363 pages
File Size : 42,7 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

Introduction to Tropical Geometry

Diane Maclagan,Bernd Sturmfels
Introduction to Tropical Geometry Book PDF

Author : Diane Maclagan,Bernd Sturmfels
Publisher : American Mathematical Soc.
Page : 363 pages
File Size : 51,8 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.

Facets of Algebraic Geometry

Paolo Aluffi,David Anderson,Milena Hering,Mircea Mustaţă,Sam Payne
Facets of Algebraic Geometry Book PDF

Author : Paolo Aluffi,David Anderson,Milena Hering,Mircea Mustaţă,Sam Payne
Publisher : Cambridge University Press
Page : 395 pages
File Size : 45,6 Mb
Release : 2022-04-07
Category : Mathematics
ISBN : 9781108792516

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Facets of Algebraic Geometry by Paolo Aluffi,David Anderson,Milena Hering,Mircea Mustaţă,Sam Payne Pdf

Written to honor the enduring influence of William Fulton, these articles present substantial contributions to algebraic geometry.

Combinatorial Algebraic Geometry

Gregory G. Smith,Bernd Sturmfels
Combinatorial Algebraic Geometry Book PDF

Author : Gregory G. Smith,Bernd Sturmfels
Publisher : Springer
Page : 390 pages
File Size : 44,9 Mb
Release : 2017-11-17
Category : Mathematics
ISBN : 9781493974863

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Combinatorial Algebraic Geometry by Gregory G. Smith,Bernd Sturmfels Pdf

This volume consolidates selected articles from the 2016 Apprenticeship Program at the Fields Institute, part of the larger program on Combinatorial Algebraic Geometry that ran from July through December of 2016. Written primarily by junior mathematicians, the articles cover a range of topics in combinatorial algebraic geometry including curves, surfaces, Grassmannians, convexity, abelian varieties, and moduli spaces. This book bridges the gap between graduate courses and cutting-edge research by connecting historical sources, computation, explicit examples, and new results.

Tropical and Logarithmic Methods in Enumerative Geometry

Renzo Cavalieri,Hannah Markwig,Dhruv Ranganathan
Tropical and Logarithmic Methods in Enumerative Geometry Book PDF

Author : Renzo Cavalieri,Hannah Markwig,Dhruv Ranganathan
Publisher : Springer Nature
Page : 163 pages
File Size : 54,8 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.

Tropical and Idempotent Mathematics

Grigoriĭ Lazarevich Litvinov,S. N. Sergeev
Tropical and Idempotent Mathematics Book PDF

Author : Grigoriĭ Lazarevich Litvinov,S. N. Sergeev
Publisher : American Mathematical Soc.
Page : 395 pages
File Size : 46,8 Mb
Release : 2009
Category : Mathematics
ISBN : 9780821847824

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Tropical and Idempotent Mathematics by Grigoriĭ Lazarevich Litvinov,S. N. Sergeev Pdf

This volume is a collection of papers from the International Conference on Tropical and Idempotent Mathematics, held in Moscow, Russia in August 2007. This is a relatively new branch of mathematical sciences that has been rapidly developing and gaining popularity over the last decade. Tropical mathematics can be viewed as a result of the Maslov dequantization applied to 'traditional' mathematics over fields. Importantly, applications in econophysics and statistical mechanics lead to an explanation of the nature of financial crises. Another original application provides an analysis of instabilities in electrical power networks. Idempotent analysis, tropical algebra, and tropical geometry are the building blocks of the subject. Contributions to idempotent analysis are focused on the Hamilton-Jacobi semigroup, the max-plus finite element method, and on the representations of eigenfunctions of idempotent linear operators. Tropical algebras, consisting of plurisubharmonic functions and their germs, are examined. The volume also contains important surveys and research papers on tropical linear algebra and tropical convex geometry.

Extended Abstracts February 2016

Maria Alberich-Carramiñana,Carlos Galindo,Alex Küronya,Joaquim Roé
Extended Abstracts February 2016 Book PDF

Author : Maria Alberich-Carramiñana,Carlos Galindo,Alex Küronya,Joaquim Roé
Publisher : Springer
Page : 118 pages
File Size : 40,7 Mb
Release : 2018-11-03
Category : Mathematics
ISBN : 9783030000271

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Extended Abstracts February 2016 by Maria Alberich-Carramiñana,Carlos Galindo,Alex Küronya,Joaquim Roé Pdf

This volume contains extended abstracts outlining selected talks and other selected presentations given by participants of the workshop "Positivity and Valuations", held at the Centre de Recerca Matemàtica (CRM) in Barcelona from February 22nd to 26th, 2016. They include brief research articles reporting new results, descriptions of preliminary work or open problems, and the outcome of work in groups initiated during the workshop. The general subject is the application of valuation theory to positivity questions in algebraic geometry. The topics covered range from purely algebraic problems like finite generation of semigroups and algebras defined by valuations, and properties of the associated Poincaré series, to more geometric questions like resolution of singularities and properties of Newton-Okounkov bodies, linked with non-archimedean geometry and tropical geometry. The book is intended for established researchers, as well as for PhD and postdoctoral students who want to learn more about the latest advances in these highly active areas of research.

Algebraic Geometry

Richard Thomas
Algebraic Geometry Book PDF

Author : Richard Thomas
Publisher : American Mathematical Soc.
Page : 635 pages
File Size : 44,8 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.

Essentials of Tropical Combinatorics

Michael Joswig
Essentials of Tropical Combinatorics Book PDF

Author : Michael Joswig
Publisher : American Mathematical Society
Page : 398 pages
File Size : 54,5 Mb
Release : 2021-12-08
Category : Mathematics
ISBN : 9781470466534

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Essentials of Tropical Combinatorics by Michael Joswig Pdf

The goal of this book is to explain, at the graduate student level, connections between tropical geometry and optimization. Building bridges between these two subject areas is fruitful in two ways. Through tropical geometry optimization algorithms become applicable to questions in algebraic geometry. Conversely, looking at topics in optimization through the tropical geometry lens adds an additional layer of structure. The author covers contemporary research topics that are relevant for applications such as phylogenetics, neural networks, combinatorial auctions, game theory, and computational complexity. This self-contained book grew out of several courses given at Technische Universität Berlin and elsewhere, and the main prerequisite for the reader is a basic knowledge in polytope theory. It contains a good number of exercises, many examples, beautiful figures, as well as explicit tools for computations using $texttt{polymake}$.

Tropical Geometry and Integrable Systems

Chris Athorne,Diane Maclagan,Ian Strachan
Tropical Geometry and Integrable Systems Book PDF

Author : Chris Athorne,Diane Maclagan,Ian Strachan
Publisher : American Mathematical Soc.
Page : 170 pages
File Size : 43,7 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821875537

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Tropical Geometry and Integrable Systems by Chris Athorne,Diane Maclagan,Ian Strachan Pdf

This volume contains the proceedings of the conference on tropical geometry and integrable systems, held July 3-8, 2011, at the University of Glasgow, United Kingdom. One of the aims of this conference was to bring together researchers in the field of tropical geometry and its applications, from apparently disparate ends of the spectrum, to foster a mutual understanding and establish a common language which will encourage further developments of the area. This aim is reflected in these articles, which cover areas from automata, through cluster algebras, to enumerative geometry. In addition, two survey articles are included which introduce ideas from researchers on one end of this spectrum to researchers on the other. This book is intended for graduate students and researchers interested in tropical geometry and integrable systems and the developing links between these two areas.

Algebraic and Combinatorial Aspects of Tropical Geometry

Erwan Brugalle,Maria Angelica Cueto,Alicia Dickenstein,Eva-Maria Feichtner,Ilia Itenberg
Algebraic and Combinatorial Aspects of Tropical Geometry Book PDF

Author : Erwan Brugalle,Maria Angelica Cueto,Alicia Dickenstein,Eva-Maria Feichtner,Ilia Itenberg
Publisher : American Mathematical Soc.
Page : 363 pages
File Size : 47,7 Mb
Release : 2013-05-23
Category : Mathematics
ISBN : 9780821891469

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Algebraic and Combinatorial Aspects of Tropical Geometry by Erwan Brugalle,Maria Angelica Cueto,Alicia Dickenstein,Eva-Maria Feichtner,Ilia Itenberg Pdf

This volume contains the proceedings of the CIEM workshop on Tropical Geometry, held December 12-16, 2011, at the International Centre for Mathematical Meetings (CIEM), Castro Urdiales, Spain. Tropical geometry is a new and rapidly developing field of mat