Counting Surfaces
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Counting Surfaces
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Author : Bertrand Eynard
language : en
Publisher: Springer Science & Business Media
Release Date : 2016-03-21
Counting Surfaces written by Bertrand Eynard and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-03-21 with Mathematics categories.
The problem of enumerating maps (a map is a set of polygonal "countries" on a world of a certain topology, not necessarily the plane or the sphere) is an important problem in mathematics and physics, and it has many applications ranging from statistical physics, geometry, particle physics, telecommunications, biology, ... etc. This problem has been studied by many communities of researchers, mostly combinatorists, probabilists, and physicists. Since 1978, physicists have invented a method called "matrix models" to address that problem, and many results have been obtained. Besides, another important problem in mathematics and physics (in particular string theory), is to count Riemann surfaces. Riemann surfaces of a given topology are parametrized by a finite number of real parameters (called moduli), and the moduli space is a finite dimensional compact manifold or orbifold of complicated topology. The number of Riemann surfaces is the volume of that moduli space. Mor e generally, an important problem in algebraic geometry is to characterize the moduli spaces, by computing not only their volumes, but also other characteristic numbers called intersection numbers. Witten's conjecture (which was first proved by Kontsevich), was the assertion that Riemann surfaces can be obtained as limits of polygonal surfaces (maps), made of a very large number of very small polygons. In other words, the number of maps in a certain limit, should give the intersection numbers of moduli spaces. In this book, we show how that limit takes place. The goal of this book is to explain the "matrix model" method, to show the main results obtained with it, and to compare it with methods used in combinatorics (bijective proofs, Tutte's equations), or algebraic geometry (Mirzakhani's recursions). The book intends to be self-contained and accessible to graduate students, and provides comprehensive proofs, several examples, and give s the general formula for the enumeration of maps on surfaces of any topology. In the end, the link with more general topics such as algebraic geometry, string theory, is discussed, and in particular a proof of the Witten-Kontsevich conjecture is provided.
Counting Surfaces
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Author : Bertrand Eynard
language : en
Publisher: Birkhäuser
Release Date : 2016-02-07
Counting Surfaces written by Bertrand Eynard and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-02-07 with Mathematics categories.
The problem of enumerating maps (a map is a set of polygonal "countries" on a world of a certain topology, not necessarily the plane or the sphere) is an important problem in mathematics and physics, and it has many applications ranging from statistical physics, geometry, particle physics, informatics, biology, ... etc. This problem has been studied by many communities of researchers, mostly combinatorists, probabilists, and physicists. In 1978+, physicists have invented a method called "matrix models" to address that problem, and many results have been obtained. Besides, another important problem in mathematics and physics (in particular string theory), is to count Riemann surfaces. Riemann surfaces of a given topology are parametrized by a finite number of real parameters (called moduli), and the moduli space is a finite dimensional compact manifold of complicated topology. The number of Riemann surfaces is the volume of that moduli space. More generally, an important problem in algebraic geometry is to characterize the moduli spaces, by computing not only their volumes, but also their intersection numbers. The so called Witten's conjecture (which was first proved by Kontsevich), was the assertion that Riemann surfaces can be obtained as limits of polygonal surfaces (maps), made of a very large number of very small polygons. In other words, the number of maps in a certain limit, should give the intersection numbers of moduli spaces. In this book, we show how that limit takes place. The goal of this book is to explain the "matrix model" method, to show the main results obtained with it, and to compare it with methods used in combinatorics (bijective proofs, Tutte's equations), or algebraic geometry (Mirzakhani's recursions). The book intends to be self-contained and pedagogical, and will provide comprehensive proofs, several examples, and will give the general formula for the enumeration of maps on surfaces of any topology. In the end, the link with more general topics such as algebraic geometry, string theory, will be discussed, and in particular we give a proof of the Witten-Kontsevich conjecture.
Counting Rational Points On Curves And Surfaces
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Author : Niklas Broberg
language : en
Publisher:
Release Date : 2002
Counting Rational Points On Curves And Surfaces written by Niklas Broberg and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with categories.
Counting Points On K3 Surfaces And Other Arithmetic Geometric Objects
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Author :
language : en
Publisher:
Release Date : 2018
Counting Points On K3 Surfaces And Other Arithmetic Geometric Objects written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018 with categories.
Translation Surfaces
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Author : Jayadev S. Athreya
language : en
Publisher: American Mathematical Society
Release Date : 2024-04-17
Translation Surfaces written by Jayadev S. Athreya and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-04-17 with Mathematics categories.
This textbook offers an accessible introduction to translation surfaces. Building on modest prerequisites, the authors focus on the fundamentals behind big ideas in the field: ergodic properties of translation flows, counting problems for saddle connections, and associated renormalization techniques. Proofs that go beyond the introductory nature of the book are deftly omitted, allowing readers to develop essential tools and motivation before delving into the literature. Beginning with the fundamental example of the flat torus, the book goes on to establish the three equivalent definitions of translation surface. An introduction to the moduli space of translation surfaces follows, leading into a study of the dynamics and ergodic theory associated to a translation surface. Counting problems and group actions come to the fore in the latter chapters, giving a broad overview of progress in the 40 years since the ergodicity of the Teichmüller geodesic flow was proven. Exercises are included throughout, inviting readers to actively explore and extend the theory along the way. Translation Surfaces invites readers into this exciting area, providing an accessible entry point from the perspectives of dynamics, ergodicity, and measure theory. Suitable for a one- or two-semester graduate course, it assumes a background in complex analysis, measure theory, and manifolds, while some familiarity with Riemann surfaces and ergodic theory would be beneficial.
Counting Rational Points On Cubic Surfaces
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Author : Efthymios Sofos
language : en
Publisher:
Release Date : 2011
Counting Rational Points On Cubic Surfaces written by Efthymios Sofos and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Rational points (Geometry) categories.
Counting Rational Points On Curves And Surfaces
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Author : Timothy Daniel Browning
language : en
Publisher:
Release Date : 2001
Counting Rational Points On Curves And Surfaces written by Timothy Daniel Browning and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Curves, Algebraic categories.
Counting Curves Of Any Genus On Rational Ruled Surfaces
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Author :
language : en
Publisher:
Release Date : 1997
Counting Curves Of Any Genus On Rational Ruled Surfaces written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997 with categories.
Counting Points Of Bounded Height On Del Pezzo Surfaces
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Author : Stephanie Kleven
language : en
Publisher:
Release Date : 2006
Counting Points Of Bounded Height On Del Pezzo Surfaces written by Stephanie Kleven and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with categories.
Counting Rational Points On Smooth Cubic Surfaces
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Author : Efthymios Sofos
language : en
Publisher:
Release Date : 2015
Counting Rational Points On Smooth Cubic Surfaces written by Efthymios Sofos and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015 with categories.