Brauer Groups Tamagawa Measures And Rational Points On Algebraic Varieties
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Brauer Groups Tamagawa Measures And Rational Points On Algebraic Varieties
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Author : Jorg Jahnel
language : en
Publisher: American Mathematical Soc.
Release Date : 2014-12-02
Brauer Groups Tamagawa Measures And Rational Points On Algebraic Varieties written by Jorg Jahnel and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-12-02 with Mathematics categories.
The central theme of this book is the study of rational points on algebraic varieties of Fano and intermediate type--both in terms of when such points exist and, if they do, their quantitative density. The book consists of three parts. In the first part, the author discusses the concept of a height and formulates Manin's conjecture on the asymptotics of rational points on Fano varieties. The second part introduces the various versions of the Brauer group. The author explains why a Brauer class may serve as an obstruction to weak approximation or even to the Hasse principle. This part includes two sections devoted to explicit computations of the Brauer-Manin obstruction for particular types of cubic surfaces. The final part describes numerical experiments related to the Manin conjecture that were carried out by the author together with Andreas-Stephan Elsenhans. The book presents the state of the art in computational arithmetic geometry for higher-dimensional algebraic varieties and will be a valuable reference for researchers and graduate students interested in that area.
Rational Points On Varieties
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Author : Bjorn Poonen
language : en
Publisher: American Mathematical Society
Release Date : 2023-08-10
Rational Points On Varieties written by Bjorn Poonen 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 2023-08-10 with Mathematics categories.
This book is motivated by the problem of determining the set of rational points on a variety, but its true goal is to equip readers with a broad range of tools essential for current research in algebraic geometry and number theory. The book is unconventional in that it provides concise accounts of many topics instead of a comprehensive account of just one—this is intentionally designed to bring readers up to speed rapidly. Among the topics included are Brauer groups, faithfully flat descent, algebraic groups, torsors, étale and fppf cohomology, the Weil conjectures, and the Brauer-Manin and descent obstructions. A final chapter applies all these to study the arithmetic of surfaces. The down-to-earth explanations and the over 100 exercises make the book suitable for use as a graduate-level textbook, but even experts will appreciate having a single source covering many aspects of geometry over an unrestricted ground field and containing some material that cannot be found elsewhere. The origins of arithmetic (or Diophantine) geometry can be traced back to antiquity, and it remains a lively and wide research domain up to our days. The book by Bjorn Poonen, a leading expert in the field, opens doors to this vast field for many readers with different experiences and backgrounds. It leads through various algebraic geometric constructions towards its central subject: obstructions to existence of rational points. —Yuri Manin, Max-Planck-Institute, Bonn It is clear that my mathematical life would have been very different if a book like this had been around at the time I was a student. —Hendrik Lenstra, University Leiden Understanding rational points on arbitrary algebraic varieties is the ultimate challenge. We have conjectures but few results. Poonen's book, with its mixture of basic constructions and openings into current research, will attract new generations to the Queen of Mathematics. —Jean-Louis Colliot-Thélène, Université Paris-Sud A beautiful subject, handled by a master. —Joseph Silverman, Brown University
Unramified Brauer Group And Its Applications
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Author : Sergey Gorchinskiy
language : en
Publisher: American Mathematical Soc.
Release Date : 2018-09-10
Unramified Brauer Group And Its Applications written by Sergey Gorchinskiy and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-09-10 with Associative algebras categories.
This book is devoted to arithmetic geometry with special attention given to the unramified Brauer group of algebraic varieties and its most striking applications in birational and Diophantine geometry. The topics include Galois cohomology, Brauer groups, obstructions to stable rationality, Weil restriction of scalars, algebraic tori, the Hasse principle, Brauer-Manin obstruction, and étale cohomology. The book contains a detailed presentation of an example of a stably rational but not rational variety, which is presented as series of exercises with detailed hints. This approach is aimed to help the reader understand crucial ideas without being lost in technical details. The reader will end up with a good working knowledge of the Brauer group and its important geometric applications, including the construction of unirational but not stably rational algebraic varieties, a subject which has become fashionable again in connection with the recent breakthroughs by a number of mathematicians.
A Study In Derived Algebraic Geometry Volume I Correspondences And Duality
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Author : Dennis Gaitsgory
language : en
Publisher: American Mathematical Soc.
Release Date : 2017
A Study In Derived Algebraic Geometry Volume I Correspondences And Duality written by Dennis Gaitsgory and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017 with Algebraic geometry -- (Co)homology theory -- Differentials and other special sheaves categories.
Derived algebraic geometry is a far-reaching generalization of algebraic geometry. It has found numerous applications in various parts of mathematics, most prominently in representation theory. This volume develops the theory of ind-coherent sheaves in the context of derived algebraic geometry. Ind-coherent sheaves are a “renormalization” of quasi-coherent sheaves and provide a natural setting for Grothendieck-Serre duality as well as geometric incarnations of numerous categories of interest in representation theory. This volume consists of three parts and an appendix. The first part is a survey of homotopical algebra in the setting of -categories and the basics of derived algebraic geometry. The second part builds the theory of ind-coherent sheaves as a functor out of the category of correspondences and studies the relationship between ind-coherent and quasi-coherent sheaves. The third part sets up the general machinery of the -category of correspondences needed for the second part. The category of correspondences, via the theory developed in the third part, provides a general framework for Grothendieck's six-functor formalism. The appendix provides the necessary background on -categories needed for the third part.
Women In Numbers Europe Ii
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Author : Irene I. Bouw
language : en
Publisher: Springer
Release Date : 2018-06-01
Women In Numbers Europe Ii written by Irene I. Bouw and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-06-01 with Mathematics categories.
Inspired by the September 2016 conference of the same name, this second volume highlights recent research in a wide range of topics in contemporary number theory and arithmetic geometry. Research reports from projects started at the conference, expository papers describing ongoing research, and contributed papers from women number theorists outside the conference make up this diverse volume. Topics cover a broad range of topics such as arithmetic dynamics, failure of local-global principles, geometry in positive characteristics, and heights of algebraic integers. The use of tools from algebra, analysis and geometry, as well as computational methods exemplifies the wealth of techniques available to modern researchers in number theory. Exploring connections between different branches of mathematics and combining different points of view, these papers continue the tradition of supporting and highlighting the contributions of women number theorists at a variety of career stages. Perfect for students and researchers interested in the field, this volume provides an easily accessible introduction and has the potential to inspire future work.
Foundations Of Free Noncommutative Function Theory
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Author : Dmitry S. Kaliuzhnyi-Verbovetskyi
language : en
Publisher: American Mathematical Soc.
Release Date : 2014-11-19
Foundations Of Free Noncommutative Function Theory written by Dmitry S. Kaliuzhnyi-Verbovetskyi and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-11-19 with Mathematics categories.
In this book the authors develop a theory of free noncommutative functions, in both algebraic and analytic settings. Such functions are defined as mappings from square matrices of all sizes over a module (in particular, a vector space) to square matrices over another module, which respect the size, direct sums, and similarities of matrices. Examples include, but are not limited to, noncommutative polynomials, power series, and rational expressions. Motivation and inspiration for using the theory of free noncommutative functions often comes from free probability. An important application area is "dimensionless" matrix inequalities; these arise, e.g., in various optimization problems of system engineering. Among other related areas are those of polynomial identities in rings, formal languages and finite automata, quasideterminants, noncommutative symmetric functions, operator spaces and operator algebras, and quantum control.
Partial Dynamical Systems Fell Bundles And Applications
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Author : Ruy Exel
language : en
Publisher: American Mathematical Soc.
Release Date : 2017-09-20
Partial Dynamical Systems Fell Bundles And Applications written by Ruy Exel and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-09-20 with Banach spaces categories.
Partial dynamical systems, originally developed as a tool to study algebras of operators in Hilbert spaces, has recently become an important branch of algebra. Its most powerful results allow for understanding structural properties of algebras, both in the purely algebraic and in the C*-contexts, in terms of the dynamical properties of certain systems which are often hiding behind algebraic structures. The first indication that the study of an algebra using partial dynamical systems may be helpful is the presence of a grading. While the usual theory of graded algebras often requires gradings to be saturated, the theory of partial dynamical systems is especially well suited to treat nonsaturated graded algebras which are in fact the source of the notion of “partiality”. One of the main results of the book states that every graded algebra satisfying suitable conditions may be reconstructed from a partial dynamical system via a process called the partial crossed product. Running in parallel with partial dynamical systems, partial representations of groups are also presented and studied in depth. In addition to presenting main theoretical results, several specific examples are analyzed, including Wiener–Hopf algebras and graph C*-algebras.
Kolmogorov Complexity And Algorithmic Randomness
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Author : A. Shen
language : en
Publisher: American Mathematical Society
Release Date : 2022-05-18
Kolmogorov Complexity And Algorithmic Randomness written by A. Shen 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 2022-05-18 with Mathematics categories.
Looking at a sequence of zeros and ones, we often feel that it is not random, that is, it is not plausible as an outcome of fair coin tossing. Why? The answer is provided by algorithmic information theory: because the sequence is compressible, that is, it has small complexity or, equivalently, can be produced by a short program. This idea, going back to Solomonoff, Kolmogorov, Chaitin, Levin, and others, is now the starting point of algorithmic information theory. The first part of this book is a textbook-style exposition of the basic notions of complexity and randomness; the second part covers some recent work done by participants of the “Kolmogorov seminar” in Moscow (started by Kolmogorov himself in the 1980s) and their colleagues. This book contains numerous exercises (embedded in the text) that will help readers to grasp the material.
The Ricci Flow Techniques And Applications
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Author : Bennett Chow
language : en
Publisher: American Mathematical Soc.
Release Date : 2015-10-19
The Ricci Flow Techniques And Applications written by Bennett Chow and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-10-19 with categories.
Ricci flow is a powerful technique using a heat-type equation to deform Riemannian metrics on manifolds to better metrics in the search for geometric decompositions. With the fourth part of their volume on techniques and applications of the theory, the authors discuss long-time solutions of the Ricci flow and related topics. In dimension 3, Perelman completed Hamilton's program to prove Thurston's geometrization conjecture. In higher dimensions the Ricci flow has remarkable properties, which indicates its usefulness to understand relations between the geometry and topology of manifolds. This book discusses recent developments on gradient Ricci solitons, which model the singularities developing under the Ricci flow. In the shrinking case there is a surprising rigidity which suggests the likelihood of a well-developed structure theory. A broader class of solutions is ancient solutions; the authors discuss the beautiful classification in dimension 2. In higher dimensions they consider both ancient and singular Type I solutions, which must have shrinking gradient Ricci soliton models. Next, Hamilton's theory of 3-dimensional nonsingular solutions is presented, following his original work. Historically, this theory initially connected the Ricci flow to the geometrization conjecture. From a dynamical point of view, one is interested in the stability of the Ricci flow. The authors discuss what is known about this basic problem. Finally, they consider the degenerate neckpinch singularity from both the numerical and theoretical perspectives. This book makes advanced material accessible to researchers and graduate students who are interested in the Ricci flow and geometric evolution equations and who have a knowledge of the fundamentals of the Ricci flow.
Geometry And Dynamics In Gromov Hyperbolic Metric Spaces
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Author : Tushar Das
language : en
Publisher: American Mathematical Soc.
Release Date : 2017-04-14
Geometry And Dynamics In Gromov Hyperbolic Metric Spaces written by Tushar Das and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-04-14 with Geometry, Hyperbolic categories.
This book presents the foundations of the theory of groups and semigroups acting isometrically on Gromov hyperbolic metric spaces. Particular emphasis is paid to the geometry of their limit sets and on behavior not found in the proper setting. The authors provide a number of examples of groups which exhibit a wide range of phenomena not to be found in the finite-dimensional theory. The book contains both introductory material to help beginners as well as new research results, and closes with a list of attractive unsolved problems.