Algebraic Groups And Differential Galois Theory
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Algebraic Groups And Differential Galois Theory
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Author : Teresa Crespo
language : en
Publisher: American Mathematical Soc.
Release Date : 2011
Algebraic Groups And Differential Galois Theory written by Teresa Crespo 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 2011 with Differential algebraic groups categories.
Differential Galois theory has seen intense research activity during the last decades in several directions: elaboration of more general theories, computational aspects, model theoretic approaches, applications to classical and quantum mechanics as well as to other mathematical areas such as number theory. This book intends to introduce the reader to this subject by presenting Picard-Vessiot theory, i.e. Galois theory of linear differential equations, in a self-contained way. The needed prerequisites from algebraic geometry and algebraic groups are contained in the first two parts of the book. The third part includes Picard-Vessiot extensions, the fundamental theorem of Picard-Vessiot theory, solvability by quadratures, Fuchsian equations, monodromy group and Kovacic's algorithm. Over one hundred exercises will help to assimilate the concepts and to introduce the reader to some topics beyond the scope of this book. This book is suitable for a graduate course in differential Galois theory. The last chapter contains several suggestions for further reading encouraging the reader to enter more deeply into different topics of differential Galois theory or related fields.
Lectures On Differential Galois Theory
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Author : Andy R. Magid
language : en
Publisher: American Mathematical Soc.
Release Date : 1994
Lectures On Differential Galois Theory written by Andy R. Magid 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 1994 with Mathematics categories.
Differential Galois theory studies solutions of differential equations over a differential base field. In much the same way that ordinary Galois theory is the theory of field extensions generated by solutions of (one variable) polynomial equations, differential Galois theory looks at the nature of the differential field extension generated by the solutions of differential equations. An additional feature is that the corresponding differential Galois groups (of automorphisms of the extension fixing the base and commuting with the derivation) are algebraic groups. This book deals with the differential Galois theory of linear homogeneous differential equations, whose differential Galois groups are algebraic matrix groups.In addition to providing a convenient path to Galois theory, this approach also leads to the constructive solution of the inverse problem of differential Galois theory for various classes of algebraic groups. Providing a self-contained development and many explicit examples, this book provides a unique approach to differential Galois theory and is suitable as a textbook at the advanced graduate level.
Galois Theory Of Linear Differential Equations
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Author : Marius van der Put
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Galois Theory Of Linear Differential Equations written by Marius van der Put 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 2012-12-06 with Mathematics categories.
From the reviews: "This is a great book, which will hopefully become a classic in the subject of differential Galois theory. [...] the specialist, as well as the novice, have long been missing an introductory book covering also specific and advanced research topics. This gap is filled by the volume under review, and more than satisfactorily." Mathematical Reviews
Differential Algebra And Related Topics
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Author : Li Guo
language : en
Publisher: World Scientific
Release Date : 2002-05-30
Differential Algebra And Related Topics written by Li Guo and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002-05-30 with Mathematics categories.
Differential algebra explores properties of solutions to systems of (ordinary or partial, linear or nonlinear) differential equations from an algebraic point of view. It includes as special cases algebraic systems as well as differential systems with algebraic constraints. This algebraic theory of Joseph F Ritt and Ellis R Kolchin is further enriched by its interactions with algebraic geometry, Diophantine geometry, differential geometry, model theory, control theory, automatic theorem proving, combinatorics, and difference equations. Differential algebra now plays an important role in computational methods such as symbolic integration, and symmetry analysis of differential equations. This volume includes tutorial and survey papers presented at workshop. Contents:The Ritt–Kolchin Theory for Differential Polynomials (W Y Sit)Differential Schemes (J J Kovacic)Differential Algebra — A Scheme Theory Approach (H Gillet)Model Theory and Differential Algebra (T Scanlon)Inverse Differential Galois Theory (A R Magid)Differential Galois Theory, Universal Rings and Universal Groups (M van der Put)Cyclic Vectors (R C Churchill & J J Kovacic)Differential Algebraic Techniques in Hamiltonian Mechanics (R C Churchill)Moving Frames and Differential Algebra (E L Mansfield)Baxter Algebras and Differential Algebras (L Guo) Readership: Graduate students, pure mathematicians, logicians, algebraic geometers, applied mathematicians and physicists. Keywords:Differential Algebra;Mathematical Logic;Algebraic Geometry;Mathematical Physics
Valuations And Differential Galois Groups
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Author : Guillaume Duval
language : en
Publisher: American Mathematical Soc.
Release Date : 2011
Valuations And Differential Galois Groups written by Guillaume Duval 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 2011 with Differential algebraic groups categories.
In this paper, valuation theory is used to analyse infinitesimal behaviour of solutions of linear differential equations. For any Picard-Vessiot extension $(F / K, \partial)$ with differential Galois group $G$, the author looks at the valuations of $F$ which are left invariant by $G$. The main reason for this is the following: If a given invariant valuation $\nu$ measures infinitesimal behaviour of functions belonging to $F$, then two conjugate elements of $F$ will share the same infinitesimal behaviour with respect to $\nu$. This memoir is divided into seven sections.
Differential Algebra Algebraic Groups
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Author :
language : en
Publisher: Academic Press
Release Date : 1973-06-15
Differential Algebra Algebraic Groups written by and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1973-06-15 with Mathematics categories.
Differential Algebra & Algebraic Groups
Differential Algebra And Algebraic Groups
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Author : Ellis Robert Kolchin
language : en
Publisher:
Release Date : 1973
Differential Algebra And Algebraic Groups written by Ellis Robert Kolchin and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1973 with Algebra, Differential categories.
Differential Galois Theory Through Riemann Hilbert Correspondence
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Author : Jacques Sauloy
language : en
Publisher: American Mathematical Soc.
Release Date : 2016-12-07
Differential Galois Theory Through Riemann Hilbert Correspondence written by Jacques Sauloy 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 2016-12-07 with Galois theory categories.
Differential Galois theory is an important, fast developing area which appears more and more in graduate courses since it mixes fundamental objects from many different areas of mathematics in a stimulating context. For a long time, the dominant approach, usually called Picard-Vessiot Theory, was purely algebraic. This approach has been extensively developed and is well covered in the literature. An alternative approach consists in tagging algebraic objects with transcendental information which enriches the understanding and brings not only new points of view but also new solutions. It is very powerful and can be applied in situations where the Picard-Vessiot approach is not easily extended. This book offers a hands-on transcendental approach to differential Galois theory, based on the Riemann-Hilbert correspondence. Along the way, it provides a smooth, down-to-earth introduction to algebraic geometry, category theory and tannakian duality. Since the book studies only complex analytic linear differential equations, the main prerequisites are complex function theory, linear algebra, and an elementary knowledge of groups and of polynomials in many variables. A large variety of examples, exercises, and theoretical constructions, often via explicit computations, offers first-year graduate students an accessible entry into this exciting area.
Galois Theory Rings Algebraic Groups And Their Applications
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Author : Simeon Ivanov
language : en
Publisher: American Mathematical Soc.
Release Date : 1992
Galois Theory Rings Algebraic Groups And Their Applications written by Simeon Ivanov 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 1992 with Mathematics categories.
This collection consists of original work on Galois theory, rings and algebras, algebraic geometry, group representations, algebraic K—theory and some of their applications.
Essays In The History Of Lie Groups And Algebraic Groups
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Author : Armand Borel
language : en
Publisher: American Mathematical Soc.
Release Date : 2001
Essays In The History Of Lie Groups And Algebraic Groups written by Armand Borel 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 2001 with Mathematics categories.
Algebraic groups and Lie groups are important in most major areas of mathematics, occuring in diverse roles such as the symmetries of differential equations and as central figures in the Langlands program for number theory. In this book, Professor Borel looks at the development of the theory of Lie groups and algebraic groups, highlighting the evolution from the almost purely local theory at the start to the global theory that we know today. As the starting point of this passagefrom local to global, the author takes Lie's theory of local analytic transformation groups and Lie algebras. He then follows the globalization of the process in its two most important frameworks: (transcendental) differential geometry and algebraic geometry. Chapters II to IV are devoted to the former,Chapters V to VIII, to the latter.The essays in the first part of the book survey various proofs of the full reducibility of linear representations of $SL 2M$, the contributions H. Weyl to representation and invariant theory for Lie groups, and conclude with a chapter on E. Cartan's theory of symmetric spaces and Lie groups in the large.The second part of the book starts with Chapter V describing the development of the theory of linear algebraic groups in the 19th century. Many of the main contributions here are due to E. Study, E. Cartan, and above all, to L. Maurer. After being abandoned for nearly 50 years, the theory was revived by Chevalley and Kolchin and then further developed by many others. This is the focus of Chapter VI. The book concludes with two chapters on various aspects of the works of Chevalley on Lie groupsand algebraic groups and Kolchin on algebraic groups and the Galois theory of differential fields.The author brings a unique perspective to this study. As an important developer of some of the modern elements of both the differential geometric and the algebraic geometric sides of the theory, he has a particularly deep appreciation of the underlying mathematics. His lifelong involvement and his historical research in the subject give him a special appreciation of the story of its development.