Differential Galois Theory through Riemann-Hilbert Correspondence - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Differential Galois Theory through Riemann-Hilbert Correspondence write by Jacques Sauloy. This book was released on 2016-12-07. Differential Galois Theory through Riemann-Hilbert Correspondence available in PDF, EPUB and Kindle. 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.
Noncommutative Geometry, Quantum Fields and Motives
Noncommutative Geometry, Quantum Fields and Motives - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Noncommutative Geometry, Quantum Fields and Motives write by Alain Connes. This book was released on 2019-03-13. Noncommutative Geometry, Quantum Fields and Motives available in PDF, EPUB and Kindle. The unifying theme of this book is the interplay among noncommutative geometry, physics, and number theory. The two main objects of investigation are spaces where both the noncommutative and the motivic aspects come to play a role: space-time, where the guiding principle is the problem of developing a quantum theory of gravity, and the space of primes, where one can regard the Riemann Hypothesis as a long-standing problem motivating the development of new geometric tools. The book stresses the relevance of noncommutative geometry in dealing with these two spaces. The first part of the book deals with quantum field theory and the geometric structure of renormalization as a Riemann-Hilbert correspondence. It also presents a model of elementary particle physics based on noncommutative geometry. The main result is a complete derivation of the full Standard Model Lagrangian from a very simple mathematical input. Other topics covered in the first part of the book are a noncommutative geometry model of dimensional regularization and its role in anomaly computations, and a brief introduction to motives and their conjectural relation to quantum field theory. The second part of the book gives an interpretation of the Weil explicit formula as a trace formula and a spectral realization of the zeros of the Riemann zeta function. This is based on the noncommutative geometry of the adèle class space, which is also described as the space of commensurability classes of Q-lattices, and is dual to a noncommutative motive (endomotive) whose cyclic homology provides a general setting for spectral realizations of zeros of L-functions. The quantum statistical mechanics of the space of Q-lattices, in one and two dimensions, exhibits spontaneous symmetry breaking. In the low-temperature regime, the equilibrium states of the corresponding systems are related to points of classical moduli spaces and the symmetries to the class field theory of the field of rational numbers and of imaginary quadratic fields, as well as to the automorphisms of the field of modular functions. The book ends with a set of analogies between the noncommutative geometries underlying the mathematical formulation of the Standard Model minimally coupled to gravity and the moduli spaces of Q-lattices used in the study of the zeta function.
Transcendence in Algebra, Combinatorics, Geometry and Number Theory
Transcendence in Algebra, Combinatorics, Geometry and Number Theory - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Transcendence in Algebra, Combinatorics, Geometry and Number Theory write by Alin Bostan. This book was released on 2021-11-02. Transcendence in Algebra, Combinatorics, Geometry and Number Theory available in PDF, EPUB and Kindle. This proceedings volume gathers together original articles and survey works that originate from presentations given at the conference Transient Transcendence in Transylvania, held in Brașov, Romania, from May 13th to 17th, 2019. The conference gathered international experts from various fields of mathematics and computer science, with diverse interests and viewpoints on transcendence. The covered topics are related to algebraic and transcendental aspects of special functions and special numbers arising in algebra, combinatorics, geometry and number theory. Besides contributions on key topics from invited speakers, this volume also brings selected papers from attendees.
Galois Theories of Linear Difference Equations: An Introduction
Galois Theories of Linear Difference Equations: An Introduction - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Galois Theories of Linear Difference Equations: An Introduction write by Charlotte Hardouin. This book was released on 2016-04-27. Galois Theories of Linear Difference Equations: An Introduction available in PDF, EPUB and Kindle. This book is a collection of three introductory tutorials coming out of three courses given at the CIMPA Research School “Galois Theory of Difference Equations” in Santa Marta, Columbia, July 23–August 1, 2012. The aim of these tutorials is to introduce the reader to three Galois theories of linear difference equations and their interrelations. Each of the three articles addresses a different galoisian aspect of linear difference equations. The authors motivate and give elementary examples of the basic ideas and techniques, providing the reader with an entry to current research. In addition each article contains an extensive bibliography that includes recent papers; the authors have provided pointers to these articles allowing the interested reader to explore further.
Lectures on Differential Galois Theory
Lectures on Differential Galois Theory - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Lectures on Differential Galois Theory write by Andy R. Magid. This book was released on . Lectures on Differential Galois Theory available in PDF, EPUB and Kindle. Providing a self-contained development with many explicit examples, this book is a unique approach to Galois theory and is suitable as a textbook at the advanced graduate level.
