Galois Groups And Fundamental Groups On Riemann Surfaces

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Galois Groups and Fundamental Groups on Riemann Surfaces

Matthias Himmelmann
Galois Groups and Fundamental Groups on Riemann Surfaces Book PDF

Author : Matthias Himmelmann
Publisher : GRIN Verlag
Page : 40 pages
File Size : 54,5 Mb
Release : 2018-10-17
Category : Mathematics
ISBN : 9783668818965

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Galois Groups and Fundamental Groups on Riemann Surfaces by Matthias Himmelmann Pdf

Bachelor Thesis from the year 2018 in the subject Mathematics - Algebra, grade: 1,0, Free University of Berlin (Mathematik), language: English, abstract: This thesis deals with the correlation of the fundamental group and the Galois group, using their corresponding entities of covering spaces and field extensions. First it is viewed in the general setting of categories, using the language of Galois categories. It is shown that the categories of the finite étale algebras and the category of covering spaces are correlated, which gives the fact that the profinite completion of the fundamental group and the absolute Galois group are similar. More specifically, on Riemann surfaces it is shown that there exists an anti-equivalence of categories between the finite field extensions of the meromorphic functions of a compact, connected Riemann Surface X and the category of branched coverings of X. A more explicit theorem, that provides an isomorphism between a specific Galois Group and the profinite Completion of the Fundamental Group of a pointed X, gives more insight on the behaviour of these two groups.

Galois Groups and Fundamental Groups

Leila Schneps
Galois Groups and Fundamental Groups Book PDF

Author : Leila Schneps
Publisher : Cambridge University Press
Page : 486 pages
File Size : 55,8 Mb
Release : 2003-07-21
Category : Mathematics
ISBN : 0521808316

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Galois Groups and Fundamental Groups by Leila Schneps Pdf

Table of contents

Galois Groups and Fundamental Groups

Tamás Szamuely
Galois Groups and Fundamental Groups Book PDF

Author : Tamás Szamuely
Publisher : Cambridge University Press
Page : 281 pages
File Size : 43,8 Mb
Release : 2009-07-16
Category : Mathematics
ISBN : 9781139481144

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Galois Groups and Fundamental Groups by Tamás Szamuely Pdf

Ever since the concepts of Galois groups in algebra and fundamental groups in topology emerged during the nineteenth century, mathematicians have known of the strong analogies between the two concepts. This book presents the connection starting at an elementary level, showing how the judicious use of algebraic geometry gives access to the powerful interplay between algebra and topology that underpins much modern research in geometry and number theory. Assuming as little technical background as possible, the book starts with basic algebraic and topological concepts, but already presented from the modern viewpoint advocated by Grothendieck. This enables a systematic yet accessible development of the theories of fundamental groups of algebraic curves, fundamental groups of schemes, and Tannakian fundamental groups. The connection between fundamental groups and linear differential equations is also developed at increasing levels of generality. Key applications and recent results, for example on the inverse Galois problem, are given throughout.

Galois Theory, Coverings, and Riemann Surfaces

Askold Khovanskii
Galois Theory Coverings and Riemann Surfaces Book PDF

Author : Askold Khovanskii
Publisher : Springer Science & Business Media
Page : 86 pages
File Size : 47,5 Mb
Release : 2013-09-11
Category : Mathematics
ISBN : 9783642388415

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Galois Theory, Coverings, and Riemann Surfaces by Askold Khovanskii Pdf

The first part of this book provides an elementary and self-contained exposition of classical Galois theory and its applications to questions of solvability of algebraic equations in explicit form. The second part describes a surprising analogy between the fundamental theorem of Galois theory and the classification of coverings over a topological space. The third part contains a geometric description of finite algebraic extensions of the field of meromorphic functions on a Riemann surface and provides an introduction to the topological Galois theory developed by the author. All results are presented in the same elementary and self-contained manner as classical Galois theory, making this book both useful and interesting to readers with a variety of backgrounds in mathematics, from advanced undergraduate students to researchers.

Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces

William Mark Goldman
Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces Book PDF

Author : William Mark Goldman
Publisher : Unknown
Page : 69 pages
File Size : 46,5 Mb
Release : 2008
Category : Geometry, Algebraic
ISBN : 1470405105

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Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces by William Mark Goldman Pdf

Details the theory of rank one Higgs bundles over a closed Riemann surface $X$ and their relation to representations of the fundamental group of $X$. This work constructs an equivalence between the deformation theories of flat connections and Higgs pairs, providing an identification of moduli spaces arising in different contexts.

Inverse Galois Theory

Gunter Malle,B.H. Matzat
Inverse Galois Theory Book PDF

Author : Gunter Malle,B.H. Matzat
Publisher : Springer Science & Business Media
Page : 450 pages
File Size : 42,5 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9783662121238

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Inverse Galois Theory by Gunter Malle,B.H. Matzat Pdf

A consistent and near complete survey of the important progress made in the field over the last few years, with the main emphasis on the rigidity method and its applications. Among others, this monograph presents the most successful existence theorems known and construction methods for Galois extensions as well as solutions for embedding problems combined with a collection of the existing Galois realizations.

Groups as Galois Groups

Helmut Völklein
Groups as Galois Groups Book PDF

Author : Helmut Völklein
Publisher : Cambridge University Press
Page : 270 pages
File Size : 54,6 Mb
Release : 1996-08-13
Category : Mathematics
ISBN : 0521562805

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Groups as Galois Groups by Helmut Völklein Pdf

Develops the mathematical background and recent results on the Inverse Galois Problem.

Discontinuous Groups and Riemann Surfaces (AM-79), Volume 79

Leon Greenberg
Discontinuous Groups and Riemann Surfaces AM 79 Volume 79 Book PDF

Author : Leon Greenberg
Publisher : Princeton University Press
Page : 452 pages
File Size : 43,7 Mb
Release : 2016-03-02
Category : Mathematics
ISBN : 9781400881642

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Discontinuous Groups and Riemann Surfaces (AM-79), Volume 79 by Leon Greenberg Pdf

Study 79 contains a collection of papers presented at the Conference on Discontinuous Groups and Ricmann Surfaces at the University of Maryland, May 21-25, 1973. The papers, by leading authorities, deal mainly with Fuchsian and Kleinian groups, Teichmüller spaces, Jacobian varieties, and quasiconformal mappings. These topics are intertwined, representing a common meeting of algebra, geometry, and analysis.

Algebraic Geometry I

V.I. Danilov,V.V. Shokurov
Algebraic Geometry I Book PDF

Author : V.I. Danilov,V.V. Shokurov
Publisher : Springer Science & Business Media
Page : 322 pages
File Size : 51,7 Mb
Release : 2006-12-15
Category : Mathematics
ISBN : 3540519955

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Algebraic Geometry I by V.I. Danilov,V.V. Shokurov Pdf

"... To sum up, this book helps to learn algebraic geometry in a short time, its concrete style is enjoyable for students and reveals the beauty of mathematics." --Acta Scientiarum Mathematicarum

Lectures on Riemann Surfaces

Otto Forster
Lectures on Riemann Surfaces Book PDF

Author : Otto Forster
Publisher : Springer Science & Business Media
Page : 262 pages
File Size : 41,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461259619

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Lectures on Riemann Surfaces by Otto Forster Pdf

This book grew out of lectures on Riemann surfaces given by Otto Forster at the universities of Munich, Regensburg, and Münster. It provides a concise modern introduction to this rewarding subject, as well as presenting methods used in the study of complex manifolds in the special case of complex dimension one. From the reviews: "This book deserves very serious consideration as a text for anyone contemplating giving a course on Riemann surfaces."—-MATHEMATICAL REVIEWS

Topics in Galois Theory

Jean-Pierre Serre
Topics in Galois Theory Book PDF

Author : Jean-Pierre Serre
Publisher : CRC Press
Page : 120 pages
File Size : 44,5 Mb
Release : 2016-04-19
Category : Mathematics
ISBN : 9781439865255

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Topics in Galois Theory by Jean-Pierre Serre Pdf

This book is based on a course given by the author at Harvard University in the fall semester of 1988. The course focused on the inverse problem of Galois Theory: the construction of field extensions having a given finite group as Galois group. In the first part of the book, classical methods and results, such as the Scholz and Reichardt constructi

Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces

William Mark Goldman,Eugene Zhu Xia
Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces Book PDF

Author : William Mark Goldman,Eugene Zhu Xia
Publisher : American Mathematical Soc.
Page : 86 pages
File Size : 52,7 Mb
Release : 2008
Category : Geometry, Algebraic
ISBN : 9780821841365

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Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces by William Mark Goldman,Eugene Zhu Xia Pdf

This expository article details the theory of rank one Higgs bundles over a closed Riemann surface $X$ and their relation to representations of the fundamental group of $X$. The authors construct an equivalence between the deformation theories of flat connections and Higgs pairs. This provides an identification of moduli spaces arising in different contexts. The moduli spaces are real Lie groups. From each context arises a complex structure, and the different complex structures define a hyperkähler structure. The twistor space, real forms, and various group actions are computed explicitly in terms of the Jacobian of $X$. The authors describe the moduli spaces and their geometry in terms of the Riemann period matrix of $X$.

Moduli Spaces of Riemann Surfaces

Benson Farb,Richard Hain,Eduard Looijenga
Moduli Spaces of Riemann Surfaces Book PDF

Author : Benson Farb,Richard Hain,Eduard Looijenga
Publisher : American Mathematical Soc.
Page : 371 pages
File Size : 54,8 Mb
Release : 2013-08-16
Category : Mathematics
ISBN : 9780821898871

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Moduli Spaces of Riemann Surfaces by Benson Farb,Richard Hain,Eduard Looijenga Pdf

Mapping class groups and moduli spaces of Riemann surfaces were the topics of the Graduate Summer School at the 2011 IAS/Park City Mathematics Institute. This book presents the nine different lecture series comprising the summer school, covering a selection of topics of current interest. The introductory courses treat mapping class groups and Teichmüller theory. The more advanced courses cover intersection theory on moduli spaces, the dynamics of polygonal billiards and moduli spaces, the stable cohomology of mapping class groups, the structure of Torelli groups, and arithmetic mapping class groups. The courses consist of a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures do not duplicate standard courses available elsewhere. The book should be a valuable resource for graduate students and researchers interested in the topology, geometry and dynamics of moduli spaces of Riemann surfaces and related topics. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

Dessins d'Enfants on Riemann Surfaces

Gareth A. Jones,Jürgen Wolfart
Dessins d Enfants on Riemann Surfaces Book PDF

Author : Gareth A. Jones,Jürgen Wolfart
Publisher : Springer
Page : 259 pages
File Size : 46,7 Mb
Release : 2016-03-23
Category : Mathematics
ISBN : 9783319247113

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Dessins d'Enfants on Riemann Surfaces by Gareth A. Jones,Jürgen Wolfart Pdf

This volume provides an introduction to dessins d'enfants and embeddings of bipartite graphs in compact Riemann surfaces. The first part of the book presents basic material, guiding the reader through the current field of research. A key point of the second part is the interplay between the automorphism groups of dessins and their Riemann surfaces, and the action of the absolute Galois group on dessins and their algebraic curves. It concludes by showing the links between the theory of dessins and other areas of arithmetic and geometry, such as the abc conjecture, complex multiplication and Beauville surfaces. Dessins d'Enfants on Riemann Surfaces will appeal to graduate students and all mathematicians interested in maps, hypermaps, Riemann surfaces, geometric group actions, and arithmetic.

Love and Math

Edward Frenkel
Love and Math Book PDF

Author : Edward Frenkel
Publisher : Basic Books
Page : 314 pages
File Size : 41,8 Mb
Release : 2013-10-01
Category : Mathematics
ISBN : 9780465069958

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Love and Math by Edward Frenkel Pdf

An awesome, globe-spanning, and New York Times bestselling journey through the beauty and power of mathematics What if you had to take an art class in which you were only taught how to paint a fence? What if you were never shown the paintings of van Gogh and Picasso, weren't even told they existed? Alas, this is how math is taught, and so for most of us it becomes the intellectual equivalent of watching paint dry. In Love and Math, renowned mathematician Edward Frenkel reveals a side of math we've never seen, suffused with all the beauty and elegance of a work of art. In this heartfelt and passionate book, Frenkel shows that mathematics, far from occupying a specialist niche, goes to the heart of all matter, uniting us across cultures, time, and space. Love and Math tells two intertwined stories: of the wonders of mathematics and of one young man's journey learning and living it. Having braved a discriminatory educational system to become one of the twenty-first century's leading mathematicians, Frenkel now works on one of the biggest ideas to come out of math in the last 50 years: the Langlands Program. Considered by many to be a Grand Unified Theory of mathematics, the Langlands Program enables researchers to translate findings from one field to another so that they can solve problems, such as Fermat's last theorem, that had seemed intractable before. At its core, Love and Math is a story about accessing a new way of thinking, which can enrich our lives and empower us to better understand the world and our place in it. It is an invitation to discover the magic hidden universe of mathematics.