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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
|---|---|---|---|---|---|
Searching... | 30000003863432 | QA174.2 M52 1993 | Open Access Book | Book | Searching... |
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Summary
Summary
First year, undergraduate, mathematics students in Japan have for many years had the opportunity of a unique experience---an introduction, at an elementary level, to some very advanced ideas in mathematics from one of the leading mathematicians of the world. Michio Kuga's lectures on Group Theory and Differential Equations are a realization of two dreams---one to see Galois groups used to attack the problems of differential equations---the other to do so in such a manner as to take students from a very basic level to an understanding of the heart of this fascinating mathematical problem. English reading students now have the opportunity to enjoy this lively presentation, from elementary ideas to cartoons to funny examples, and to follow the mind of an imaginative and creative mathematician into a world of enduring mathematical creations.
Reviews 1
Choice Review
The theory of Fuchsian differential equations, that cornerstone of 19th-century mathematics so closely associated with the names Gauss, Klein, Schwarz, Hermite, Riemann, Hilbert, Poincar'e, and others, seems to have passed entirely from fashion, its traces absorbed into more abstract theories such as those of Riemann surfaces and automorphic forms. In fact, although this beautiful subject has been undergoing something of a revival at the frontiers of research, sadly, it has all but disappeared from the curriculum. Likewise, it has received scant modern exposition (but see Masaaki Yoshida's Fuchsian Differential Equations, 1987, and Jeremy Gray's Linear Differential Equations and Group Theory from Riemann to Poincar'e, 1985). By contrast, Galois's theory of equations remains the standard culmination of advanced undergraduate algebra courses. Students usually meet examples from number fields and algebraic objects; the analogy between Galois groups and fundamental groups from topology that puts Galois' ideas on a geometric, visually intuitive footing is seldom mentioned prior to graduate school. In the context of Riemann surfaces and function fields this connection is better than a mere analogy, and Kuga makes this the basis for an idiosyncratic geometric exposition of Galois theory. Here the question of the solvability of Fuchsian differential equations is the main application, rather than the usual question of solvability of polynomials by radicals. Translated from the 1968 Japanese original, there is nothing like this book in the literature. Highly recommended. Undergraduate; graduate. D. V. Feldman; University of New Hampshire
Table of Contents
| Preface |
| Pre-Mathematics |
| No Prerequisites |
| Sets and Maps |
| Equivalence Classes |
| The Story of Free Groups |
| Heave Ho! (Pull it Tight) |
| Fundamental Groups of Surfaces |
| Fundamental Groups |
| Examples of Fundamental Groups |
| Examples of Fundamental Groups, Continued |
| Men Who Don't Realize That Their Wives Have Been Interchanged |
| Coverings |
| Covering surfaces and Fundamental Groups |
| Covering Surfaces and Fundamental Groups, Continued |
| The Group of Covering Transformations |
| Everyone has a Tail |
| The Universal Covering Space |
| The Correspondence Between Coverings of (D;O) and Subgroups of pi1(D;O).-Seeing Galois Theory |
| Continuous Functions of Covering Surfaces |
| Solvable or Not? |
| Differential Equations |
| Elementary methods of Solving Differential Equations |
| Regular Singularities |
| Differential Equations of Fuchsian Type |
| References |
| Notation |
| Index |
