Galois' Dream: Group Theory and Differential Equations: Group Theory and Differential Equations

ISBN-13: 9781461267102 / Angielski / Miękka / 2012 / 150 str.

Michio Kuga; Susan Addington; Motohico Mulase

Opis

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. 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.

Recenzje

"Galois' theory of equations remains the standard culmination of advanced undergraduate algebra courses...and Kuga makes this the basis for an idiosyncratic geometric exposition of Galois theory... There is nothing like this book in the literature. Highly recommended."

-CHOICE

Spis treści

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.

Blurb

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.

From elementary ideas to cartoons to funny examples (considered "undignified" by many of his colleagues,) the author provided his students with a book that was considered "hip" just to own, to be seen reading, and perhaps to be learning from. Many of his students went on to become good mathematicians, having fallen into the "crevasse" of mathematical curiosity.

English reading students now have the opportunity to enjoy this lively presentation and to follow the mind of an imaginative and creative mathematician into a world---not really so far removed---of enduring mathematical creations.