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
CONTENT
Pre-Mathematics -- 0th Week No prerequisites -- 1st Week Sets and Maps -- 2nd Week Equivalence Classes -- 3rd Week The Story of Free Groups -- Heave Ho! (Pull it Tight) -- 4th Week Fundamental Groups of Surfaces -- 5th Week Fundamental Groups -- 6th Week Examples of Fundamental Groups -- 7th Week Examples of Fundamental Groups, continued -- Men Who Don't Realize That Their Wives Have Been Interchanged -- 8th Week Coverings -- 9th Week Covering Surfaces and Fundamental Groups -- 10th Week Covering Surfaces and Fundamental Groups, continued -- 11th Week The Group of Covering Transformations -- Everyone Has a Tail -- 12th Week The Universal Covering Space -- 13th Week The Correspondence Between Coverings of (D; O) and Subgroups of ?1 (D; O) -- Seeing Galois Theory -- 14th Week Continuous Functions on Covering Surfaces -- 15th Week Function Theory on Covering Surfaces -- Solvable or Not? -- 16th Week Differential Equations -- 17th Week Elementary Methods of Solving Differential Equations -- 18th Week Regular Singularities -- 19th Week Differential Equations of Fuchsian Type -- References -- Notation
