Author: John B. Conway
Publisher: American Mathematical Society
Mathematics degrees lead students to study some branches of the subject in considerable detail. As a result, graduates will have developed valuable transferable skills and will be well equipped in terms of the problem-solving and analytical abilities that in-depth study of mathematics requires. They will understand some areas of mathematics extremely well, but may have only limited knowledge of topics that were covered in the options they did not choose or which were not available.
Furthermore, the increasing modularisation of university curricula has meant that the connections between different areas of mathematics are not always made apparent. Indeed, for many students who attended schools in England, their pre-university experience of mathematics is that it is presented as a range of topics to be mastered for assessment, and which can then be completely forgotten as the learner moves on to the next module. Undergraduates today often express surprise when they discover that a topic covered earlier in the curriculum is required elsewhere later. This attitude is profoundly depressing: do students really think that we are encouraging them to put enormous effort into learning things that are going to be of no future value?
Mathematical Connections attempts to address this issue by providing a textbook for a course that brings out the interactions between different parts of mathematics - as the author describes it, a "capstone course". (The books recommended below also provide material that presents a bigger picture.) John Conway introduces and develops six topics spanning geometry, algebra and analysis, showing how mathematical ideas have diverse applications.
The topics he covers are: the trisection of angles; polyhedra, tessellations and map colouring; Hilbert spaces; the spectral theorem; matrices, topology and the general linear group; and modules. Brief historical notes provide context. The author's tone is genial; the exposition is clear and appropriate for advanced undergraduates (reasonable mathematical sophistication is expected from the reader); there are exercises (but no solutions); and an appendix furnishes the necessary background material on topics such as groups, rings and lattices for those to whom these concepts are new. There are notes on how to read mathematics, which, although one might hope that student readers will have come across such guidance earlier in their studies, are useful and have wide application.
One could base a final-year undergraduate or master's module on this course. Alternatively, and perhaps this is likely in practice to be how the book is most often used, the material could be used in a wide variety of places in a pure mathematics degree, since the chapters are largely independent.
It is to be hoped that Conway's book will inspire university teachers to seek opportunities to draw attention to the rich interconnections between different areas of mathematics and to help students appreciate the power of the subject.
Who is it for? Advanced undergraduates.
Presentation: Clear, focused and logical.
Would you recommend it? Lecturers will find inspiring examples of mathematical connections.
Authors: J. Mason, L. Burton and K. Stacey
Edition: Second revised
Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century
Author: Jeremy Gray