The authors tell us what they are up to on the back cover: “The reader will learn numerous results that often belong to neither the standard undergraduate nor graduate curriculum and will discover connections between classical and contemporary ideas…” This is a pretty ambitious goal, but Mathematical Omnibus certainly delivers. Although the title is somewhat unusual, the subtitle is accurate. This book contains thirty lectures on classic mathematics. The authors mention in the preface that beauty was one of the criteria for the contents of the book. For the most part, I would agree that the topics meet that criterion.
This is an enjoyable book with suggested uses ranging from a text for an undergraduate Honors Mathematics Seminar to a coffee table book. It is appropriate for either. It could also be used as a starting point for undergraduate research topics or a place to find a short undergraduate seminar talk.
The lectures vary in length, difficulty and rigor, but almost every one introduced something new that is not part of typical coursework. The exercises also vary in difficulty, but many are interesting and there are answers and solutions provided to some of the exercises. The topics are often woven into the context of the mathematical community in which it evolved and many interesting figures, pictures and photos are incorporated into the text. This book gives students a much broader view of mathematics and mathematicians.
I found the lectures quite accessible and would judge them to be reasonable for an undergraduate to read and understand. They also use results covered in a standard undergraduate mathematics curriculum, but take a step further, by using them to discuss less standard topics. They model an investigative approach to problems very nicely and give students a glimpse into the types of questions a mathematician might ask. At times the prose can be abrupt, and some of the ideas presented in the lectures and exercises are challenging. There are some lectures where the definitions are not included, but they can be found in other lectures in the book.
This is a wonderful book that is not only fun to read, but gives the reader new ideas to think about.
Suzanne Caulk is an assistant professor of mathematics at Regis University in Denver, CO. She is very interested in modular forms. You can email her at firstname.lastname@example.org.