If you are reading this blog, you probably have a science degree ;) after graduating the opportunities to use mathematics in the day job are often limited to a small subset of what was learnt. The books below, varying in difficulty, are an occasion to practice or to have “mathematical recreations”.
I try to rate the level of technicality of the book *** are the math books you would expect to read to prepare for a master’s degree exam, ** still require a pencil and a paper to explore the details and * or less contain very few technicalities.
These are among my favorites: they require very little knowledge about any specific topic in mathematics and are pure moments of cleverness, smart arguments and nice illustrations.
(**) Proofs from the book also available in PDF. When Erdos proved something and the proof looked clumsy to him, he was saying “This is not the proof from the book” or “Let’s look for the proof of the book”. And indeed, when proving statements, some proofs seem more natural than others: usually, the shortest and most convincing ones. This book is an attempt - a successful one - to gather them. The infinity of prime numbers, D’Alembert Gauss theorem, the Law of quadratic reciprocity are just a few examples of all the results presented in this book.
Some proofs may be profitable to the reading of Number Theory 1: Fermat’s Dream.
(**) The Art of Mathematics: Coffee Time in Memphis more than a hundred exercises, with hints and detailed solutions. The difficulty varies greatly from an exercise to another and the solutions come from many different fields. Ideal for long trips ;)
Algebra an number theory
If you like number theory, the following books are a must have. The first volume is easily accessible, however, the following ones will require a working knowledge of Galois theory. I love the way authors present the intuitions behind the proofs and the main steps to go through before actually “jumping” into the proof.
(***) Number Theory 2: Introduction to Class Field Theory If the previous book presented some facts that looked “magic”, this one focuses on explaining why they happen. It is much harder than the previous read. I would strongly advise this read to those who loved the previous one.
(***) Number Theory 3: Iwasawa Theory and Modular Forms The same advice applies ;)
If you do not know about Galois theory, these two references may help. They do not qualify as “casual readings” but they will help understand the “Number theory saga”.
(***) Galois Theory by Emil Artin Though this course is quite old, the book gives a clear presentation of the topic
Applications of mathematics in real life
(**) Music: A Mathematical Offering by Dave Benson covers many topics about the interplay between mathematics and music.
(*) A Practitioner’s Guide to Asset Allocation by William Kinlaw, Mark P. Kritzman, David Turkington. The mathematical details are very light in this book, the focus is put on the models, the history and controversies around models and some actual data about typical correlations, returns of asset classes (this is scarcer than one would think for a book of finance!). If you want to invest by yourself and know about how to diversify, this is a very good starting point.
(**) Portfolio Optimization and Performance Analysis by Jean-Luc Prigent. If you liked the previous book and want to dig (much deeper) in portfolio optimization, this book is a detailed analysis of the existing models.
(*) Euler: The Master of Us All great book about the works of Euler, as the title indicates. The chapter are organized according to the branches of mathematics Euler contributed to (all of them, at his time), and the proofs are the proofs he presented at the time.
Théorème vivant (in French) this one is hard to describe. Do not expect to understand precisely the contents of Cedric Villani’s work by reading this book. Likewise, the equations come with few explanations. It is more like a diary of a researcher.