What's Happening in the Mathematical Sciences, Volume 11

The prolific Dana Mackenzie has done it again! The 11th volume of What’s Happening in the Mathematical Sciences, recently published by AMS, gives the reader a comprehensive overview of exactly what its title suggests and what is badly needed, a readable overview of some of the (arguably) major contributions of math to the real world in the past few years.

 

What’s Happening in the Mathematical Sciences has 9 short chapters (135 pages), so it is very accessible for those who are short of time but want to get a glimpse of some recent advancements. Anyone with high-level math and above who reads the book will be able to understand the essence of the topics discussed, while those with an academic background in math or the physical sciences will be able to understand most of the material, so this book is really for anyone who is math-inclined and curious. Every chapter is self-contained, divided into sub-sections going from easy to hard, with examples, illustrations and color images that all help to make the content understandable. 

 

The topics are really broad and range from the popular, such as the math of dieting or a challenging problem from the game of “SET”, to the important role math plays in the top scientific discoveries and research of today - the detection of gravitational waves, data science, and quantum computers. Most chapters deal with the intersection between math and real-world problems, like the chapter that discusses the mathematics of commuting, or the very relevant issue of gerrymandering - redrawing election districts to favor one party. All include up-to-date results, even for historical problems like the chapter on the asymptotic Fermat’s last theorem.

 

The chapter that I liked the most, although it was very hard to choose since all chapters are so well-written, was the chapter on expanding graphs, also known as expanders. The chapter begins by describing the importance of studying these graphs in various contexts such as computer science and brain modeling. Then, the basics are covered, taking painstaking trouble to make sure all definitions are understood. After reading this part, readers will know what Heawood graphs are and will be able to grasp the concepts of adjacency matrices, eigenvectors and spectral gaps. Once this is in place, Mackenzie presents the works of the pioneers in the field, such as Kolmagorov and Barzdin, Lubotzky, Phillips and Sarnak, and others. Then, the chapter shows the many uses of expanders and the current state of research up to the present day - from error codes to knot theory to face recognition and other implementations in computer science. The chapter ends with a discussion of expanders in high dimensions. Mackenzie does justice to many contemporary researchers whose work he describes. What I really liked about this chapter is how Mackenzie takes a topic from beginning to end and shows all the milestones of its evolution. I also liked how elegantly and seamlessly he alternates between applied math and theoretical math.

 

Two things are missing from the book, and I recommend reconsidering for upcoming volumes: an index and references. Although we live in the Google age, good indexing and referencing still help, especially for those who want to delve deeper into the subjects.

 

For me as an educator and a recreational mathematician always looking for new, interesting subjects, this book was exactly what I needed and filled what I always felt was a void in my knowledge. I would really like to know how Mackenzie chooses the topics - perhaps this can also be addressed in future volumes.

 

Dana Mackenzie is a well-known author of many popular science books, including five of the previous volumes in the series, two of them co-authored with Barry Cipra. He has the impressive ability to write in a way that is accessible yet not too simple, making What’s Happening in the Mathematical Sciences not just a book for those who want to broaden their horizons in math and science, but also a book that is useful for people who are looking for exciting topics to continue studying. I, for one, lapped up every page of the book!

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Yossi Elran specializes in recreational math and works at the Davidson Institute of Science Education, the educational arm of the Weizmann Institute of Science, Rehovot, Israel. He is the co-author of the Paper Puzzle Book. His email address is: yossi.elran@weizmann.ac.il