This intermittent series published by the American Mathematical Society began in 1993. The newest entry, the thirteenth, continues the practice of choosing eight to ten topics of broader interest that made news in some way over the last few years. The “mathematical sciences” in this series generally include strictly pure mathematical topics as well as applications and often something in the realm of computational science. Mackenzie notes in his introduction that “what’s new” is often related to “what’s old” in illuminating ways, something that is amply demonstrated here.

The current book is well-written with its articles all of broad interest. In this volume perhaps more chapters than usual address questions in geometry (broadly considered), but as a whole the book addresses a pleasing variety of topics ranging from number theory, deep learning, and new approaches to causality in chaotic dynamical systems to singularities in fluid mechanics. The individual chapters are written at a level mostly accessible to undergraduates in mathematics. Some subjects touch on more advanced theory, but each of those makes accommodations for readers with less background.

Rather than listing the subjects of all the chapters, I will point to some of the items that seem particularly attention-worthy to me. One chapter deals with fascinating questions about the packing of spheres, and – in particular – the proof in 2016 describing the densest packing of spheres in 8 and 24 dimensions. Two other chapters explore visualizations of the geometry of the universe in Thurston’s eight possible three-dimensional geometries, and then describe how a determination of where Lincoln stood at Gettysburg during his address there was reconstructed from photographs taken at the time.

Number theory appears too. James Maynard’s theorem of 2016 says, for example, that there are an infinite number of primes that omit the digit 7, for example, or any other single digit that you might choose. He found that, despite the the large collection of powerful tools at his disposal, the classic “circle method” of Hardy and Littlewood, gave what he needed and provided a very elegant proof.

Artificial intelligence in the form of ChatGPT and GPT-4 emerged with big stories in 2022. The article here on deep learning involves a very interesting mix of mathematics, computer science, and statistics, and it attempts to separate the hype from the reality. It discusses how the neural networks can be inclined to make mistakes, how they might be improved, and how AI-created documents could be identified by digital watermarking.

This is an appealing and attractively presented collection, well worth your attention.

Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.