What's Happening in the Mathematical Sciences : Volume 13

Exploring topics ranging from artificial intelligence to non-Euclidean geometry, the 13th volume of What's Happening in the Mathematical Sciences by Dana Mackenzie and Leila Sloman is an exquisitely crafted and skillfully authored examination of cutting-edge advancements in mathematics.

The What’s Happening in the Mathematical Sciences series presents a selection of recent discoveries and exciting fields of research in mathematics, explained in depth but in a slow-paced, reader-friendly way.

In the first few months of 2023, artificial “brains” like ChatGPT and GPT-4 were constantly in the news, and they have already turned into big business. One chapter in this book, “Deep Learning: Part Math, Part Alchemy”, explains how math disentangles hype from reality and explains some of the remarkable advances of machine learning. Meanwhile, “Organizing the Chaos Inside the Brain” explores animal brains, and describes how biologists can apply chaos theory to simulate the wanderings of a fly from firing data on neurons within its brain.

This issue of What's Happening also includes many treats for readers who like pure math—especially those who are interested in geometry. In recent months and years, there have been unexpected discoveries in tiling (“One Stone to Rule Them All”), sphere-packing in more than three dimensions (“A Fascination of Spheres”) and the reconstruction of three-dimensional scenes from two-dimensional images (“Multi-View Geometry: E Pluribus Unum”). The chapter “How to Draw an Alternate Universe” will, as promised, open a door to a completely different, non-Euclidean universe—or several of them. Shakespeare’s words, “something rich and strange”, only begin to describe them.

In “How Mathematicians Unearthed the Stubborn Secrets of Fano Varieties”, readers will learn about one of the building blocks of algebraic geometry, the branch of geometry that deals with surfaces defined by polynomial equations. The chapter “Missing One Digit” addresses a seemingly elementary problem in number theory: how many prime numbers do not have a “7” in them? The answer is easy to guess—but hard to prove. “Fluid Flow: Two Paths to a Singularity” discusses another guess that is hard to prove: can fluids in an enclosed region develop “singularities” akin to a breaking wave? Computer evidence is mounting that they can—including some evidence from machine learning algorithms. (Which brings us full circle back to the “Deep Learning” chapter.)

Dana Mackenzie has written for the What's Happening series since Volume 6, published in 2006. In this volume he is joined by Leila Sloman, whose name will be familiar to many readers from her work for Quanta Magazine.