What’s Happening in the Mathematical Sciences, Volume 12

As always, What's Happening in the Mathematical Sciences presents a selection of topics in mathematics that have attracted particular attention in recent years. This volume is dominated by an event that shook the world in 2020 and 2021, the coronavirus (or COVID-19) pandemic. While the world turned to politicians and physicians for guidance, mathematicians played a key role in the background, forecasting the epidemic and providing rational frameworks for making decisions. The first three chapters of this book highlight several of their contributions, ranging from advising governors and city councils to predicting the effect of vaccines to identifying possibly dangerous “escape variants” that could re-infect people who already had the disease. In recent years, scientists have sounded louder and louder alarms about another global threat: climate change. Climatologists predict that the frequency of hurricanes and waves of extreme heat will change. But to even define an “extreme” or a “change,” let alone to predict the direction of change, is not a climate problem: it's a math problem. Mathematicians have been developing new techniques, and reviving old ones, to help climate modelers make such assessments. In a more light-hearted vein, “Descartes' Homework” describes how a famous mathematician's blunder led to the discovery of new properties of foam-like structures called Apollonian packings. “Square Pegs and Squiggly Holes” shows that square pegs fit virtually any kind of hole, not just circular ones. “Much Ado About Zero” explains how difficult problems about eigenvalues of matrices can sometimes be answered by playing a simple game that involves coloring dots on a grid or a graph. Finally, “Dancing on the Edge of the Impossible” provides a progress report on one of the oldest and still most important challenges in number theory: to devise an effective algorithm for finding all of the rational-number points on an algebraic curve. In the great majority of cases, number theorists know that the number of solutions is finite, yet they cannot tell when they have found the last one. However, two recently proposed methods show potential for breaking the impasse.