What's Happening in the Mathematical Sciences (What's Happening in the Mathermatical Sciences)

The goal of the series is to shed light on topics on the leading edge of mathematical research in a way that is accessible to the mathematical layperson. The articles frequently combine mathematics with physics, and are written in a lively style that should be accessible to anyone with genuine interest and some college-level experience in mathematics and science. --Choice What's Happening in the Mathematical Sciences showcases the remarkable recent progress in pure and applied mathematics. Once again, there are some surprises, where we discover new properties of familiar things, in this case tightly-packed tetrahedra or curious turtle-like shapes that right themselves. Mathematics also has played significant roles in current events, most notably the financial crisis, but also in screening for breast cancer. The Netflix competition to find a better algorithm for recommending videos to subscribers demonstrated how deeply mathematics is used behind the scenes in our everyday lives. Mathematicians have settled several important conjectures in the past few years. In topology, the recently solved Kervaire invariant conjecture tells us about exotic spheres in high dimension. The Weinstein conjecture, proved by Cliff Taubes, guarantees periodicity in certain important dynamical systems. A very old dynamical system--the game of billiards--received two innovative makeovers. First, mathematicians proved the existence of "wandering" trajectories in an inside-out version of the game, called "outer billiards," which some researchers consider a toy model for planetary motion. Second, mathematicians proved two different versions of the Quantum Unique Ergodicity conjecture, which says that a quantum-mechanical billiard ball behaves, in the long term (and at high energies) similarly to a classical billiard ball. The proof uses ideas from pure number theory dating back to Ramanujan. Finally, in another area of statistical physics, mathematicians showed that the transition from an unmixed to a mixed system often happens, relatively speaking, in the blink of an eye. Dana Mackenzie, a science and mathematics writer, makes the mathematics and the applications easily comprehensible, by calling on common sense or on similar but familiar phenomena. The stories invite you into the exciting world of modern mathematics, with its thrill of discovery and the anticipation of what is still to come. Anyone with an interest in mathematics, from high school teachers and college students to engineers and computer scientists, will find something of interest here. The stories are well told and the mathematics is gripping.