This is the eighth volume of the series "What's Happening in the Mathematical Sciences". The goal of this book, and of the whole series, is to give account for some recent progress in mathematics. The topics covered in the nine chapters of this book range from the high-dimensional topology to quantum chaos and include applications in computer science, medicine, financial markets, ...
This is the eighth volume of the series "What's Happening in the Mathematical Sciences". The series, published by the American Mathematical Society, started in 1993 and its goal is to shed light on some of the outstanding recent progress in both pure and applied mathematics.
The book is divided into nine chapters which present some remarkable mathematical achievements.
The first chapter "Accounting for Taste" describes how Netflix, a movie rental company, offered a million-dollar prize for a computer algorithm to recommend videos to customers. The first year of competition identified matrix factorization as the best single approach. However to factor matrices with unknown elements the winner team had to devise their own strategy combining matrix factorization with regularization and gradient descent. After three years of competition the award was given to the team called BellKor’s Pragmatic Chaos. This is an example of the use of mathematics behind the scenes in everyday life.
The second chapter "A Brave New Symplectic World" is devoted to the conjecture of Weinstein saying that certain kinds of dynamical systems with two degrees of freedom always have periodic solutions. The conjecture was proposed in the late 1970s as a problem in symplectic topology and solved thirty years later by Cliff Taubes. The remarkable thing is that Taube's solution does not stay within the original discipline and borrows some ideas from string theory, developed by physicist Edward Witten.
"Mathematics and the Financial Crisis" described the collapse of the world's financial markets in 2008. The Black-Scholes formula to estimate the value of call options is explained in detail. For some time this formula was almost perfect but a mathematical model is only as good as its assumptions.
"The Ultimate Billiard Shot" deals with the game of outer billiards proposed in 1959 by Bernhard Neumann. The outer billiard table is infinitely large and it has a hole in the center. The question is: Does the table need to be infinitely large? In other words, is there any way a ball that starts near the central region can spiral out to infinity? The answer depends on the shape of the hole. In 2007, Schwartz proved that for certain shapes, an outer billiards shot cannot be contained in any bounded region. The game of outer billiards may seem a bit restricted but is of interest to mathematicians as a toy model of planetary motion.
The fifth chapter, "Simpatient", deals with the controversial recommendation in 2009 by the U.S. Preventive Services Task Force that women aged 40-49 should no longer be advised to have an annual mammogram. A public health panel used six breast cancer model to take this decision. This is an example of the growing acceptance of mathematical models for medical decision-making, at least behind the scenes.
"Instant Randomness" addresses questions of the following type: How long does it take to mix milk in a coffee cup, neutrons in an atomic reactor, atoms in a gas, or electron spins in a magnet? In many systems the onset of randomness is quite sudden. This abrupt mixing behavior is the "cutoff phenomenon", and the time when it occurs is called the mixing time.
Quantum chaos is the topic of the seventh chapter "In Search of Quantum Chaos". In the 1970s and 1980s chaos theory revolutionized the study of classical dynamical systems. In the atomic and subatomic realm chaos seems to be absent. However, there is a gray zone, the semiclassical limit, between he quantum world and the macroscopic world. Mathematicians have recently confirmed the occurrence of quantum chaos in this zone.
Even in the twenty-first century mathematics reveal new phenomena in the ordinary three-dimensional space. This is the topic of the chapter "3-D Surprises". In 2008 and 2009, some new ways to pack tetrahedra extremely densely were discovered. In 2005, two engineers in Hungary discovered a new three-dimensional object similar to a tetrahedron but with curvy sides. It is the first homogeneous, self-righting (and self-wronging!) object.
Last chapter is "As One Heroic Age Ends, a New One Begins". In the 1950s John Milnor constructed 7-dimensional "exotic spheres" which are identical to normal spheres from the viewpoint of continuous topology, but different from the viewpoint of smooth topology. This was the starting point of a new era of high-dimensional topology. But one question, the Kervaire Invariant One problem was open for more than forty years. In 2009 three mathematicians, Mike Hill, Michael Hopkins and Doug Ravenel, answered this question. But this may be just the beginning of what topologists will learn from the new machinery used to solved this problem.
The book is well written and can be of interest to both mathematicians and general public with some background in mathematics. Many pictures and illustrative diagrams are included in the book.