The hugely influential book on how the understanding of causality revolutionized science and the world, by the pioneer of artificial intelligence 'Wonderful ... illuminating and fun to read' Daniel Kahneman, Nobel Prize-winner and author of Thinking, Fast and Slow 'Correlation does not imply causation.' For decades, this mantra was invoked by scientists in order to avoid taking positions as to whether one thing caused another, such as smoking and cancer, or carbon dioxide and global warming. But today, that taboo is dead. The causal revolution, sparked by world-renowned computer scientist Judea Pearl and his colleagues, has cut through a century of confusion and placed cause and effect on a firm scientific basis. Now, Pearl and science journalist Dana Mackenzie explain causal thinking to general readers for the first time, showing how it allows us to explore the world that is and the worlds that could have been. It is the essence of human and artificial intelligence. And just as Pearl's discoveries have enabled machines to think better, The Book of Why explains how we too can think better. 'Pearl's accomplishments over the last 30 years have provided the theoretical basis for progress in artificial intelligence and have redefined the term "thinking machine"' Vint Cerf

This book presents Maslov's canonical operator method for finding asymptotic solutions of pseudo differential equations. The classical WKB method, so named in honor of its authors: Wentzel, Kramers and Brillouin, was created for finding quasi classical approximations in quantum mechanics. The simplicity, obviousness and "physicalness" of this method quickly made it popular: specialists in mathematical physics accepted it unequivocally as one of the weapons in their arsenal. The number of publications which are connected with the WKB method in one way or another can probably no longer be counted. The alternative name of the WKB method in diffraction problem- the ray method or the method of geometric optics - indicates that the approximations in the WKB method are constructed by means of rays. More precisely, the first approximation of the WKB method is constructed by means of rays (isolating the singular part), after which the usual methods of the (regular) theory of perturbations are applied. However, the ray method is not applicable at the points of space where the rays focus or form a caustic. Mathematically this fact expresses itself in the fact that the amplitude of the waves at such points become infinite.

GENESIS REVISED

"It takes a certain amount of courage to step beyond one’s day-to-day experiments and look at the big picture–and the origin of the Moon is a ‘big picture’ question par excellence. Perhaps it makes sense that William Hartmann, one of the two scientists who unraveled the Moon’s biggest mystery, is not only a scientist but also a part-time artist and science fiction writer. It took someone with an artist’s eye and a fiction writer’s speculative temperament to see the big picture.
"This is a book about that big picture: the origin of the Moon, as interpreted by Hartmann and Alastair Cameron, the second patriarch of ‘The Big Splat.’ It is also about a doomed planet called Theia, and a familiar one called Earth that used to look vastly different from today’s Earth. But, most of all, it is about a long lineage of intellectual voyagers who began exploring the Moon long before Neil Armstrong planted his boot into the lunar dust."
–– From the Introduction

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.

What's Happening in the Mathematical Sciences is a collection of articles highlighting some of the most recent developments in mathematics. These include important achievements in pure mathematics, as well as its fascinating applications. On the pure mathematics side, ``Prime Clusters and Gaps: Out-Experting the Experts'' talks about new insights into the distribution of prime numbers, the perpetual source of new problems, and new results. Recently, several mathematicians (including Yitang Zhang and James Maynard) significantly improved our knowledge of the distribution of prime numbers. Advances in the so-called Kadison-Singer problem and its applications in signal processing algorithms used to analyze and synthesize signals are described in ``The Kadison-Singer Problem: A Fine Balance''. ``Quod Erat Demonstrandum'' presents two examples of perseverance in mathematicians' pursuit of truth using, in particular, computers to verify their arguments. And ``Following in Sherlock Holmes' Bike Tracks'' shows how an episode in one of Sir Arthur Conan Doyle's stories about Sherlock Holmes naturally led to very interesting problems and results in the theory of completely integrable systems. On the applied side, ``Climate Past, Present, and Future'' shows the importance of mathematics in the study of climate change and global warming phenomena. Mathematical models help researchers to understand the past, present, and future changes of climate, and to analyze their consequences. ``The Truth Shall Set Your Fee'' talks about algorithms of information exchange in cyberspace. Economists have known for a long time that trust is a cornerstone of commerce, and this becomes even more important nowadays when a lot of transactions, big and small, are done over the Internet. Recent efforts of theoretical computer scientists led to the development of so-called ``rational protocols'' for information exchange, where the parties in the information exchange process find that lies do not pay off. Over the last 100 years many professional mathematicians and devoted amateurs contributed to the problem of finding polygons that can tile the plane, e.g., used as floor tiles in large rooms and walls. Despite all of these efforts, the search is not yet complete, as the very recent discovery of a new plane-tiling pentagon shows in ``A Pentagonal Search Pays Off''. Mathematics can benefit coaches and players in some of the most popular team sports as shown in ``The Brave New World of Sports Analytics''. The increased ability to collect and process statistics, big data, or ``analytics'' has completely changed the world of sports analytics. The use of modern methods of statistical modeling allows coaches and players to create much more detailed game plans as well as create many new ways of measuring a player's value. Finally, ``Origami: Unfolding the Future'' talks about the ancient Japanese paper-folding art and origami's unexpected connections to a variety of areas including mathematics, technology, and education.

The ""AMS"" series ""What's Happening in the Mathematical Sciences"" distills the amazingly rich brew of current research in mathematics down to a few choice samples. This volume leads off with an update on the Poincare Conjecture, a hundred-year-old problem that has apparently been solved by Grigory Perelman of St. Petersburg, Russia. So what did topologists do when the oldest and most famous problem about closed manifolds was vanquished? As the second chapter describes, they confronted a suite of problems concerning the 'ends' of open manifolds...and solved those, too. Not to be outdone, number theorists accomplished several unexpected feats in the first five years of the new century, from computing a trillion digits of pi to finding arbitrarily long equally-spaced sequences of prime numbers.Undergraduates made key discoveries, as explained in the chapters on Venn diagrams and primality testing. In applied mathematics, the Navier-Stokes equations of fluid mechanics continued to stir up interest. One team proved new theorems about the long-term evolution of vortices, while others explored the surprising ways that insects use vortices to move around. The random jittering of Brownian motion became a little less mysterious. Finally, an old and trusted algorithm of computer science had its trustworthiness explained in a novel way. Barry Cipra explains these new developments in his wry and witty style, familiar to readers of Volumes 1-5, and is joined in this volume by Dana Mackenzie. Volume 6 of ""What's Happening"" will convey to all readers - from mathematical novices to experts - the beauty and wonder that is mathematics.