What's Happening in the Mathematical Sciences is a series of publications that report on the latest mathematical research in a way that both mathematicians and non-mathematicians can appreciate. Volume 10 is co-authored by Dana Mackenzie and Barry Cipra; Volumes 7 through 9 are by Dana Mackenzie; Volume 6 is co-authored by Dana Mackenzie and Barry Cipra; Volumes 1 through 5 were written by Barry Cipra.
The table of contents of each and the full text of selected articles are below.
The ancient Japanese art of paper-folding is going high-tech, as engineers invent new devices that deploy or undeply folding. These inventions lead in turn to challenging mathematical problems about assembly pathways, defects, and curved folds in flat materials.
Mathematics got its real-life Walter Mitty story in 2013, when Yitang Zhang shocked number theorists with the first finite upper bound on the minimum size of prime gaps. One of the oldest problems in number theory, the Twin Prime Conjecture, may now be within reach.
When you pay a stranger, especially online, for help, how can you be sure you're getting honest answers? A new theory in computer science shows how rational self-interest dovetails with the pressing need for trustworthy computation.
Change is everywhere you look in Earth’s climate, and always has been. Throughout climate science, mathematical models help sort out what did happen (mass extinctions), what is happening (melting ice sheets), and what might happen (tipping points).
In a story published in 1905, Sherlock Holmes incorrectly deduced which way a bicycle went, on the basis of its tracks. The subtle relationship between a bike’s front and rear tracks recently helped mathematicians solve another Victorian-era problem on the operation of planimeters.
A proof is a kind of mathematical poem—and sometimes an epic one, at that. Two recent proofs, each years in the making, show the lengths to which mathematicians will go in the dogged pursuit of truth, including, these days, enlisting computers to double check their logic.
Great problems come in many disguises. The Kadison-Singer problem, first posed as a problem in theoretical physics, popped up in many other mathematical contexts over more than half a century until it was finally solved in 2013 graph theorists.
Finding shapes that tile the plane isn’t hard to do. Finding all of them is trickier. Mathematicians still don't know how many different convex pentagons are capable of tiling the plane. But the list, long stalled at 14, just inched up, thanks to a new algorithm and a computer search.
In the last few years, professional sports have been swept a new wave of statistical methods, or “analytics.” These methods, coupled with new data sources like video capture, quantify elusive skills and challenge cherished assumptions about in-game strategy.
In 1963, Peter Higgs predicted the existence of the Higgs field, which explains why many subatomic particles have nonzero mass. This prediction was based on a mere page of mathematical calculation and an inspired analogy with the Landau-Ginzburg theory of phase transitions. Almost half a century and more than $9 billion later, experimental physicists at CERN finally chased down the Higgs boson, the most elusive quarry in physics. The discovery filled in the last missing piece of the Standard Model of quantum physics, and spectacularly vindicated the use of abstract symmetry principles to discover new physical phenomena.
In 2012, geometers in rapid succession solved three open problems concerning optimal geometries of tori (inner tubes). The Willmore Conjecture identified the torus with least bending energy; the Lawson Conjecture identified the torus in the hypersphere (a sphere in four-dimensional space) with the least surface area; and the Pinkall-Sterling Conjecture classified all of the tori in the hypersphere that minimize area subject to a volume constraint. Tubing has never been so much fun!
The great Japanese tsunami of 2011 killed more than 15,000 people in a country that had been better prepared for tsunamis than any other in the world. The tragedy highlighted gaps in our scientific understanding of this hugely destructive natural phenomenon, and even more so in the public’s understanding. It also pointed out some ways in which mathematical models got it right, but not fast enough.
An interdisciplinary team at UCLA discovered how to adapt a mathematical model developed for earthquake prediction to identify likely crime “hot spots.” Two field tests of their software were resounding successes, and predictive policing was recognized in Time magazine and other media outlets as one of the top scientific discoveries of 2011.
Three decades ago, three-dimensional topology seemed like a wild, untamed jungle of disparate examples. Then William Thurston proposed a series of conjectures that brought some order to the chaos. In 2012, just months before Thurston’s death, the Virtual Haken Conjecture and Virtual Fibering Conjecture were finally proved, showing that almost all three-dimensional manifolds are descended from templates constructed in an elementary fashion.
Rubik’s cube, the mathematician’s favorite toy, continues to attract new fans and inspire new research. While “speedcubers” developed new algorithms (and manual dexterity) to solve the cube in less than 10 seconds, mathematicians proved that an omniscient being could always solve the classic, 3-by-3-by-3 cube in 20 moves or less.
In 2009 the world experienced its first flu pandemic in forty years. Fortunately it turned out much milder than the three great pandemics of the twentieth century, but it provided an ideal opportunity to test a variety of mathematical simulations in real time. Conclusions: The simulations worked pretty well, and communications between modelers and field workers were excellent, but the late delivery of vaccine would have been a fiasco in a worse epidemic.
Topic modeling is a new statistical technique named after its ability to identify topics (such as genetics or climate change) in a large body of documents. While still in its early days, it has proved hugely popular in the fields of “digital humanities” and it might enable social-networking websites to respond automatically and anonymously to cyber-bullying.
Blessed (or perhaps cursed?) with a catchy name, tropical geometry enables mathematicians to solve difficult problems in classical algebraic geometry by making simple combinatorial models that look a lot like stick figures. This new type of geometry also has surprising applications to string theory in physics, evolutionary trees in biology, and the scheduling of trains.
Unfortunately, Volume 9 is no longer available.
Netflix, a movie rental company, offered a million-dollar prize for a computer algorithm that could significantly upgrade the company's ability to predict its customers' likes and dislikes. The wildly successful competition upset some conventional wisdom in the field of machine learning and vividly demonstrated the power of "crowdsourcing."
Thirty years ago, Alan Weinstein conjectured that certain kinds of dynamical systems with two degrees of freedom, such as a pendulum that is free to stretch as well as swing, always have periodic (repeating) solutions. At the time, no one had any idea how to prove his conjecture. But that was before "symplectic camels" and a "crazy" homology theory based on the equations for a magnetic monopole. Conclusion: Periodic orbits exist. (Magnetic monopoles still don't.)
In the first decade of the 2000s, investors embraced credit derivatives: a clever, formula-based method to parcel out the risk of subprime loans and profit from a booming housing market. But they forgot--or chose to ignore--that a mathematical model is only as good as its assumptions.
In an inside-out version of billiards that you can't quite fit into your garage (or even the Milky Way), two groups of researchers find different ways to sink the cue ball into a pocket at infinity.
Mathematical models of patient populations have begun to supplement or replace clinical trials in cases where the trials would be difficult or impossible to perform. In 2009, a public health panel used six breast cancer models to arrive at a controversial recommendation that women aged 40-49 should no longer be advised to have an annual mammogram.
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 the Ising model of magnetism, mathematicians have calculated a unique cutoff time when the system abruptly goes from essentially unmixed to almost completely mixed. The same behavior is expected for other systems described by the theory of Markov chains.
A classical billiard ball, on a table with curved, "dispersing" sides, travels on a chaotic trajectory that essentially randomizes the ball's position over the long term. However, at low energies, quantum billiard balls are not chaotic. In a tour de force combining pure number theory with physics, mathematicians proved that quantum chaos does emerge at high energies.
Even in the twenty-first century, mathematics can still reveal new phenomena in ordinary three-dimensional space. Item 1: a stunningly efficient way to pack tetrahedra. Item 2: the Gömböc, a homogeneous body that automatically rights (and wrongs) itself.
John Milnor's discovery of "exotic spheres" in 1956 ushered in a new era of high-dimensional topology, with powerful new tools like framed cobordism theory, stable homotopy theory, and surgery. But one question, called the Kervaire Invariant One problem, remained stubbornly unanswered for more than 40 years, until three topologists found "a shortcut to Mount Everest."
Unfortunately, Volume 8 is no longer available.Order What's Happening in the Mathematical Sciences, Volume 7 on the AMS Bookstore
Two seemingly unrelated kinds of knots--modular knots and Lorenz knots--turn out to be the same. For number theorists, the payoff is a new way to look at an old concept, the modular surface. For dynamical systems theorists, it's a new way to understand the beginnings of chaos.
Fifteen years ago, Andrew Wiles proved Fermat's Last Theorem, thereby solving the most famous problem in mathematics. The aftershocks are still being felt. A team of number theorists, including Wiles collaborator, Richard Taylor, adapts Wiles' machinery to polish off another celebrated unsolved problem in number theory.
Persi Diaconis, Susan Holmes and Richard Montgomery tell you how to get the odds in your favor the next time you flip a coin. And no, you don't have to cheat.
An innocent puzzle about tiling a checkerboard with dominos leads, ultimately, to a new model of random surfaces. The theory is both simple enough to be exactly solvable, and complex enough to include phase transitions between "solid," "liquid," and "gaseous" states--a combination that has never occurred previously in statistical physics.
New "metamaterials" may soon bring invisibility cloaks--and inaudibility cloaks, too--out of the realm of fantasy and into reality. Unlike Harry Potter's cloak, these devices will work strictly on the principles of mathematics, not magic.
In algebraic geometry, the Mori Minimal Model Program is an ambitious effort to match the geometry of surfaces defined by polynomial equations to their algebra. After 100 years of carrying out the program one dimension at a time, algebraic geometers race to the top of the ladder.
More than 1000 years ago, an unknown scribe copied some of Archimedes' works onto parchment. But 200 years later, another scribe erased them and wrote a prayer book over the top. Over the last ten years, high-tech imaging methods have penetrated through the damage done by centuries of neglect and abuse to reveal what Archimedes, the greatest mathematician of ancient Greece, really wrote.
Racing against time and the incurable illness of one of their colleagues, a team of twenty group theorists puts together a detailed survey of "the most beautiful structure in mathematics"-- the exceptional Lie group, E8.
Proving that more is not always better, mathematicians show that "sparse" or "compressible" signals, such as digital photographs and cell phone messages, can be reconstructed from many fewer measurements than engineers previously supposed. As an extreme proof of principle, engineers design the world's first single-pixel camera.
See links to Cipra and Mackenzie talking about the subject of each chapter.
In 2002, Grigory Perelman announced a solution to the Poincaré Conjecture, a problem in topology selected in 2000 as one of the seven leading math challenges of the millennium. After more than three years of scrutiny, mathematicians are cautiously accepting his proof, which uses a geometric partial differential equation called "Ricci flow," first studied by Richard Hamilton.
While three-dimensional topologists awaited a verdict on the Poincaré Conjecture (which deals with closed manifolds), several of them solved a suite of long-standing conjectures about the ends of open manifolds.
Pi lovers rejoice! You can now stick your finger into pi and pluck out any digit (say, the trillionth one) without having to compute all the preceding digits. The only catch is that you have to count in sixteens.
A question posed by an undergraduate student--can a Venn diagram be rotationally symmetric?--leads to some beautiful "doilies," intricate mathematics, and (after forty years) a solution found by another student.
numbers surprised number theorists. A team of Indian mathematicians, including two students, discovers the first polynomial-time algorithm for testing to see if a number is prime. American, Turkish, and Hungarian mathematicians collaborate to show that small gaps between consecutive primes occur much more often than anyone had previously proved. And another international collaboration finds evenly spaced (but non-consecutive) prime sequences of any desired length.
Manjul Bhargava started with a simple idea of putting numbers into a box. What came out of the box, eventually, was a whole new way to combine and to count objects in algebraic number theory.
From Jupiter's Great Red Spot to the eddies in a stream, vortices are a familiar feature of fluid flow. Mathematicians have gained new insights into the formation and long-term survival of vortices in both two- and three-dimensional fluids.
How do water striders move on a nearly frictionless surface? How do dragonflies hover? These and other conundrums of biology can be explained by the mathematics of the Navier-Stokes equations.
A new random process called Schramm-Loewner evolution turns out to be a good model for a variety of physical phenomena, from the random jitters of air molecules to phase transitions of a magnetic material. The key mathematical property of these systems, proven in some cases and still conjectural in others, is conformal invariance.
Some algorithms of computer science, such as the simplex algorithm for solving linear programming problems, work better than they are supposed to. A new measure of complexity, called smoothed analysis, shows why the standard worst-case scenarios are so misleading.
Progress proceeds apace in the post-Fermat world of elliptic curves and modular forms.
Powerful statistical methods are helping researchers elucidate the three-dimensional structure of life's most important molecules.
A centuries old problem--the Kepler conjecture--has yielded to new insights and some dogged computation.
Is the universe finite? Observations of the cosmic microwave background and a new mathematical algorithm may provide an answer.
Computer models are helping researchers understand where traffic jams come from--and maybe what to do about them.
Plimpton 322 is one of mathematicians' favorite cuneiform tablets. But what did it mean to the scribe who composed it?
Researchers have found a short distance from theory to applications in the study of small world networks.
New methods have revealed a multitude of solutions to an old problem in celestial mechanics: the orbital motion of three bodies.
The Clay Mathematics Institute has singled out seven important problems in mathematics, with a $1 million dollar prize for each.
A new theorem helps explain why statistical physics has had such a hard time with one of its central problems.
Deep Blue's victory over Garry Kasparov is the end of a long road in computer chess, but the mathematical study of "perfect" play in combinatorial games has an even longer ways to go.
Quantum chaologists and analytic number theorists have their sights set on a mysterious mathematical object: the Riemann zeta function. (Please Note: This article is available in PDF format.)
Paul Erdős, the "Johnny Appleseed" of mathematics, inspired hundreds of colleagues and left a rich legacy of mathematical problems--and solutions.
Computer power has brought algorithmic questions in algebraic geometry back to the fore.
An automated theorem prover succeeeds in settling a decades-old conjecture in symbolic logic. What's left for human minds to do?
A collaboration between mathematicians and biologists has led to the first experimental evidence for chaotic dynamics in a population.
Mathematical discoveries are shaping research in a potentially revolutionary kind of computing, based on principles of quantum mechanics.
A breakthrough in the theory of computational complexity has implications for cryptographic systems with "guaranteed" security.
Mathematics and art have more in commmon than is commonly supposed. Two twentieth-century artists, M. C. Escher and Marcel Duchamp, used mathematics as an inspiration for works of art.
Henri Poincaré's thoughts on thinking, written near the beginning of the twentieth century, are well worth repeating at century's end.
Andrew Wiles has completed the astonishing tour de force that resolves the most famous problem in mathematics. His proof is the realization of a boyhood dream.
A breakthrough in theoretical physics has simplified a notoriously difficult theory in 4-dimensional geometry, and given mathematicians a lot to think about.
Will computers of the future be bio-engineered? It's a possibility.
Thomas Nicely set out to study prime numbers that occur in pairs. Along the way, he discovered that Intel's Pentium chip couldn't divide. (Please Note: This article is available in PDF format.)
Modern technology relies on mathematical control theory to keep things on an even keel. How do the equations know what to do?
Mathematical techniques, faster computers, and better algorithms are gaining ground in the study of complex fluid flows. For some researchers, computtational turbulence is literally a pipe dream.
The continuously increasing power of computers has enabled researchers to take a discrete look at the world. Theorists seek to explain the complex patterns that are often seen.
Researchers are working hard to simplify one of the most complicated proofs in mathematical history--the classification of simple groups.
A computational challenge in number theory has been met, considerably sooner than the 20,000 years it was expected to take.
A theorem about multivariate integration may find a home on Wall Street. You could call it a get-rich-quick scheme.
The announcement last year of a proof of Fermat's Last Theorem stunned the mathematical world. Andrew Wiles's proof, though currently incomplete, has nonetheless drawn rave reviews.
What's the quickest way to untie a knot? Researchers have untangled a good part of the answer.
Will compact waves cruise the information superhighways of the future? In theory, it's possible.
A team of mathematicians, computer scientists, and engineers has designed a new medical imaging technology based on the safe application of electric current
Mathematicians and scientists are rapidly learning to speak a new language. The results are making a big splash.
Computer scientists will do anything to avoid bottlenecks and speed up computations. But gamble on the results? You bet!
Undergraduate students at a summer mathematics research program have found some slick answers to some old problems about the geometry of soap bubbles.
A complicated class of error-correcting codes has suddenly gotten much easier to use.
A combinatorial problem, long thought to be difficult, has finally been solved--with surprising ease.
A clever construction "pulls the plug" on a 40-year old conjecture about the topology of vector fields.
Unfortunately, Volume 2 is no longer available.
Mathematicians are working with biologists to delve into some of the most challenging problems in biology today, from understanding the human immune system to "computing" the human heart.
Can a computer be trusted when it produces a proof so long and complicated that no human can check the details? Theorists have cooked up a new way to tell whether or not a computer proof is right.
Can you hear the shape of a drum? is a famous problem that asks if two drums that look different can make the same sound. After decades of head-scratching, mathematicians have come up with the answer.
Mathematicians have been teaming up with scientists to work on solving environmental problems, from ocean modeling to dealing with hazardous waste.
Intuition about our three-dimensional world can be surprisingly misleading when it comes to higher dimensions, as two recent results in geometry show.
An unusual blend of differential geometry and dynamical systems has led to an important theoretical result about the number of closed "geodesic" curves on distorted spheres.
Growing crystals--on a computer? Mathematicians are helping materials scientists to better understand the nature of crystals, while picking up some challenging mathematical problems along the way.
A group of talented and inquisitive undergraduates "camped out" last summer at the Geometry Center. Using sophisticated computer graphics and their own imaginations, they came up with some fascinating mathematics.
Strange as it may sound, there are composite numbers that "masquerade" as primes. A group of mathematicians trying to hunt down these prime imposters ended up proving there are infinitely many of them.
How many colors are needed to distinguish neighboring colors on a map? The famous Four Color Theorem notwithstanding, this is a challenging problem in graph theory--especially when your maps aren't flat.
Unfortunately, Volume 1 is no longer available.