The Navier-Stokes equations capture in a few succinct terms one of the most ubiquitous features of the physical world: the flow of fluids. The equations, which date to the 1820s, are today used to model everything from ocean currents to turbulence in the wake of an airplane to the flow of blood in the heart.
While physicists consider the equations to be as reliable as a hammer, mathematicians eye them warily. To a mathematician, it means little that the equations appear to work. They want proof that the equations are unfailing: that no matter the fluid, and no matter how far into the future you forecast its flow, the mathematics of the equations will still hold. Such a guarantee has proved elusive. The first person (or team) to prove that the Navier-Stokes equations will always work—or to provide an example where they don’t—stands to win one of seven Millennium Prize Problems endowed by the Clay Mathematics Institute, along with the associated $1 million reward.
Mathematicians have developed many ways of trying to solve the problem. New work posted online in September raises serious questions about whether one of the main approaches pursued over the years will succeed. The paper, by Tristan Buckmaster and Vlad Vicol of Princeton University, is the first result to find that under certain assumptions, the Navier-Stokes equations provide inconsistent descriptions of the physical world.
“We’re figuring out some of the inherent issues with these equations and why it’s quite possible [that] people have to rethink them,” said Buckmaster.
Buckmaster and Vicol’s work shows that when you allow solutions to the Navier-Stokes equations to be very rough (like a sketch rather than a photograph), the equations start to output nonsense: They say that the same fluid, from the same starting conditions, could end up in two (or more) very different states. It could flow one way or a completely different way. If that were the case, then the equations don’t reliably reflect the physical world they were designed to describe.
Blowing Up the Equations
To see how the equations can break down, first imagine the flow of an ocean current. Within it there may be a multitude of crosscurrents, with some parts moving in one direction at one speed and other areas moving in other directions at other speeds. These crosscurrents interact with one another in a continually evolving interplay of friction and water pressure that determines how the fluid flows.
Mathematicians model that interplay using a map that tells you the direction and magnitude of the current at every position in the fluid. This map, which is called a vector field, is a snapshot of the internal dynamics of a fluid. The Navier-Stokes equations take that snapshot and play it forward, telling you exactly what the vector field will look like at every subsequent moment in time.
The equations work. They describe fluid flows as reliably as Newton’s equations predict the future positions of the planets; physicists employ them all the time, and they’ve consistently matched experimental results. Mathematicians, however, want more than anecdotal confirmation—they want proof that the equations are inviolate, that no matter what vector field you start with, and no matter how far into the future you play it, the equations always give you a unique new vector field.
This is the subject of the Millennium Prize problem, which asks whether the Navier-Stokes equations have solutions (where solutions are in essence a vector field) for all starting points for all moments in time. These solutions have to provide the exact direction and magnitude of the current at every point in the fluid. Solutions that provide information at such infinitely fine resolution are called “smooth” solutions. With a smooth solution, every point in the field has an associated vector that allows you to travel “smoothly” over the field without ever getting stuck at a point that has no vector—a point from which you don’t know where to move next.
Smooth solutions are a complete representation of the physical world, but mathematically speaking, they may not always exist. Mathematicians who work on equations like Navier-Stokes worry about this kind of scenario: You’re running the Navier-Stokes equations and observing how a vector field changes. After some finite amount of time, the equations tell you a particle in the fluid is moving infinitely fast. That would be a problem. The equations involve measuring changes in properties like pressure, friction, and velocity in the fluid — in the jargon, they take “derivatives” of these quantities — but you can’t take the derivative of an infinite value any more than you can divide by zero. So if the equations produce an infinite value, you can say they’ve broken down, or “blown up.” They can no longer describe subsequent states of your fluid.
Blowup is also a strong hint that your equations are missing something about the physical world they’re supposed to describe. “Maybe the equation is not capturing all the effects of the real fluid because in a real fluid we don’t expect” particles to ever start moving infinitely fast, said Buckmaster.
Solving the Millennium Prize problem involves either showing that blowup never happens for the Navier-Stokes equations or identifying the circumstances under which it does. One strategy mathematicians have pursued to do that is to first relax just how descriptive they require solutions to the equations to be.
From Weak to Smooth
When mathematicians study equations like Navier-Stokes, they sometimes start by broadening their definition of what counts as a solution. Smooth solutions require maximal information — in the case of Navier-Stokes, they require that you have a vector at every point in the vector field associated with the fluid. But what if you slackened your requirements and said that you only needed to be able to compute a vector for some points or only needed to be able to approximate vectors? These kinds of solutions are called “weak” solutions. They allow mathematicians to start feeling out the behavior of an equation without having to do all the work of finding smooth solutions (which may be impossible to do in practice).
“From a certain point of view, weak solutions are even easier to describe than actual solutions because you have to know much less,” said Camillo De Lellis, coauthor with László Székelyhidi of several important papers that laid the groundwork for Buckmaster and Vicol’s work.
Weak solutions come in gradations of weakness. If you think of a smooth solution as a mathematical image of a fluid down to infinitely fine resolution, weak solutions are like the 32-bit, or 16-bit, or 8-bit version of that picture (depending on how weak you allow them to be).
In 1934 the French mathematician Jean Leray defined an important class of weak solutions. Rather than working with exact vectors, “Leray solutions” take the average value of vectors in small neighborhoods of the vector field. Leray proved that it’s always possible to solve the Navier-Stokes equations when you allow your solutions to take this particular form. In other words, Leray solutions never blow up.
Leray’s achievement established a new approach to the Navier-Stokes problem: Start with Leray solutions, which you know always exist, and see if you can convert them into smooth solutions, which you want to prove always exist. It’s a process akin to starting with a crude picture and seeing if you can gradually dial up the resolution to get a perfect image of something real.
“One possible strategy is to show these weak Leray solutions are smooth, and if you show they’re smooth, you’ve solved the original Millennium Prize problem,” said Buckmaster.
There’s one more catch. Solutions to the Navier-Stokes equations correspond to real physical events, and physical events happen in just one way. Given that, you’d like your equations to have only one set of unique solutions. If the equations give you multiple possible solutions, they’ve failed.
Because of this, mathematicians will be able to use Leray solutions to solve the Millennium Prize problem only if Leray solutions are unique. Nonunique Leray solutions would mean that, according to the rules of Navier-Stokes, the exact same fluid from the exact same starting conditions could end up in two distinct physical states, which makes no physical sense and implies that the equations aren’t really describing what they’re supposed to describe.
Buckmaster and Vicol’s new result is the first to suggest that, for certain definitions of weak solutions, that might be the case.
In their new paper, Buckmaster and Vicol consider solutions that are even weaker than Leray solutions—solutions that involve the same averaging principle as Leray solutions but also relax one additional requirement (known as the “energy inequality”). They use a method called “convex integration,” which has its origins in work in geometry by the mathematician John Nash and was imported more recently into the study of fluids by De Lellis and Székelyhidi.
Using this approach, Buckmaster and Vicol prove that these very weak solutions to the Navier-Stokes equations are nonunique. They demonstrate, for example, that if you start with a completely calm fluid, like a glass of water sitting still by your bedside, two scenarios are possible. The first scenario is the obvious one: The water starts still and remains still forever. The second is fantastical but mathematically permissible: The water starts still, erupts in the middle of the night, then returns to stillness.
“This proves nonuniqueness because from zero initial data you can construct at least two objects,” said Vicol.
Buckmaster and Vicol prove the existence of many nonunique weak solutions (not just the two described above) to the Navier-Stokes equations. The significance of this remains to be seen. At a certain point, weak solutions might become so weak that they stop really bearing on the smoother solutions they’re meant to imitate. If that’s the case, then Buckmaster and Vicol’s result might not lead far.
“Their result is certainly a warning, but you could argue it’s a warning for the weakest notion of weak solutions. There are many layers [of stronger solutions] on which you could still hope for much better behavior” in the Navier-Stokes equations, said De Lellis.
Buckmaster and Vicol are also thinking in terms of layers, and they have their sights set on Leray solutions—proving that those, too, allow for a multitrack physics in which the same fluid from the same position can take on more than one future form.
“Tristan and I think Leray solutions are not unique. We don’t have that yet, but our work is laying the foundation for how you’d attack the problem,” said Vicol.
Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.
Glitzy ceremony honours work including that on mapping post-big bang primordial light, cell biology, plant science and neurodegenerative diseases
The most glitzy event on the scientific calendar took place on Sunday night when the Breakthrough Foundation gave away $22m (16.3m) in prizes to dozens of physicists, biologists and mathematicians at a ceremony in Silicon Valley.
The winners this year include five researchers who won $3m (2.2m) each for their work on cell biology, plant science and neurodegenerative diseases, two mathematicians, and a team of 27 physicists who mapped the primordial light that warmed the universe moments after the big bang 13.8 billion years ago.
Now in their sixth year, the Breakthrough prizes are backed by Yuri Milner, a Silicon Valley tech investor, Mark Zuckerberg of Facebook and his wife Priscilla Chan, Anne Wojcicki from the DNA testing company 23andMe, and Googles Sergey Brin. Launched by Milner in 2012, the awards aim to make rock stars of scientists and raise their profile in the public consciousness.
The annual ceremony at Nasas Ames Research Center in California provides a rare opportunity for some of the worlds leading minds to rub shoulders with celebrities, who this year included Morgan Freeman as host, fellow actors Kerry Washington and Mila Kunis, and Miss USA 2017 Kra McCullough. When Joe Polchinski at the University of California in Santa Barbara shared the physics prize last year, he conceded his nieces and nephews would know more about the A-list attendees than he would.
Oxford University geneticist Kim Nasmyth won for his work on chromosomes but said he had not worked out what to do with the windfall. Its a wonderful bonus, but not something you expect, he said. Its a huge amount of money, I havent had time to think it through. On being recognised for what amounts to his lifes work, he added: You have to do science because you want to know, not because you want to get recognition. If you do what it takes to please other people, youll lose your moral compass. Nasmyth has won lucrative awards before and channelled some of his winnings into Gregor Mendels former monastery in Brno.
Another life sciences prizewinner, Joanne Chory at the Salk Institute in San Diego, was honoured for three decades of painstaking research into the genetic programs that flip into action when plants find themselves plunged into shade. Her work revealed that plants can sense when a nearby competitor is about to steal their light, sparking a growth spurt in response. The plants detect threatening neighbours by sensing a surge in the particular wavelengths of red light that are given off by vegetation.
Chory now has ambitious plans to breed plants that can suck vast quantities of carbon dioxide out of the atmosphere in a bid to combat climate change. She believes that crops could be selected to absorb 20 times more of the greenhouse gas than they do today, and convert it into suberin, a waxy material found in roots and bark that breaks down incredibly slowly in soil. If we can do this on 5% of the landmass people are growing crops on, we can take out 50% of global human emissions, she said.
Three other life sciences prizes went to Kazutoshi Mori at Kyoto University and Peter Walter for their work on quality control mechanisms that keep cells healthy, and to Don Cleveland at the University of California, San Diego, for his research on motor neurone disease.
The $3m Breakthrough prize in mathematics was shared by two British-born mathematicians, Christopher Hacon at the University of Utah and James McKernan at the University of California in San Diego. The pair made major contributions to a field of mathematics known as birational algebraic geometry, which sets the rules for projecting abstract objects with more than 1,000 dimensions onto lower-dimensional surfaces. It gets very technical, very quickly, said McKernan.
Speaking before the ceremony, Hacon was feeling a little unnerved. Its really not a mathematician kind of thing, but Ill probably survive, he said. Ive got a tux ready, but Im not keen on wearing it. Asked what he might do with his share of the winnings, Hacon was nothing if not realistic. Ill start by paying taxes, he said. And I have six kids, so the rest will evaporate.
Chuck Bennett, an astrophysicist at Johns Hopkins University in Baltimore, led a Nasa mission known as the Wilkinson Microwave Anisotropy Probe (WMAP) to map the faint afterglow of the big bangs radiation that now permeates the universe. The achievement, now more than a decade old, won the 27-strong science team the $3m Breakthrough prize in fundamental physics. When we made our first maps of the sky, I thought these are beautiful, Bennett told the Guardian. It is still absolutely amazing to me. We can look directly back in time.
Bennett believes that the prizes may help raise the profile of science at a time when it is sorely needed. The point is not to make rock stars of us, but of the science itself, he said. I dont think people realise how big a role science plays in their lives. In everything you do, from the moment you wake up to the moment you go to sleep, theres something about what youre doing that involves scientific advances. I dont think people think about that at all.
Using a single thread roughly 1-2 km long (0.6 – 1.2 mi), Petros Vrellis continuously wraps the thread in straight, continuous lines, from one peg to its direct opposite peg in a circular, 28″ loom with 200 evenly spaced anchor pegs on its circumference. Thus each artwork is made from 3,000 – 4,000 continuously intersecting straight lines of a single thread.
Interestingly, knitting is done by hand, with step-by-step instructions dictated by a computer algorithm designed by the new media artist. Vrellis explains:
“The pattern is generated from a specially designed algorithm, coded in openframeworks. The algorithm takes as input a digital photograph and outputs the knitting pattern. Over 2 billion calculations are needed to produce each pattern.”
For ‘inputs’, Vrellis used famous portraits by the famous Spanish Renaissance artist El Greco. Below you can see a timelapse video along with close-ups of Petros’ experimental knitting project. For more information check out his official website. If you’re interested in purchasing any of the original artworks you can see what’s currently available on Saatchi Art.
At the heart of every financial services firm’s operations is a team of data scientists whose job it is to take all of the information that comes in and structure it in a way that the rocket scientists and genius mathematicians on staff can turn into something useful for their equations and analysis.
It’s a time-consuming, labor-intensive and difficult job that only a select few can handle. Those select few have now launched Crux Informatics to take over the data processing that big banks need done.
The company is emerging from stealth with a $10 million investment led by Goldman Sachs, to manage what the company calls the “information supply chain” that’s beginning to get out of hand for most big institutions that live and die by data analysis.
Goldman Sachs Principal Strategic Investments Group led the new round, which included additional undisclosed institutional investors, and will be used to expand the product suite for its large customers in the financial services industry.
“The emergence of unstructured data as an important input into the investment process creates a great opportunity for financial institutions, but only if actionable insights can be extrapolated from it,” said Darren Cohen, global head of Goldman Sachs’ Principal Strategic Investments group, in a statement. “Crux’s innovative approach — coupled with their deep expertise in financial services and capital markets — brings economies of scale that will allow companies to be more agile, inventive and effective with data.”
Think of Crux as a Switzerland for data storage and services. The company won’t reveal any information or resell to anyone else the proprietary information it processes and holds for its clients. It’s merely a processing engine for taking the data that big banks and businesses that depend on big data sets need, and crunches that data — reducing it to the metrics that matter most for the clients it serves.
“Gathering information about the world, doing analysis, and driving unique insight is the life-blood of the financial industry”, said Philip Brittan, the chief executive officer of Crux, in a statement. “It is a hard process filled with numerous pain points. Crux is a unique new offering, created to help our clients much more easily find, explore, and make use of relevant data. We take on the burdensome and non-differentiating aspects of our customers’ information supply chains, so they can focus on what really matters for their business. In doing so, we strive to make data delightful.”
Brittan and his founding executive team have a deep knowledge of the ways in which these data sets work. The former chief technology officer of Thomson Reuters financial risk division, Brittan previously worked as the former head of foreign exchange at Bloomberg and as the former chief of Google Finance before founding Crux. He’s joined by Larry Leibowitz, who previously worked as the COO and head of global equities listing and trading at Euronext, and Elizabeth Pritchard, who heads the company’s go-to-market strategy and previously worked at Goldman Sachs for 19 years as the company’s head of Market Data Services.
On top of the deep data expertise of Crux’s founding team, the company has also rolled out a toolset for creating data pipelines, a set of APIs to connect with existing financial services applications and a provisioning system to provide audit trails to track who used the data in an organization.
In an ongoing series of artworks entitled ‘Ciclotramas‘, Brazilian artist Janaina Mello Landini unravels ropes into incredible fractal patterns that evoke tree roots, river basins, lightning strikes and circulatory systems.
Landini has been developing this concept since 2010, using threads and strings to create site-specific installations that occupy the space in an immersive way. She adds:
The idea is to â€œunstitchâ€ Time from its inside, unraveling the threads of the same rope in constant bifurcations, until the last indivisible stage is reached, a point that holds everything together in perfect equilibrium.
At the headquarters of Cloudflare, in San Francisco, there’s a wall of lava lamps: the Entropy Wall. They’re used to generate random numbers and keep a good bit of the internet secure: here’s how.
For a technical overview of the Entropy Wall click here.
Video by YouTuber Tom Scott
Exclusive: one of the greatest conceptual breakthroughs in mathematics has been traced to the Bakhshali manuscript, dating from the 3rd or 4th century
Nowt, nada, zilch: there is nothing new about nothingness. But the moment that the absence of stuff became zero, a number in its own right, is regarded as one of the greatest breakthroughs in the history of mathematics.
Now scientists have traced the origins of this conceptual leap to an ancient Indian text, known as the Bakhshali manuscript a text which has been housed in the UK since 1902.
Radiocarbon dating reveals the fragmentary text, which is inscribed on 70 pieces of birch bark and contains hundreds of zeroes, dates to as early as the 3rd or 4th century about 500 years older than scholars previously believed. This makes it the worlds oldest recorded origin of the zero symbol that we use today.
Dating from 1,000 years before Pythagorass theorem, the Babylonian clay tablet is a trigonometric table more accurate than any today, say researchers
At least 1,000 years before the Greek mathematician Pythagoras looked at a right angled triangle and worked out that the square of the longest side is always equal to the sum of the squares of the other two, an unknown Babylonian genius took a clay tablet and a reed pen and marked out not just the same theorem, but a series of trigonometry tables which scientists claim are more accurate than any available today.
The 3,700-year-old broken clay tablet survives in the collections of Columbia University, and scientists now believe they have cracked its secrets.
The team from the University of New South Wales in Sydney believe that the four columns and 15 rows of cuneiform wedge shaped indentations made in the wet clay represent the worlds oldest and most accurate working trigonometric table, a working tool which could have been used in surveying, and in calculating how to construct temples, palaces and pyramids.
The fabled sophistication of Babylonian architecture and engineering is borne out by excavation. The Hanging Gardens of Babylon, believed by some archaeologists to have been a planted step pyramid with a complex artificial watering system, was written of by Greek historians as one of the seven wonders of the ancient world.
Daniel Mansfield, of the universitys school of mathematics and statistics, described the tablet which may unlock some of their methods as a fascinating mathematical work that demonstrates undoubted genius with potential modern application because the base 60 used in calculations by the Babylonians permitted many more accurate fractions than the contemporary base 10.
Finally, the world is safe from Italian economists doing mathematics on a plane.
Alarm bells were rung last Thursday on a flight from Philadelphia to Ontario, after a passenger saw aman suspiciously writing down a complicated looking formula on a piece of paper and notified cabin crew. The passenger told flight attendants she was feeling ill, causing the flight to turnaround on the runway.
After some confusion, the mysterious mathematicsenthusiastwas taken off the flight and questioned by security agents.
Fortunately for international security, the man was actually Guido Menzio, an Italian-born associate professor in Economics at the University of Pennsylvania, who also happened to be a young, dark-haired, bearded, and slightly tanned male with a foreign accent on a plane.
Menzio told theAssociated Press: “I thought they were trying to get clues about her illness. Instead, they tell me that the woman was concerned that I was a terrorist because I was writing strange things on a pad of paper.”
His scrawlings were actually some last minute work on a differential equation that he was preparing for a lecture on Search Theory in Canada.
Aftertwo hours of questioning,Menzio,who said he was treated with respect,was able to explain himself to the security officials andwas allowed back onto the flight. The passenger who complained, however, did not return to the flight.