What are tariffs, and how do they work?

In 1930, as the Great Depression ravaged the United States, President Herbert Hoover made a bold economic decision. With the goal of supporting American farmers and manufacturers, he approved a sweeping 20% tax on all imported agricultural and industrial goods. He believed making foreign goods more expensive would encourage consumers to buy domestic, thus funneling cash into American businesses and creating more jobs. But Hoover’s plan completely backfired. Other countries retaliated by taxing American goods, leading exports to fall and global trade to decline. In the end, this policy led American prices to spike and caused even more unemployment. These disastrous outcomes of the Smoot-Hawley Tariff Act were no surprise to the over 1,000 economists who publicly condemned the bill. But this wouldn’t be the last time a world leader was shocked by a tariff’s consequences.

So, what exactly is a tariff? A tariff is a tax an importer pays to bring foreign goods into a country. For example, if they import $1,000 worth of goods at a 10% tariff rate, they’ll pay $100 to the government in addition to $1,000 to the seller. Foreign exporters could respond to a tariff by reducing their product’s price to keep an importer’s business. But if they don’t, importers bear this cost and then increase their prices so consumers will make up the difference. This may sound straightforward, but enacting a tariff kicks off a complicated economic chain reaction. So to keep things simple, let’s look at some examples from just one country to explore some of the factors that make a tariff’s outcomes tricky to predict.

First, tariffs can be applied to different countries at different rates. For example, in 2024, America had a 6% tariff on all wool entering the country. However, if that wool came from Mexico or South Korea, it would be tariff-free due to trade agreements. Meanwhile, importers getting wool from Russia could face a hefty 55.5% tariff. Policies like these might help domestic wool producers, or they might just encourage importers to buy from specific foreign producers.

Things get even trickier when trying to predict how tariffs impact jobs. Sometimes the jobs they create come at a high cost. In 2018, the US imposed a tariff on washing machines that created 1,800 jobs and raised roughly $82 million in government revenue. However, it also increased the average price of washing machines by $86, costing consumers roughly $1.5 billion. This means consumers paid roughly $817,000 to create each of those jobs.

In other cases, the jobs tariffs create come at the cost of existing jobs. In 2002, the Bush administration imposed tariffs as high as 30% on imported steel. Employment in US steel production had been decaying for decades, so by making foreign steel more expensive, the tariff sought to create new jobs for US steel manufacturers. Initially, the policy did slow down job loss. But since US steel still cost more than pre-tariff foreign steel, domestic industries that used this material had to increase their prices. This made their products less competitive internationally, where foreign manufacturers were still using cheap steel. Within a year of enacting the tariff, the US lost far more jobs in steel-related industries than they gained in steel-producing industries.

Clearly, how other countries respond to tariffs is another big part of the ripple effect these policies have. One clear example comes from 2018, when the US attempted to address its trade deficit with China by taxing Chinese goods. While these tariffs protected domestic businesses and raised government revenue, they also set off a trade war. China retaliated by raising tariffs on US exporters, including farmers relying on Chinese markets for soybean sales. In the end, this led to higher prices for US consumers and a slight drop in America’s GDP.

Trade wars aren’t new. And even in the 18th and early 19th centuries, when certain tariffs did help struggling industries find their domestic footing, they were mostly used to raise government funds. But over the last 200 years, our global economy has become far more interdependent. Today, there are more manufacturing components being imported than ever before, many of which are used to make a wide variety of consumer goods. This complexity means that even a seemingly simple tariff can trigger a global chain reaction, creating more consequences than anyone can plan for.

The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2026 was awarded to Robert E. Hall and Michael Woodford “for microeconomic foundations of macroeconomic stabilisation.”

How does the Nobel Peace Prize work?

What do a 17-year-old Pakistani, a Norwegian explorer, a Tibetan monk, and an American pastor have in common? They were all awarded the Nobel Peace Prize. Among the top prestigious awards in the world, this prize has honored some of the most celebrated and revered international figures and organizations in history.

To understand how it all got started, we have to go back to the 1800s. Swedish chemist Alfred Nobel was then mostly known for the invention of dynamite, a breakthrough which launched his career as a successful inventor and businessman. 30 years later, he had become extremely wealthy, but never married, and had no children. When his will was opened after his death, it came as a surprise that his fortune was to be used for five prizes in physics, chemistry, medicine, literature, and peace. These prizes illustrated his lifelong commitment to sciences and his passion for literature.

But what about peace? Because Nobel’s name was tied to inventions used in the war industry, many have assumed that he created the peace prize out of regret. However, this is all speculation as he never expressed any such sentiments, and his inventions were also used for constructive purposes. Instead, many historians connect Alfred Nobel’s interest for the peace cause to his decade-long friendship and correspondence with an Austrian pacifist named Bertha Von Suttner. Von Suttner was one of the leaders of the international peace movement, and in 1905, after Nobel’s death, she became the first woman to be awarded the Nobel Peace Prize.

Nobel’s will outlined three criteria for the Peace Prize, which unlike the other Sweden-based prizes, would be administered in Norway. Disarmament, peace congresses, and brotherhood between nations. These standards have since been expanded to include other ways of promoting peace, such as human rights and negotiations. And the prize doesn’t just have to go to one person. About a third of Noble Peace Prizes have been shared by two or three laureates.

So how do nominations for the prize work? According to the Nobel Foundation, a valid nomination can come from a member of a national assembly, state government, or an international court. Eligible nominators also include university rectors, professors of the social sciences, history, philosophy, law, and theology, and previous recipients of the Peace Prize. But if you want to know more about who was recently nominated, you’ll have to be patient. All information about nominations remains secret for 50 years. Take Martin Luther King Jr. We didn’t actually know who nominated him until 2014. His nominators turned out to be the Quakers, who had won the prize previously, and eight members of the Swedish Parliament.

There’s no limit to the number of times a person or organization can be nominated. In fact, Jane Addams, recognized as the founder of social work in the United States, was nominated 91 times before finally being awarded the prize. The absence of a laureate can also be symbolic. The 1948 decision not to award the prize following the death of Mahatma Gandhi has been interpreted as an attempt to respectfully honor the so-called missing laureate. As with the other Nobel Prizes, the Peace Prize can’t be awarded posthumously.

The secret selection process takes almost a year, and is carried out by the five appointed members of the Norwegian Nobel Committee who are forbidden from having any official political function in Norway. Starting with a large pool of nominations, exceeding 300 in recent years, they access each candidate’s work and create a short list. Finally, the chairman of the Nobel Committee publicly announces the laureate in October. The awards ceremony takes place on December 10th, the anniversary of Alfred Nobel‘s death. The prize itself includes a gold medal inscribed with the Latin words, “Pro pace et fraternitate gentium,” or “For the peace and brotherhood of men,” as well as a diploma and a large cash prize. Recently, it’s been 8 million Swedish kronor, or roughly a million US dollars, which is split in the case of multiple laureates. And while laureates can use the prize money as they choose, in recent years, many have donated it to humanitarian or social causes.

For many years, the Nobel Peace Prize was predominately awarded to European and North American men. But in recent years, significant changes have been taking place, making the prize more global than ever. 23 organizations and 103 individuals, that’s 87 men and 16 women, have made up the 126 Nobel Peace Prize laureates in history. They include Desmond Tutu for his nonviolent campaign against apartheid in South Africa, Jody Williams for her campaign to ban and clear anti-personnel mines, Rigoberta Menchú Tum for her work for social justice and reconciliation based on respect for the rights of Indigenous peoples, Martti Ahtisaari for his efforts to resolve international conflicts in Namibia, Kosovo, and Indonesia, and Aung San Suu Kyi for her nonviolent struggle for democracy and human rights in Myanmar.

They’re just a few examples of the people who have inspired us, challenged us, and demonstrated through their actions that there are many paths to peace.

The Nobel Peace Prize 2026 was awarded to Navanethem “Navi” Pillay “for her efforts to promote peace and international law.”

Short Talk on Pain

by Anne Carson

Lawns and fields and hills and wide old velvet
sleeves, green things. They stretch, fold, roll away,
unfurl and calm the eye. Look lush in paintings.
Battles are fought on greens. Or you could spread
a meal and sup. How secretly they lie, floors of
distant forests. Next comes the grave, in many a
poem about green. But this is not a poem. This is a
billboard for frozen green peas. Frozen green peas
are good for pain.

Copyright © 2023 by Anne Carson. Originally published in Poem-a-Day on August 25, 2023, by the Academy of American Poets.


A crash course in organic chemistry - Jakob Magolan - TEDxUIdaho

I’d like you to ask yourself, what do you feel when you hear the words “organic chemistry?” What comes to mind? There is a course offered at nearly every university, and it’s called Organic Chemistry, and it is a grueling, heavy introduction to the subject, a flood of content that overwhelms students, and you have to ace it if you want to become a doctor or a dentist or a veterinarian. And that is why so many students perceive this science like this … as an obstacle in their path, and they fear it and they hate it and they call it a weed-out course. What a cruel thing for a subject to do to young people, weed them out. And this perception spread beyond college campuses long ago. There is a universal anxiety about these two words.

I happen to love this science, and I think this position in which we have placed it is inexcusable. It’s not good for science, and it’s not good for society, and I don’t think it has to be this way. And I don’t mean that this class should be easier. It shouldn’t. But your perception of these two words should not be defined by the experiences of premed students who frankly are going through a very anxious time of their lives. So I’m here today because I believe that a basic knowledge of organic chemistry is valuable, and I think that it can be made accessible to everybody, and I’d like to prove that to you today. Would you let me try?

Audience: Yeah!

Jakob Magolan: All right, let’s go for it.

Here I have one of these overpriced EpiPens. Inside it is a drug called epinephrine. Epinephrine can restart the beat of my heart, or it could stop a life-threatening allergic reaction. An injection of this right here will do it. It would be like turning the ignition switch in my body’s fight-or-flight machinery. My heart rate, my blood pressure would go up so blood could rush to my muscles. My pupils would dilate. I would feel a wave of strength. Epinephrine has been the difference between life and death for many people. This is like a little miracle that you can hold in your fingers.

Here is the chemical structure of epinephrine. This is what organic chemistry looks like. It looks like lines and letters … No meaning to most people. I’d like to show you what I see when I look at that picture. I see a physical object that has depth and rotating parts, and it’s moving. We call this a compound or a molecule, and it is 26 atoms that are stitched together by atomic bonds. The unique arrangement of these atoms gives epinephrine its identity,
but nobody has ever actually seen one of these, because they’re very small, so we’re going to call this an artistic impression, and I want to explain to you how small this is. In here, I have less than half a milligram of it dissolved in water. It’s the mass of a grain of sand. The number of epinephrine molecules in here is one quintillion. That’s 18 zeroes. That number is hard to visualize. Seven billion of us on this planet? Maybe 400 billion stars in our galaxy? You’re not even close. If you wanted to get into the right ballpark, you’d have to imagine every grain of sand on every beach, under all the oceans and lakes, and then shrink them all so they fit in here.

Epinephrine is so small we will never see it, not through any microscope ever, but we know what it looks like, because it shows itself through some sophisticated machines with fancy names like “nuclear magnetic resonance spectrometers.” So visible or not, we know this molecule very well. We know it is made of four different types of atoms, hydrogen, carbon, oxygen and nitrogen. These are the colors we typically use for them. Everything in our universe is made of little spheres that we call atoms. There’s about a hundred of these basic ingredients, and they’re all made from three smaller particles: protons, neutrons, electrons. We arrange these atoms into this familiar table. We give them each a name and a number. But life as we know it doesn’t need all of these, just a smaller subset, just these. And there are four atoms in particular that stand apart from the rest as the main building blocks of life, and they are the same ones that are found in epinephrine: hydrogen, carbon, nitrogen and oxygen. Now what I tell you next is the most important part. When these atoms connect to form molecules, they follow a set of rules. Hydrogen makes one bond, oxygen always makes two, nitrogen makes three and carbon makes four. That’s it. HONC – one, two, three, four. If you can count to four, and you can misspell the word “honk,” you’re going to remember this for the rest of your lives.

Now here I have four bowls with these ingredients. We can use these to build molecules. Let’s start with epinephrine. Now, these bonds between atoms, they’re made of electrons. Atoms use electrons like arms to reach out and hold their neighbors. Two electrons in each bond, like a handshake, and like a handshake, they are not permanent. They can let go of one atom and grab another. That’s what we call a chemical reaction, when atoms exchange partners and make new molecules. The backbone of epinephrine is made mostly of carbon atoms, and that’s common. Carbon is life’s favorite structural building material, because it makes a good number of handshakes with just the right grip strength. That’s why we define organic chemistry as the study of carbon molecules.

Now, if we build the smallest molecules we can think of that follow our rules, they highlight our rules, and they have familiar names: water, ammonia and methane, \( H_{2}0 \) and \( NH_{3} \) and \( CH_{4} \). The words “hydrogen,” “oxygen” and “nitrogen” – we use the same words to name these three molecules that have two atoms each. They still follow the rules, because they have one, two and three bonds between them. That’s why oxygen gets called \( O_2 \).

I can show you combustion. Here’s carbon dioxide, \( CO_2 \). Above it, let’s place water and oxygen, and beside it, some flammable fuels. These fuels are made of just hydrogen and carbon. That’s why we call them hydrocarbons. We’re very creative.

So when these crash into molecules of oxygen, as they do in your engine or in your barbecues, they release energy and they reassemble, and every carbon atom ends up at the center of a \( CO_2 \) molecule, holding on to two oxygens, and all the hydrogens end up as parts of waters, and everybody follows the rules. They are not optional, and they’re not optional for bigger molecules either, like these three. This is our favorite vitamin sitting next to our favorite drug, and morphine is one of the most important stories in medical history. It marks medicine’s first real triumph over physical pain, and every molecule has a story, and they are all published. They’re written by scientists, and they’re read by other scientists, so we have handy representations to do this quickly on paper, and I need to teach you how to do that.

So we lay epinephrine flat on a page, and then we replace all the spheres with simple letters, and then the bonds that lie in the plane of the page, they just become regular lines, and the bonds that point forwards and backwards, they become little triangles, either solid or dashed to indicate depth. We don’t actually draw these carbons. We save time by just hiding them. They’re represented by corners between the bonds, and we also hide every hydrogen that’s bonded to a carbon. We know they’re there whenever a carbon is showing us any fewer than four bonds. The last thing that’s done is the bonds between OH and NH. We just get rid of those to make it cleaner, and that’s all there is to it. This is the professional way to draw molecules. This is what you see on Wikipedia pages.

It takes a little bit of practice, but I think everyone here could do it, but for today, this is epinephrine. This is also called adrenaline. They’re one and the same. It’s made by your adrenal glands. You have this molecule swimming through your body right now. It’s a natural molecule. This EpiPen would just give you a quick quintillion more of them.

We can extract epinephrine from the adrenal glands of sheep or cattle, but that’s not where this stuff comes from. We make this epinephrine in a factory by stitching together smaller molecules that come mostly from petroleum. And this is 100 percent synthetic. And that word, “synthetic,” makes some of us uncomfortable. It’s not like the word “natural,” which makes us feel safe. But these two molecules, they cannot be distinguished. We’re not talking about two cars that are coming off an assembly line here. A car can have a scratch on it, and you can’t scratch an atom. These two are identical in a surreal, almost mathematical sense. At this atomic scale, math practically touches reality. And a molecule of epinephrine … it has no memory of its origin. It just is what it is, and once you have it, the words “natural” and “synthetic,” they don’t matter, and nature synthesizes this molecule just like we do, except nature is much better at this than we are.

Before there was life on earth, all the molecules were small, simple: carbon dioxide, water, nitrogen, just simple things. The emergence of life changed that. Life brought biosynthetic factories that are powered by sunlight, and inside these factories, small molecules crash into each other and become large ones: carbohydrates, proteins, nucleic acids, multitudes of spectacular creations. Nature is the original organic chemist, and her construction also fills our sky with the oxygen gas we breathe, this high-energy oxygen.

All of these molecules are infused with the energy of the sun. They store it like batteries. So nature is made of chemicals. Maybe you guys can help me to reclaim this word, “chemical,” because it has been stolen from us. It doesn’t mean toxic, and it doesn’t mean harmful, and it doesn’t mean man-made or unnatural. It just means “stuff,” OK?

You can’t have chemical-free lump charcoal. That is ridiculous.

And I’d like to do one more word. The word “natural” doesn’t mean “safe,” and you all know that. Plenty of nature’s chemicals are quite toxic, and others are delicious, and some are both … toxic and delicious.

The only way to tell whether something is harmful is to test it, and I don’t mean you guys. Professional toxicologists: we have these people. They’re well-trained, and you should trust them like I do.

So nature’s molecules are everywhere, including the ones that have decomposed into these black mixtures that we call petroleum. We refine these molecules. There’s nothing unnatural about them. We purify them. Now, our dependence on them for energy – that means that every one of those carbons gets converted into a molecule of \( CO_2 \). That’s a greenhouse gas that is messing up our climate. Maybe knowing this chemistry will make that reality easier to accept for some people, I don’t know, but these molecules are not just fossil fuels. They’re also the cheapest available raw materials for doing something that we call synthesis. We’re using them like pieces of LEGO. We have learned how to connect them or break them apart with great control. I have done a lot of this myself, and I still think it’s amazing it’s even possible. What we do is kind of like assembling LEGO by dumping boxes of it into washing machines, but it works.

We can make molecules that are exact copies of nature, like epinephrine, or we can make creations of our own from scratch, like these two. One of these eases the symptoms of multiple sclerosis; the other one cures a type of blood cancer that we call T-cell lymphoma. A molecule with the right size and shape, it’s like a key in a lock, and when it fits, it interferes with the chemistry of a disease. That’s how drugs work. Natural or synthetic, they’re all just molecules that happen to fit snugly somewhere important.

But nature is much better at making them than we are, so hers look more impressive than ours, like this one. This is called vancomycin. She gave this majestic beast two chlorine atoms to wear like a pair of earrings. We found vancomycin in a puddle of mud in a jungle in Borneo in 1953. It’s made by a bacteria. We can’t synthesize this cost-efficiently in a lab. It’s too complicated for us, but we can harvest it from its natural source, and we do, because this is one of our most powerful antibiotics. And new molecules are reported in our literature every day. We make them or we find them in every corner of this planet. And that’s where drugs come from, and that’s why your doctors have amazing powers … to cure deadly infections and everything else.

Being a physician today is like being a knight in shining armor. They fight battles with courage and composure, but also with good equipment. So let’s not forget the role of the blacksmith in this picture, because without the blacksmith, things would look a little different …

But this science is bigger than medicine. It is oils and solvents and flavors, fabrics, all plastics, the cushions that you’re sitting on right now – they’re all manufactured, and they’re mostly carbon, so that makes all of it organic chemistry. This is a rich science.

I left out a lot today: phosphorus and sulfur and the other atoms, and why they all bond the way they do, and symmetry and non-bonding electrons, and atoms that are charged, and reactions and their mechanisms, and it goes on and on and on, and synthesis takes a long time to learn.

But I didn’t come here to teach you guys organic chemistry – I just wanted to show it to you, and I had a lot of help with that today from a young man named Weston Durland, and you’ve already seen him. He’s an undergraduate student in chemistry, and he also happens to be pretty good with computer graphics. So Weston designed all the moving molecules that you saw today. He and I wanted to demonstrate through the use of graphics like these to help someone talk about this intricate science. But our main goal was just to show you that organic chemistry is not something to be afraid of. It is, at its core, a window through which the beauty of the natural world looks richer.

Thank you.

The Nobel Prize in Chemistry 2026 was awarded to Henri B. Kagan and Kenso Soai “for the discovery of non-linear effects and autocatalysis in asymmetric organic synthesis.”

Why neutrinos matter

They’re everywhere, but you will never see one. Trillions of them are flying through you right this second, but you can’t feel them. These ghost particles are called neutrinos and if we can catch them, they can tell us about the furthest reaches and most extreme environments of the universe. Neutrinos are elementary particles, meaning that they can’t be subdivided into other particles the way atoms can. Elementary particles are the smallest known building blocks of everything in the universe, and the neutrino is one of the smallest of the small. A million times less massive than an electron, neutrinos fly easily through matter, unaffected by magnetic fields. In fact, they hardly ever interact with anything. That means that they can travel through the universe in a straight line for millions, or even billions, of years, safely carrying information about where they came from.

So where do they come from? Pretty much everywhere. They’re produced in your body from the radioactive decay of potassium. Cosmic rays hitting atoms in the Earth’s atmosphere create showers of them. They’re produced by nuclear reactions inside the sun and by radioactive decay inside the Earth. And we can generate them in nuclear reactors and particle accelerators.

But the highest energy neutrinos are born far out in space in environments that we know very little about. Something out there, maybe supermassive black holes, or maybe some cosmic dynamo we’ve yet to discover, accelerates cosmic rays to energies over a million times greater than anything human-built accelerators have achieved. These cosmic rays, most of which are protons, interact violently with the matter and radiation around them, producing high-energy neutrinos, which propagate out like cosmic breadcrumbs that can tell us about the locations and interiors of the universe’s most powerful cosmic engines. That is, if we can catch them. Neutrinos’ limited interactions with other matter might make them great messengers, but it also makes them extremely hard to detect.

One way to do so is to put a huge volume of pure transparent material in their path and wait for a neutrino to reveal itself by colliding with the nucleus of an atom. That’s what’s happening in Antarctica at IceCube, the world’s largest neutrino telescope. It’s set up within a cubic kilometer of ice that has been purified by the pressure of thousands of years of accumulated ice and snow, to the point where it’s one of the clearest solids on Earth. And even though it’s shot through with boreholes holding over 5,000 detectors, most of the cosmic neutrinos racing through IceCube will never leave a trace. But about ten times a year, a single high-energy neutrino collides with a molecule of ice, shooting off sparks of charged subatomic particles that travel faster through the ice than light does. In a similar way to how a jet that exceeds the speed of sound produces a sonic boom, these superluminal charged particles leave behind a cone of blue light, kind of a photonic boom. This light spreads through IceCube, hitting some of its detectors located over a mile beneath the surface. Photomultiplier tubes amplify the signal, which contains information about the charged particles’ paths and energies. The data are beamed to astrophysicists around the world
who look at the patterns of light for clues about the neutrinos that produced them.

These super energetic collisions are so rare that IceCube’s scientists give each neutrino nicknames, like Big Bird and Dr. Strangepork. IceCube has already observed the highest energy cosmic neutrinos ever seen. The neutrinos it detects should finally tell us where cosmic rays come from and how they reached such extreme energies. Light, from infrared, to x-rays, to gamma rays, has given us increasingly energetic and continuously surprising views of the universe. We are now at the dawn of the age of neutrino astronomy, and we have no idea what revelations IceCube and other neutrino telescopes may bring us about the universe’s most violent, most energetic phenomena.

  • The Nobel Prize in Physics 2026 was awarded to Francis Halzen “for decisive contributions to the IceCube Neutrino Observatory and the discovery of high-energy neutrinos of astrophysical origin.”
  • We are getting our first hints that neutrinos do have a small, but not-zero, mass. This could make us rethink some of our ideas about cosmology. - Particle physics - Very Short Introductions, by Frank Close
  • That all changed when IceCube captured the blazar neutrino from the Orion constellation. “When I go out at night and look at the sky, I make a map using light beams I detect with my eye,” says IceCube principal investigator Francis Halzen. “So, what we do is exactly the same thing, but we detect neutrino beams.” Halzen describes early neutrino maps of the sky as uniform and featureless with some emerging, but statistically insignificant, signs of it individual neutrino emitters.” Then this one source started to appear which was this supermassive black hole, which we subsequently discovered in this multi-messenger campaign, using other telescopes.” As Halzen sees it, IceCube’s discovery is the first confirmed neutrino star other than the sun.” The development is very similar to gamma ray astronomy from the ground. For years, they had only one source, which was the Crab Nebula. They optimized their instruments and built more, and now they’re doing astronomy. So, I hope the same will happen with neutrinos,” said Halzen. “In fact, I have no doubt.” - Ghost Particle - In Search of the Elusive and Mysterious Neutrino, chapter 9 Lifting a Veil on the Universe, by Alan Chodos and James Riordon

A light switch for neurons - Ed Boyden - TED2011

Think about your day for a second. You woke up, felt fresh air on your face as you walked out the door, encountered new colleagues and had great discussions and felt in awe when you found something new. But I bet there’s something you didn’t think about today, something so close to home, you probably don’t think about it very often at all. And that’s that all those sensations, feelings, decisions and actions are mediated by the computer in your head called your brain.

Now, the brain may not look like much from the outside – a couple pounds of pinkish-gray flesh, amorphous. But the last 100 years of neuroscience have allowed us to zoom in on the brain and to see the intricacy of what lies within. And they’ve told us that this brain is an incredibly complicated circuit made out of hundreds of billions of cells called neurons. Now, unlike a human-designed computer, where there’s a fairly small number of different parts, and we know how they work because we humans designed them, the brain is made out of thousands of different kinds of cells, maybe tens of thousands. They come in different shapes; they’re made out of different molecules; they project and connect to different brain regions. They also change in different ways in different disease states.

Let’s make it concrete. There’s a class of cells, a fairly small cell, an inhibitory cell, that quiets its neighbors. It’s one of the cells that seems to be atrophied in disorders like schizophrenia. It’s called the basket cell. And this cell is one of the thousands of kinds of cell that we’re learning about. New ones are being discovered every day. As just a second example: these pyramidal cells, large cells, can span a significant fraction of the brain. They’re excitatory. And these are some of the cells that might be overactive in disorders such as epilepsy. Every one of these cells is an incredible electrical device. They receive inputs from thousands of upstream partners and compute their own electrical outputs, which then, if they pass a certain threshold, will go to thousands of downstream partners. And this process, which takes just a millisecond or so, happens thousands of times a minute in every one of your 100 billion cells, as long as you live and think and feel.

So how are we going to figure out what this circuit does? Ideally, we could go through this circuit and turn these different kinds of cell on and off and see whether we could figure out which ones contribute to certain functions and which ones go wrong in certain pathologies. If we could activate cells, we could see what powers they can unleash, what they can initiate and sustain. If we could turn them off, then we could try and figure out what they’re necessary for. And that’s the story I’m going to tell you about today. And honestly, where we’ve gone through over the last 11 years, through an attempt to find ways of turning circuits and cells and parts and pathways of the brain on and off, both to understand the science and also to confront some of the issues that face us all as humans.

Now, before I tell you about the technology, the bad news is that a significant fraction of us in this room, if we live long enough, will encounter, perhaps, a brain disorder. Already, a billion people have had some kind of brain disorder that incapacitates them. The numbers don’t do it justice, though. These disorders – schizophrenia, Alzheimer’s, depression, addiction – they not only steal away our time to live, they change who we are. They take our identity and change our emotions and change who we are as people.

Now, in the 20th century, there was some hope that was generated through the development of pharmaceuticals for treating brain disorders. And while many drugs have been developed that can alleviate symptoms of brain disorders, practically none of them can be considered to be cured. In part, that’s because, if you think about it, we’re bathing the brain in a chemical –
this elaborate circuit, made of thousands of different kinds of cell – is being bathed in a substance. That’s also why most of the drugs, not all, on the market can present some kind of serious side effect too.

Now some people have gotten some solace from electrical stimulators that are implanted in the brain, for Parkinson’s disease or cochlear implants. These have indeed been able to bring some kind of remedy to people with certain kinds of disorders. But electricity also will go in all directions – the path of least resistance – which is where that phrase, in part, comes from, and will also affect normal circuits, as well as the abnormal ones you want to fix. So again, we’re sent back to the idea of ultraprecise control: Could we dial in information precisely where we want it to go?

So, when I started in neuroscience 11 years ago – I had trained as an electrical engineer and a physicist – the first thing I thought about was, if these neurons are electrical devices, all we need to do is to find some way of driving those electrical changes at a distance. If we could turn on the electricity in one cell but not its neighbors, that’d give us the tool to activate and shut down these different cells to figure out what they do and how they contribute to the networks in which they’re embedded. It would also allow us to have the ultraprecise control we need to fix the circuit computations that have gone awry.

Now, how are we going to do that? Well, there are many molecules that exist in nature which are able to convert light into electricity. You can think of them as little proteins that are like solar cells. If we install these molecules in neurons somehow, then these neurons would become electrically drivable with light, and their neighbors, which don’t have this molecule, would not. There’s one other magic trick you need to make this happen: the ability to get light into the brain. The brain doesn’t feel pain. Taking advantage of all the effort that’s gone into the internet, telecommunications, etc., you can put optical fibers connected to lasers to activate – in animal models, for example, in preclinical studies – these neurons and see what they do.

So how do we do this? Around 2004, in collaboration with Georg Nagel and Karl Deisseroth, this vision came to fruition. There’s a certain alga that swims in the wild, and it needs to navigate towards light in order to photosynthesize optimally. And it senses light with a little eyespot, which works not unlike how our eye works. In its membrane, or its boundary, it contains little proteins that indeed can convert light into electricity. These molecules are called channelrhodopsins. And each of these proteins acts just like that solar cell that I told you about. When blue light hits it, it opens a little hole and allows charged particles to enter the eyespot; that allows this eyespot to have an electrical signal, just like a solar cell charging a battery.

So what we need to do is take these molecules and somehow install them in neurons. And because it’s a protein, it’s encoded for in the DNA of this organism. So all we’ve got to do is take that DNA, put it into a gene therapy vector, like a virus, and put it into neurons. And this was a very productive time in gene therapy, and lots of viruses were coming along, so this turned out to be fairly simple. Early in the morning one day in the summer of 2004, we gave it a try, and it worked on the first try. You take this DNA and put it into the neuron. The neuron uses its natural protein-making machinery to fabricate these little light-sensitive proteins and install them all over the cell, like putting solar panels on a roof. And the next thing you know, you have a neuron which can be activated with light. So this is very powerful.

One of the tricks you have to do is figure out how to deliver these genes to the cells you want and not all the other neighbors. And you can do that; you can tweak the viruses so they hit some cells and not others. And there’s other genetic tricks you can play in order to get light-activated cells. This field has now come to be known as “optogenetics.” And just as one example of the kind of thing you can do, you can take a complex network, use one of these viruses to deliver the gene just to one kind of cell in this dense network. And then when you shine light on the entire network, just that cell type will be activated.

For example, let’s consider that basket cell I told you about earlier, the one that’s atrophied in schizophrenia and the one that is inhibitory. If we can deliver that gene to these cells – they won’t be altered by the expression of the gene, of course – then flash blue light over the entire brain network, just these cells are going to be driven. And when the light turns off, these cells go back to normal; there don’t seem to be adverse events. Not only can you study what these cells do, what their power is in computing in the brain, you can also use this to try to figure out if we could jazz up the activity of these cells if indeed, they’re atrophied.

I want to tell you some short stories about how we’re using this both at the scientific clinical and preclinical levels. One of the questions that we’ve confronted is: What signals in the brain mediate the sensation of reward? Because if you could find those, those would be some of the signals that could drive learning; the brain will do more of what got that reward. These are also signals that go awry in disorders such as addiction. So if we could figure out what cells they are, we could maybe find new targets for which drugs can be designed or screened against or maybe places where electrodes could be put in for people who have severe disability. To do that, we came up with a very simple paradigm in collaboration with the Fiorillo group, where, if the animal goes to one side of this little box, it gets a pulse of light. And we’ll make different cells in the brain sensitive to light. If these cells can mediate reward, the animal should go there more and more. And that’s what happens.

The animal goes to the right-hand side and pokes his nose there and gets a flash of blue light every time he does it. He’ll do that hundreds of times. These are the dopamine neurons, in some of the pleasure centers in the brain. We’ve shown that a brief activation of these is enough to drive learning. Now we can generalize the idea. Instead of one point in the brain, we can devise devices that span the brain, that can deliver light into three-dimensional patterns – arrays of optical fibers, each coupled to its own independent miniature light source. Then we can try to do things in vivo that have only been done to date in a dish, like high-throughput screening throughout the entire brain for the signals that can cause certain things to happen or that could be good clinical targets for treating brain disorders.

One story I want to tell you about is: How can we find targets for treating post-traumatic stress disorder, a form of uncontrolled anxiety and fear? One of the things that we did was to adopt a very classical model of fear. This goes back to the Pavlovian days. It’s called Pavlovian fear conditioning, where a tone ends with a brief shock. The shock isn’t painful, but it’s a little annoying. And over time – in this case, a mouse, which is a good animal model, commonly used in such experiments – the animal learns to fear the tone. It will react by freezing, sort of like a deer in the headlights. Now the question is: What targets in the brain can we find that allow us to overcome this fear? So we play that tone again, after it’s been associated with fear. But we activate different targets in the brain, using that optical fiber array I showed on the previous slide, in order to try and figure out which targets can cause the brain to overcome that memory of fear.

This brief video shows you one of these targets that we’re working on now. This is an area in the prefrontal cortex, a region where we can use cognition to try to overcome aversive emotional states. The animal hears a tone. A flash of light occurs. There’s no audio, but you see that the animal freezes – the tone used to mean bad news. There’s a little clock in the lower left-hand corner. You can see the animal is about two minutes into this. This next clip is just eight minutes later. And the same tone is going to play, and the light is going to flash again. OK, there it goes. Right … now. And now you can see, just 10 minutes into the experiment, that we’ve equipped the brain, by photoactivating this area, to overcome the expression of this fear memory.

Over the last couple years, we’ve gone back to the tree of life, because we wanted to find ways to turn circuits in the brain off. If we could do that, this could be extremely powerful. If you can delete cells for a few milliseconds or seconds, you can figure out what role they play in the circuits in which they’re embedded. We surveyed organisms from all over the tree of life – every kingdom of life but animals; we see slightly differently. We found molecules called halorhodopsins or archaerhodopsins, that respond to green and yellow light. And they do the opposite of the molecule I told you about before, with the blue light activator, channelrhodopsin.

Let’s give an example of where we think this is going to go. Consider, for example, a condition like epilepsy, where the brain is overactive. Now, if drugs fail in epileptic treatment, one of the strategies is to remove part of the brain, but that’s irreversible, and there could be side effects. What if we could just turn off that brain for the brief amount of time until the seizure dies away, and cause the brain to be restored to its initial state, like a dynamical system that’s being coaxed down into a stable state? This animation tries to explain this concept where we made these cells sensitive to being turned off with light, and we beam light in, and just for the time it takes to shut down a seizure, we’re hoping to be able to turn it off. We don’t have data to show you on this front, but we’re very excited about this.

I want to close on one story, which we think is another possibility, which is that maybe these molecules, if you can do ultraprecise control, can be used in the brain itself to make a new kind of prosthetic, an optical prosthetic. I already told you that electrical stimulators are not uncommon. Seventy-five thousand people have Parkinson’s deep-brain stimulators implanted, maybe 100,000 people have cochlear implants, which allow them to hear. Another thing – you’ve got to get these genes into cells. A new hope in gene therapy has been developed, because viruses like the adeno-associated virus – which probably most of us around this room have; it doesn’t have any symptoms – have been used in hundreds of patients to deliver genes into the brain or the body. And so far, there have not been serious adverse events associated with the virus.

There’s one last elephant in the room: the proteins themselves, which come from algae, bacteria and funguses and all over the tree of life. Most of us don’t have funguses or algae in our brains, so what will our brain do if we put that in? Will the cells tolerate it? Will the immune system react? It’s early – these haven’t been done in humans yet – but we’re working on a variety of studies to examine this. So far, we haven’t seen overt reactions of any severity to these molecules or to the illumination of the brain with light. So it’s early days, to be upfront, but we’re excited about it.

I wanted to close with one story, which we think could potentially be a clinical application. Now, there are many forms of blindness where the photoreceptors – light sensors in the back of our eye – are gone. And the retina is a complex structure. Let’s zoom in on it so we can see it in more detail. The photoreceptor cells are shown here at the top. The signals that are detected by the photoreceptors are transformed via various computations until finally, the layer of cells at the bottom, the ganglion cells, relay the information to the brain, where we see that as perception. In many forms of blindness, like retinitis pigmentosa or macular degeneration, the photoreceptor cells have atrophied or been destroyed. Now, how could you repair this? It’s not even clear that a drug could cause this to be restored, since there’s nothing for the drug to bind to. On the other hand, light can still get into the eye. The eye is still transparent and you can get light in. So what if we could take these channelrhodopsins and other molecules and install them on some of these other spared cells and convert them into little cameras? And because there are so many of these cells in the eye, potentially, they could be very high-resolution cameras.

This is some work that we’re doing, led by one of our collaborators, Alan Horsager at USC, and being sought to be commercialized by a start-up company, Eos Neuroscience, which is funded by the NIH. What you see here is a mouse trying to solve a six-arm maze. There’s a bit of water to motivate the mouse to move or he’ll just sit there. The goal of this maze is to get out of the water and go to a little platform that’s under the lit top port. Mice are smart, so this one solves the maze eventually, but he does a brute-force search. He’s swimming down every avenue until he finally gets to the platform. He’s not using vision to do it. These different mice are different mutations that recapitulate different kinds of blindness that affect humans. So we’re being careful in trying to look at these different models so we come up with a generalized approach.

So how can we solve this? We’ll do exactly what we outlined in the previous slide. We’ll take these blue light photo sensors and install them onto a layer of cells in the middle of the retina in the back of the eye and convert them into a camera – just like installing solar cells all over those neurons to make them light-sensitive. Light is converted to electricity on them. So this mouse was blind a couple weeks before this experiment and received one dose of this photosensitive molecule on a virus. And now you can see, the animal can indeed avoid walls and go to this little platform and make cognitive use of its eyes again. And to point out the power of this: these animals can get to that platform just as fast as animals that have seen their entire lives. So this preclinical study, I think, bodes hope for the kinds of things we’re hoping to do in the future.

We’re also exploring new business models for this new field of neurotechnology. We’re developing tools and sharing them freely with hundreds of groups all over the world for them to study and try to treat different disorders. Our hope is that by figuring out brain circuits at a level of abstraction that lets us repair them and engineer them, we can take some of these intractable disorders I mentioned earlier, practically none of which are cured, and in the 21st century, make them history.

Thank you.

Juan Enriquez: So some of this stuff is a little dense.

But the implications of being able to control seizures or epilepsy with light instead of drugs and being able to target those specifically is a first step. The second thing that I think I heard you say is you can now control the brain in two colors, like an on-off switch.

Ed Boyden: That’s right.

JE: Which makes every impulse going through the brain a binary code.

EB: Right. With blue light, we can drive information, and it’s in the form of a one. And by turning things off, it’s more or less a zero. Our hope is to eventually build brain coprocessors that work with the brain so we can augment functions in people with disabilities.

JE: And in theory, that means that, as a mouse feels, smells, hears, touches, you can model it out as a string of ones and zeros.

EB: Yeah. We’re hoping to use this as a way of testing what neural codes can drive certain behaviors and certain thoughts and certain feelings and use that to understand more about the brain.

JE: Does that mean that someday you could download memories and maybe upload them?

EB: That’s something we’re starting to work on very hard. We’re now working on trying to tile the brain with recording elements, too, so we can record information and then drive information back in – sort of computing what the brain needs in order to augment its information processing.

JE: Well, that might change a couple things. Thank you.

EB: Thank you.

The Nobel Prize in Physiology or Medicine 2026 was awarded jointly to Karl Deisseroth, Peter Hegemann and Georg Nagel “for their discoveries concerning light-gated ion channels and optogenetics.”

Anecdotes and Highlights Along the Navier–Stokes Breakthrough Timeline

How did the Navier–Stokes equations become the second Millennium Prize Problem to approach a breakthrough?

Why ask for a proof? Because a proof gives not only certitude, but also understanding. - Clay Mathematics Institute - The Millennium Prize Problems — Navier-Stokes Equation

A fundamental problem in analysis is to decide whether such smooth, physically reasonable solutions exist for the Navier–Stokes equations. To give reasonable leeway to solvers while retaining the heart of the problem, we ask for a proof of one of the following four statements.

(A) Existence and smoothness of Navier–Stokes solutions on \( \mathbf{\mathbb{R^3}} \).

(B) Existence and smoothness of Navier–Stokes solutions on \( \mathbf{\mathbb{R^3/Z^3}} \).

(C) Breakdown of Navier–Stokes solutions on \( \mathbf{\mathbb{R^3}} \).

(D) Breakdown of Navier–Stokes Solutions on \( \mathbf{\mathbb{R^3/Z^3}} \).

– Existence And Smoothness Of The Navier–Stokes Equation, Official Problem Description by Charles L. Fefferman

Fluids are important and hard to understand. There are many fascinating problems and conjectures about the behavior of solutions of the Euler and Navier–Stokes equations. Since we don’t even know whether these solutions exist, our understanding is at a very primitive level. Standard methods from PDE appear inadequate to settle the problem. Instead, we probably need some deep, new ideas. - Existence And Smoothness Of The Navier–Stokes Equation, Official Problem Description by Charles L. Fefferman


1738 Daniel Bernoulli’s Hydrodynamica (relates fluid flow to pressure).

\( \displaystyle \huge p + \frac{1}{2}\rho v^2 + \rho gh = \text{constant} \)

Daniel Bernoulli was primarily a physicist.

Pressure field around an airfoil. The lines are isobars of equal pressure along their length. The arrows show the pressure differential from high (red) to low (blue) and hence also the net force which causes the air to accelerate in that direction.
Bernoulli’s principle can be used to calculate the lift force on an airfoil, if the behaviour of the fluid flow in the vicinity of the foil is known. For example, if the air flowing past the top surface of an aircraft wing is moving faster than the air flowing past the bottom surface, then Bernoulli’s principle implies that the pressure on the surfaces of the wing will be lower above than below. This pressure difference results in an upwards lifting force. Whenever the distribution of speed past the top and bottom surfaces of a wing is known, the lift forces can be calculated (to a good approximation) using Bernoulli’s equations, which were established by Bernoulli over a century before the first man-made wings were used for the purpose of flight.

The flow of air causes a lifting force on the object, allowing for non-contact adhesion
Commercially available Bernoulli grips are commonly used to handle rigid sheet like material such as silicon wafers in circuit board manufacturing, or photovoltaic cell components.


1757 Euler’s “Principes généraux du mouvement des fluides” (Euler equations, start of
modern hydrodynamics).

\(
\displaystyle \huge
\begin{aligned}
\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} &= -\frac{1}{\rho}\nabla p + \mathbf{f} \\
\nabla \cdot \mathbf{u} &= 0
\end{aligned}
\)

This makes it quite clear how far removed we are from a complete understanding of the motion of fluids and that my exposition is no more than a mere beginning. Nevertheless, everything that the Theory of Fluids contains is embodied in the two equations formulated above, so that it is not the laws of Mechanics that we lack in order to pursue this research but only the Analysis, which has not yet been sufficiently developed for this purpose. It is therefore clearly apparent what discoveries we still need to make in this branch of Science before we can arrive at a more perfect Theory of the motion of fluids. - Leonhard Euler

Indeed, it is notoriously hard to perform a reliable numerical simulation of the 3D Euler equations.


1823 Navier formulates equations now known as the Navier–Stokes equations.

\(
\displaystyle \huge
\begin{align}
\rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} \right) &= -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f} \\
\nabla \cdot \mathbf{u} &= 0
\end{align}
\)

The Navier–Stokes equations agree very well with experiments on real fluids under many and varied circumstances. Since fluids are important, so are the Navier–Stokes equations.

In principle the Navier–Stokes equations provide a complete and very accurate theory of lift, but practical prediction of lift requires that the effects of turbulence be modeled in the Reynolds-averaged Navier–Stokes equations rather than computed directly.

The character of the equation … changes simultaneously in all relevant respects: Both order and degree change. Hence bad mathematical difficulties must be expected. — John von Neumann

However, numerical and physical experiments suggest strongly that this is not the case. Instead, it seems that there exists \( T_0 > 0 \), depending on \( u^{0} \) but independent of \( ν \), such that the fluid loses at least half of its initial energy by time \( T_0 \), regardless of how small \( ν \) is (provided that \( ν > 0 \) ). We need to understand why a tiny viscosity dissipates a lot of energy.

The fundamental basis of almost all computational fluid dynamics problems is the Navier–Stokes equations.
CFD is applied to a range of research and engineering problems in multiple fields of study and industries, including aerodynamics and aerospace analysis, hypersonics, weather simulation, natural science and environmental engineering, industrial system design and analysis, biological engineering, fluid flows and heat transfer, engine and combustion analysis, and visual effects for film and games.


1883 Osborne Reynolds demonstrated Laminar–turbulent transition

Reynolds’ 1883 experiment on fluid dynamics in pipes

Reynolds’ 1883 observations of the nature of the flow in his experiments


1904 Ludwig Prandtl described the boundary layer and its importance for drag and streamlining.

Boundary layer visualization, showing transition from laminar to turbulent condition


1934 Jean Leray published an important paper that founded the study of weak solutions of the Navier–Stokes equations.

A promising idea, pioneered by J. Leray in the 1930s, is to study “weak solutions” of the Navier–Stokes equations.

A long-established idea in analysis is to prove existence and regularity of solutions of a PDE by first constructing a weak solution, then showing that any weak solution is smooth. This program has been tried for Navier–Stokes with partial success.

It has been known for a long time how to construct suitable weak solutions of 3D Navier–Stokes, but the uniqueness of these solutions has not been proved.


The global regularity problem for Navier-Stokes asks for existence of global smooth solutions to a Cauchy problem for a nonlinear PDE.
The standard response to this question is turbulence – the behaviour of three-dimensional Navier-Stokes equations at fine scales is much more nonlinear (and hence unstable) than at coarse scales. I would phrase the obstruction slightly differently, as supercriticality. Or more precisely, all of the globally controlled quantities for Navier-Stokes evolution which we are aware of (and we are not aware of very many) are either supercritical with respect to scaling, which means that they are much weaker at controlling fine-scale behaviour than controlling coarse-scale behaviour, or they are non-coercive, which means that they do not really control the solution at all, either at coarse scales or at fine.
In conclusion, while it is good to occasionally have a crack at impossible problems, just to try one’s luck, I would personally spend much more of my time on other, more tractable PDE problems than the Clay prize problem, though one should certainly keep that problem in mind if, in the course on working on other problems, one indeed does stumble upon something that smells like a breakthrough in Strategy 1, 2, or 3 above. (In particular, there are many other serious and interesting questions in fluid equations that are not anywhere near as difficult as global regularity for Navier-Stokes, but still highly worthwhile to resolve.) - Why global regularity for Navier-Stokes is hard, by Terence Tao, March 18, 2007

Most mathematicians have concentrated their energy on trying to prove that the solutions to the Navier-Stokes equations remain smooth and well-behaved forever, a property called global regularity. Purported proofs of global regularity surface every few months, but so far each one has had a fatal flaw[1] [2].
Tao’s ideas certainly aren’t standard, Friedlander said. “But results about global regularity, whether positive or negative, will almost certainly require something that is nonstandard.” Tao’s ideas are quite likely to lead to some interesting mathematics, she said, whether or not they answer the Navier-Stokes question.
Meanwhile, Tao’s new program for tackling Navier-Stokes has changed his thinking about the problem. Previously, he had viewed both sides of the problem — proving global regularity or establishing that solutions can blow up — as equally remote goals (though in the end, only one can be true). Now he believes that it should be possible to concoct special scenarios in which a solution to the Navier-Stokes equations blows up. (This wouldn’t mean that the real ocean could blow up — instead, it would suggest that in these rare cases, the Navier-Stokes equations don’t fully capture the ocean’s physics.) “My mindset has certainly changed,” he said. - A Fluid New Path in Grand Math Challenge, by Erica Klarreich , February 24, 2014

[1] Mukhtarbay Otelbaev, Prof. Dr., Academician of the NAS of the RK, Director of the Eurasian Mathematical Institute of L.N. Gumilyov Eurasian National University, completed and published paper “Existence of a strong solution of the Navier-Stokes equations” in the press.
Mukhtarbay Otelbaev is the winner of the title “Man of Science of the Year” in contest “Altyn Adam” (“Man of the Year in Kazakhstan “ in the category “Science”) 2002; in 2002-2003, 2004 – 2005 is the Winner of the state scientific grant for scientists and experts who have made outstanding contributions to the development of science and technology, winner of the Economic Cooperation Organization in the “Science and Technology“, 2004, winner of the MES grant of the RK “Best university teacher”, laureate of the State Prize of the Republic Kazakhstan in the field of science and technology.
[2] In 2014, about the decision of the sixth problem of the Millennium was announced by the author mukhtarbai Otelbayev from the Eurasian national University named after Lev Gumilev (Kazakhstan). American mathematician Terence Tao found counterexamples to refute the decision of the Kazakhstan scientist.

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.
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.
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.
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.
Buckmaster and Vicol prove the existence of many nonunique weak solutions to the Navier-Stokes equations. - Mathematicians Find Wrinkle in Famed Fluid Equations, by Kevin Hartnett, December 21, 2017

Why are these equations, which describe familiar phenomena such as water flowing through a hose, so much harder to understand mathematically than, say, Einstein’s field equations, which involve stupefying objects like black holes? The answer, I discovered, is turbulence. It’s something we’ve all experienced, whether flying through choppy air at 30,000 feet or watching a whirlpool gather in the bathtub drain. Yet familiarity hasn’t bred knowledge: Turbulence is one of the least understood parts of the physical world.
Mathematicians worry about this kind of scenario: You’re running the equations, and after some finite amount of time, they tell you a particle in the fluid is moving infinitely fast. That would be a problem because you can’t calculate the change of an infinite value any more than you can divide by zero. Mathematicians refer to such scenarios as “blowup,” and in a blowup scenario you’d say the equations break down and solutions don’t exist. - What Makes the Hardest Equations in Physics So Difficult?, by Kevin Hartnett, January 16, 2018

For centuries, mathematicians have sought to understand and model the motion of fluids. The equations that describe how ripples crease the surface of a pond have also helped researchers to predict the weather, design better airplanes, and characterize how blood flows through the circulatory system. These equations are deceptively simple when written in the right mathematical language. However, their solutions are so complex that making sense of even basic questions about them can be prohibitively difficult.
Perhaps the oldest and most prominent of these equations, formulated by Leonhard Euler more than 250 years ago, describe the flow of an ideal, incompressible fluid: a fluid with no viscosity, or internal friction, that cannot be forced into a smaller volume. “Almost all nonlinear fluid equations are kind of derived from the Euler equations,” said Tarek Elgindi, a mathematician at Duke University. “They’re the first ones, you could say.”
“It’s an amazing result,” said Tristan Buckmaster, a mathematician at the University of Maryland who was not involved in the work. “There are no results of its kind in the literature.”
The 177-page proof — the result of a decade-long research program — makes significant use of computers. This arguably makes it difficult for other mathematicians to verify it. (In fact, they are still in the process of doing so, though many experts believe the new work will turn out to be correct.) It also forces them to reckon with philosophical questions about what a “proof” is, and what it will mean if the only viable way to solve such important questions going forward is with the help of computers.
Perhaps in some situations, the equations will proceed as expected, producing precise values for the state of the fluid at any given moment, only for one of those values to suddenly skyrocket to infinity. At that point, the Euler equations are said to give rise to a “singularity” — or, more dramatically, to “blow up.”
These efforts mark a growing trend in the field of fluid dynamics: the use of computers to solve important problems.
But in fluid mechanics, computer-assisted proofs are still a relatively new technique. In fact, when it comes to statements about singularity formation, Hou and Chen’s proof is the first of its kind: Previous computer-assisted proofs were only able to tackle toy problems in the area. - Computer Proof ‘Blows Up’ Centuries-Old Fluid Equations, by Jordana Cepelewicz, November 16, 2022

And for nearly 200 years, the Navier-Stokes equations have served as an unimpeachable theory of how fluids in the real world behave — from ocean currents threading their way between the continents to air wrapping around an aircraft’s wings.
In these realistic theories, “a lot of people believe that there are singularities, but that they are unstable, so we never see them,” said Charlie Fefferman, a mathematician at Princeton University who formulated the million-dollar Navier-Stokes challenge.
“The graveyards are strewn with alleged singular solutions of 3D Euler,” Fefferman said. - Using AI, Mathematicians Find Hidden Glitches in Fluid Equations, by Charlie Wood, January 9, 2026

The Navier-Stokes equations presume that fluids are continuous substances that flow perfectly smoothly, no matter how much you zoom in on a point. But in fact, fluids are amalgamations of molecules and atoms. As you zoom in, you’ll eventually discern tiny blips in the flow due to this grainy nature, blips that the classic fluid equations lop off.
“Navier-Stokes is very much an approximation,” said Michael Landry, a physicist at the Massachusetts Institute of Technology. “It’s not an exact equation.”
They realized you could always swap two parcels of a fluid for free. You could also shuffle three parcels, or four, or any number. In contrast, you can’t exchange any regions of a solid without paying an energy toll, through rupture or serious internal stress, so solids lack these swapping symmetries.
A fluid is any material with a particular set of symmetries, no matter what it’s made of. - Theory of Fluids Enters the 21st Century, by Charlie Wood, August 17, 2026

“I was thrilled that the problem was solved,” said Charles Fefferman of Princeton University, who wrote the Clay Institute’s official description of the Navier-Stokes problem. The heroes of the story, he said, are Córdoba and Martínez-Zoroa.
Turbulence is even weirder than it appears to be.
“Ten years ago, nobody believed there was a singularity for Navier-Stokes,” said Córdoba — though many believed that the Euler equations did admit a singularity. This began to change in 2013, when Thomas Hou of the California Institute of Technology and Guo Luo, now at the Hang Seng University of Hong Kong, derived a groundbreaking result showing that the Euler equations can “blow up,” as mathematicians like to say, in a cylinder if the top and bottom halves are set spinning in opposite directions. “That’s the first really serious claim of singularity,” Córdoba remembered.
In their 2013 cylinder result, Hou and Luo used computer models to simulate a scenario that might lead the Euler equations to blow up. They then proved that blowup does occur by computationally accounting for all potential errors. In the years since, similar techniques have become the dominant mode of attack on the Euler and Navier-Stokes problems.
But in his 2021 doctoral dissertation, Martínez-Zoroa pioneered analytic techniques that don’t rely on computers at all. This made him, along with Córdoba, his doctoral adviser, an oddball in the area. By 2023, the pair had proved that a version of the Euler equations with a messy forcing function displayed singularities.
In broad outline, the pair’s technique relies on creating an infinite sequence of “layers,” each of which is a non-singular solution to the equation they are studying. (They’ve applied similar techniques to both the Euler and Navier-Stokes equations, as well as to other related systems.) They then combine those solutions in what Martínez-Zoroa calls an “infinite cascade” to produce a new solution. - AI Has Solved One of Math’s $1 Million Millennium Prize Problems, by Konstantin Kakaes, September 8, 2026

Martin Hairer talks with Tom Crawford: I remember last year in Oxford. I mean, I guess you were there. There was this workshop for the 25 years of the millennium problems. Yes. And it was very clear that the Navier-Stokes one was the one that was sort of in the end sprint, right? - Navier-Stokes, AI and the Future of Mathematics with Martin Hairer, Sep 20, 2026


  • The number of problems that have been solved is one more than I would expect. - Manjul Bhargava
  • Some problems, the tools are not there. You have to wait until there’s some sort of breakthrough… - Terence Tao
  • “Knowledge is power,” or so the adage goes. But understanding is more powerful power. But understanding is still kept a secret, at least in mathematics. - How to Bake Pi, by Eugenia Cheng
  • AI models skip to the end. At the moment, their proofs are nearly impossible to read. They elide salient details and waste pages on irrelevant concepts. They fail to demonstrate how the new work builds on or connects to other results in the mathematical literature. Mathematicians know that the proofs are true — they’re verified by a system that checks their logic — but have no idea why. What’s missing is understanding, the currency that mathematicians trade in. Mathematicians have always known that understanding is more valuable than an answer. - Is AI the End of Math As We Know It?, by Jordana Cepelewicz

My English Phrases List - October - 2026

such is life

Such is life and life is such and after all it isn’t much. First a cradle. Then a hearse. It might have been better, but it could have been worse. — Norman Wisdom

make a difference

Your help made a big difference.

by all means

I am only considering the global regularity problem for Navier-Stokes, from a purely mathematical viewpoint, and in the precise formulation given by the Clay Institute; I will not discuss at all the question as to what implications a rigorous solution (either positive or negative) to this problem would have for physics, computational fluid dynamics, or other disciplines, as these are beyond my area of expertise. But if anyone qualified in these fields wants to make a comment along these lines, by all means do so. - Why global regularity for Navier-Stokes is hard, by Terence Tao

twists and turns

The road through the mountains has many twists and turns.

My English Words List - October - 2026

morning glory

morning glory

noun

Morning glory flower, Ipomoea nil

Morning Glory

By Carolyn Marie Rodgers

the morning glories wrap themselves as they
crawl up the grey garden fence.

turning point

noun

That job was a major turning point in her career.

It was another turning point in the winding road that led me, at age 66, to earn a Ph.D. - It’s never too late to stretch your wings: Why I got a Ph.D. at age 66, by Tracy Evans

Torontonian

Torontonian

noun

Torontonians don’t feel safe, despite positive data. Here’s how the issue is shaping the city’s election race

Neurotransmitters and Neuromodulators

Balance is Key

Neurotransmitters(神经递质)

excitatory(兴奋性)

glutamic acid, glutamate(谷氨酸)

\(\huge C_{5}H_{9}NO_{4} \)

Skeletal formula of L-glutamic acid

inhibitory(抑制性)

GABA(gamma-aminobutyric acid)(γ-氨基丁酸)

\(\huge C_{4}H_{9}NO_{2} \)

γ-Aminobutyric acid

modulatory(调节性), neuromodulatory

serotonin(血清素,5-羟色胺)

\(\huge C_{10}H_{12}N_{2}O \)

Serotonin

dopamine(多巴胺)

\(\huge C_{8}H_{11}NO_{2} \)

Skeletal formula of dopamine

norepinephrine(去甲肾上腺素)

\(\huge C_{8}H_{11}NO_{3} \)

Skeletal formula of noradrenaline

epinephrine, adrenaline(肾上腺素)

\(\huge C_{9}H_{13}NO_{3} \)

Skeletal formula of adrenaline

acetylcholine(乙酰胆碱)

\(\huge [C_{7}H_{16}NO_{2}]^{+} \)

Acetylcholine

oxytocin(催产素)

\(\huge C_{43}H_{66}N_{12}O_{12}S_{2} \)

Oxytocin

Neuromodulators(神经调质)

endorphin(内啡肽)

α-Endorphin

β-Endorphin

γ-Endorphin