The suspicion, harbored by many physicists and mathematicians over the decades but rarely actively pursued, is that the peculiar panoply of forces and particles that comprise reality spring logically from the properties of eight-dimensional numbers called “octonions.”

As numbers go, the familiar real numbers — those found on the number line, like 1, π and -83.777 — just get things started. Real numbers can be paired up in a particular way to form “complex numbers,” first studied in 16th-century Italy, that behave like coordinates on a 2-D plane. Adding, subtracting, multiplying and dividing is like translating and rotating positions around the plane. Complex numbers, suitably paired, form 4-D “quaternions,” discovered in 1843 by the Irish mathematician William Rowan Hamilton, who on the spot ecstatically chiseled the formula into Dublin’s Broome Bridge. John Graves, a lawyer friend of Hamilton’s, subsequently showed that pairs of quaternions make octonions: numbers that define coordinates in an abstract 8-D space.

There the game stops. Proof surfaced in 1898 that the reals, complex numbers, quaternions and octonions are the only kinds of numbers that can be added, subtracted, multiplied and divided. The first three of these “division algebras” would soon lay the mathematical foundation for 20th-century physics, with real numbers appearing ubiquitously, complex numbers providing the math of quantum mechanics, and quaternions underlying Albert Einstein’s special theory of relativity. This has led many researchers to wonder about the last and least-understood division algebra. Might the octonions hold secrets of the universe?

“Octonions are to physics what the Sirens were to Ulysses,” Pierre Ramond, a particle physicist and string theorist at the University of Florida, said in an email.

Günaydin, the Penn State professor, was a graduate student at Yale in 1973 when he and his advisor Feza Gürsey found a surprising link between the octonions and the strong force, which binds quarks together inside atomic nuclei. An initial flurry of interest in the finding didn’t last. Everyone at the time was puzzling over the Standard Model of particle physics — the set of equations describing the known elementary particles and their interactions via the strong, weak and electromagnetic forces (all the fundamental forces except gravity). But rather than seek mathematical answers to the Standard Model’s mysteries, most physicists placed their hopes in high-energy particle colliders and other experiments, expecting additional particles to show up and lead the way beyond the Standard Model to a deeper description of reality. They “imagined that the next bit of progress will come from some new pieces being dropped onto the table, from thinking harder about the pieces we already have,” said Latham Boyle, a theoretical physicist at the Perimeter Institute of Theoretical Physics in Waterloo, Canada.

Decades on, no particles beyond those of the Standard Model have been found. Meanwhile, the strange beauty of the octonions has continued to attract the occasional independent-minded researcher, including Furey, the Canadian grad student who visited Günaydin four years ago. Looking like an interplanetary traveler, with choppy silver bangs that taper to a point between piercing blue eyes, Furey scrawled esoteric symbols on a blackboard, trying to explain to Günaydin that she had extended his and Gürsey’s work by constructing an octonionic model of both the strong and electromagnetic forces.

“Communicating the details to him turned out to be a bit more of a challenge than I had anticipated, as I struggled to get a word in edgewise,” Furey recalled. Günaydin had continued to study the octonions since the ’70s by way of their deep connections to string theory, M-theory and supergravity — related theories that attempt to unify gravity with the other fundamental forces. But his octonionic pursuits had always been outside the mainstream. He advised Furey to find another research project for her Ph.D., since the octonions might close doors for her, as he felt they had for him.

But Furey didn’t — couldn’t — give up. Driven by a profound intuition that the octonions and other division algebras underlie nature’s laws, she told a colleague that if she didn’t find work in academia she planned to take her accordion to New Orleans and busk on the streets to support her physics habit. Instead, Furey landed a postdoc at the University of Cambridge in the United Kingdom. She has since produced a number of results connecting the octonions to the Standard Model that experts are calling intriguing, curious, elegant and novel. “She has taken significant steps toward solving some really deep physical puzzles,” said Shadi Tahvildar-Zadeh, a mathematical physicist at Rutgers University who recently visited Furey in Cambridge after watching an online series of lecture videos she made about her work.

Furey has yet to construct a simple octonionic model of all Standard Model particles and forces in one go, and she hasn’t touched on gravity. She stresses that the mathematical possibilities are many, and experts say it’s too soon to tell which way of amalgamating the octonions and other division algebras (if any) will lead to success.

“She has found some intriguing links,” said Michael Duff, a pioneering string theorist and professor at Imperial College London who has studied octonions’ role in string theory. “It’s certainly worth pursuing, in my view. Whether it will ultimately be the way the Standard Model is described, it’s hard to say. If it were, it would qualify for all the superlatives — revolutionary, and so on.”

I met Furey in June, in the porter’s lodge through which one enters Trinity Hall on the bank of the River Cam. Petite, muscular, and wearing a sleeveless black T-shirt (that revealed bruises from mixed martial arts), rolled-up jeans, socks with cartoon aliens on them and Vegetarian Shoes–brand sneakers, in person she was more Vancouverite than the otherworldly figure in her lecture videos. We ambled around the college lawns, ducking through medieval doorways in and out of the hot sun. On a different day I might have seen her doing physics on a purple yoga mat on the grass.

Furey, who is 39, said she was first drawn to physics at a specific moment in high school, in British Columbia. Her teacher told the class that only four fundamental forces underlie all the world’s complexity — and, furthermore, that physicists since the 1970s had been trying to unify all of them within a single theoretical structure. “That was just the most beautiful thing I ever heard,” she told me, steely-eyed. She had a similar feeling a few years later, as an undergraduate at Simon Fraser University in Vancouver, upon learning about the four division algebras. One such number system, or infinitely many, would seem reasonable. “But four?” she recalls thinking. “How peculiar.”

After breaks from school spent ski-bumming, bartending abroad and intensely training as a mixed martial artist, Furey later met the division algebras again in an advanced geometry course and learned just how peculiar they become in four strokes. When you double the dimensions with each step as you go from real numbers to complex numbers to quaternions to octonions, she explained, “in every step you lose a property.” Real numbers can be ordered from smallest to largest, for instance, “whereas in the complex plane there’s no such concept.” Next, quaternions lose commutativity; for them, a × b doesn’t equal b × a. This makes sense, since multiplying higher-dimensional numbers involves rotation, and when you switch the order of rotations in more than two dimensions you end up in a different place. Much more bizarrely, the octonions are nonassociative, meaning (a × b) × c doesn’t equal a × (b × c). “Nonassociative things are strongly disliked by mathematicians,” said John Baez, a mathematical physicist at the University of California, Riverside, and a leading expert on the octonions. “Because while it’s very easy to imagine noncommutative situations — putting on shoes then socks is different from socks then shoes — it’s very difficult to think of a nonassociative situation.” If, instead of putting on socks then shoes, you first put your socks into your shoes, technically you should still then be able to put your feet into both and get the same result. “The parentheses feel artificial.”

The octonions’ seemingly unphysical nonassociativity has crippled many physicists’ efforts to exploit them, but Baez explained that their peculiar math has also always been their chief allure. Nature, with its four forces batting around a few dozen particles and anti-particles, is itself peculiar. The Standard Model is “quirky and idiosyncratic,” he said.

In the Standard Model, elementary particles are manifestations of three “symmetry groups” — essentially, ways of interchanging subsets of the particles that leave the equations unchanged. These three symmetry groups, SU(3), SU(2) and U(1), correspond to the strong, weak and electromagnetic forces, respectively, and they “act” on six types of quarks, two types of leptons, plus their anti-particles, with each type of particle coming in three copies, or “generations,” that are identical except for their masses. (The fourth fundamental force, gravity, is described separately, and incompatibly, by Einstein’s general theory of relativity, which casts it as curves in the geometry of space-time.)

Sets of particles manifest the symmetries of the Standard Model in the same way that four corners of a square must exist in order to realize a symmetry of 90-degree rotations. The question is, why this symmetry group — SU(3) × SU(2) × U(1)? And why this particular particle representation, with the observed particles’ funny assortment of charges, curious handedness and three-generation redundancy? The conventional attitude toward such questions has been to treat the Standard Model as a broken piece of some more complete theoretical structure. But a competing tendency is to try to use the octonions and “get the weirdness from the laws of logic somehow,” Baez said.

Furey began seriously pursuing this possibility in grad school, when she learned that quaternions capture the way particles translate and rotate in 4-D space-time. She wondered about particles’ internal properties, like their charge. “I realized that the eight degrees of freedom of the octonions could correspond to one generation of particles: one neutrino, one electron, three up quarks and three down quarks,” she said — a bit of numerology that had raised eyebrows before. The coincidences have since proliferated. “If this research project were a murder mystery,” she said, “I would say that we are still in the process of collecting clues.”

To reconstruct particle physics, Furey uses the product of the four division algebras, $latex \mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ ($latex \mathbb{R}$ for reals, $latex \mathbb{C}$ for complex numbers, $latex \mathbb{H}$ for quaternions and $latex \mathbb{O}$ for octonions) — sometimes called the Dixon algebra, after Geoffrey Dixon, a physicist who first took this tack in the 1970s and ’80s before failing to get a faculty job and leaving the field. (Dixon forwarded me a passage from his memoirs: “What I had was an out-of-control intuition that these algebras were key to understanding particle physics, and I was willing to follow this intuition off a cliff if need be. Some might say I did.”)

Whereas Dixon and others proceeded by mixing the division algebras with extra mathematical machinery, Furey restricts herself; in her scheme, the algebras “act on themselves.” Combined as $latex \mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$, the four number systems form a 64-dimensional abstract space. Within this space, in Furey’s model, particles are mathematical “ideals”: elements of a subspace that, when multiplied by other elements, stay in that subspace, allowing particles to stay particles even as they move, rotate, interact and transform. The idea is that these mathematical ideals are the particles of nature, and they manifest the symmetries of $latex \mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$.

As Dixon knew, the algebra splits cleanly into two parts: $latex \mathbb{C}\otimes\mathbb{H}$ and $latex \mathbb{C}\otimes\mathbb{O}$, the products of complex numbers with quaternions and octonions, respectively (real numbers are trivial). In Furey’s model, the symmetries associated with how particles move and rotate in space-time, together known as the Lorentz group, arise from the quaternionic $latex \mathbb{C}\otimes\mathbb{H}$ part of the algebra. The symmetry group SU(3) × SU(2) × U(1), associated with particles’ internal properties and mutual interactions via the strong, weak and electromagnetic forces, come from the octonionic part, $latex \mathbb{C}\otimes\mathbb{O}$.

Günaydin and Gürsey, in their early work, already found SU(3) inside the octonions. Consider the base set of octonions, 1, e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6} and e_{7}, which are unit distances in eight different orthogonal directions: They respect a group of symmetries called G2, which happens to be one of the rare “exceptional groups” that can’t be mathematically classified into other existing symmetry-group families. The octonions’ intimate connection to all the exceptional groups and other special mathematical objects has compounded the belief in their importance, convincing the eminent Fields medalist and Abel Prize–winning mathematician Michael Atiyah, for example, that the final theory of nature must be octonionic. “The real theory which we would like to get to,” he said in 2010, “should include gravity with all these theories in such a way that gravity is seen to be a consequence of the octonions and the exceptional groups.” He added, “It will be hard because we know the octonions are hard, but when you’ve found it, it should be a beautiful theory, and it should be unique.”

Holding e_{7} constant while transforming the other unit octonions reduces their symmetries to the group SU(3). Günaydin and Gürsey used this fact to build an octonionic model of the strong force acting on a single generation of quarks.

Furey has gone further. In her most recent published paper, which appeared in May in *The* *European Physical Journal C*, she consolidated several findings to construct the full Standard Model symmetry group, SU(3) × SU(2) × U(1), for a single generation of particles, with the math producing the correct array of electric charges and other attributes for an electron, neutrino, three up quarks, three down quarks and their anti-particles. The math also suggests a reason why electric charge is quantized in discrete units — essentially, because whole numbers are.

However, in that model’s way of arranging particles, it’s unclear how to naturally extend the model to cover the full three particle generations that exist in nature. But in another new paper that’s now circulating among experts and under review by *Physical Letters B*, Furey uses $latex \mathbb{C}\otimes\mathbb{O}$ to construct the Standard Model’s two unbroken symmetries, SU(3) and U(1). (In nature, SU(2) × U(1) is broken down into U(1) by the Higgs mechanism, a process that imbues particles with mass.) In this case, the symmetries act on all three particle generations and also allow for the existence of particles called sterile neutrinos — candidates for dark matter that physicists are actively searching for now. “The three-generation model only has SU(3) × U(1), so it’s more rudimentary,” Furey told me, pen poised at a whiteboard. “The question is, is there an obvious way to go from the one-generation picture to the three-generation picture? I think there is.”

This is the main question she’s after now. The mathematical physicists Michel Dubois-Violette, Ivan Todorov and Svetla Drenska are also trying to model the three particle generations using a structure that incorporates octonions called the exceptional Jordan algebra. After years of working solo, Furey is beginning to collaborate with researchers who take different approaches, but she prefers to stick with the product of the four division algebras, $latex \mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$, acting on itself. It’s complicated enough and provides flexibility in the many ways it can be chopped up. Furey’s goal is to find the model that, in hindsight, feels inevitable and that includes mass, the Higgs mechanism, gravity and space-time.

Already, there’s a sense of space-time in the math. She finds that all multiplicative chains of elements of $latex \mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ can be generated by 10 matrices called “generators.” Nine of the generators act like spatial dimensions, and the 10th, which has the opposite sign, behaves like time. String theory also predicts 10 space-time dimensions — and the octonions are involved there as well. Whether or how Furey’s work connects to string theory remains to be puzzled out.

So does her future. She’s looking for a faculty job now, but failing that, there’s always the ski slopes or the accordion. “Accordions are the octonions of the music world,” she said — “tragically misunderstood.” She added, “Even if I pursued that, I would always be working on this project.”

Furey mostly demurred on my more philosophical questions about the relationship between physics and math, such as whether, deep down, they are one and the same. But she is taken with the mystery of why the property of division is so key. She also has a hunch, reflecting a common allergy to infinity, that $latex \mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ is actually an approximation that will be replaced, in the final theory, with another, related mathematical system that does not involve the infinite continuum of real numbers.

That’s just intuition talking. But with the Standard Model passing tests to staggering perfection, and no enlightening new particles materializing at the Large Hadron Collider in Europe, a new feeling is in the air, both unsettling and exciting, ushering a return to whiteboards and blackboards. There’s the burgeoning sense that “maybe we have not yet finished the process of fitting the current pieces together,” said Boyle, of the Perimeter Institute. He rates this possibility “more promising than many people realize,” and said it “deserves more attention than it is currently getting, so I am very glad that some people like Cohl are seriously pursuing it.”

Boyle hasn’t himself written about the Standard Model’s possible relationship to the octonions. But like so many others, he admits to hearing their siren song. “I share the hope,” he said, “and even the suspicion, that the octonions may end up playing a role, somehow, in fundamental physics, since they are very beautiful.”

]]>One of the many women who, in a different world, might have won the physics prize in the intervening 55 years is Sau Lan Wu. Wu is the Enrico Fermi Distinguished Professor of Physics at the University of Wisconsin, Madison, and an experimentalist at CERN, the laboratory near Geneva that houses the Large Hadron Collider. Wu’s name appears on more than 1,000 papers in high-energy physics, and she has contributed to a half-dozen of the most important experiments in her field over the past 50 years. She has even realized the improbable goal she set for herself as a young researcher: to make at least three major discoveries.

Wu was an integral member of one of the two groups that observed the J/psi particle, which heralded the existence of a fourth kind of quark, now called the charm. The discovery, in 1974, was known as the November Revolution, a coup that led to the establishment of the Standard Model of particle physics. Later in the 1970s, Wu did much of the math and analysis to discern the three “jets” of energy flying away from particle collisions that signaled the existence of gluons — particles that mediate the strong force holding protons and neutrons together. This was the first observation of particles that communicate a force since scientists recognized photons of light as the carriers of electromagnetism. Wu later became one of the group leaders for the ATLAS experiment, one of the two collaborations at the Large Hadron Collider that discovered the Higgs boson in 2012, filling in the final piece of the Standard Model. She continues to search for new particles that would transcend the Standard Model and push physics forward.

Sau Lan Wu was born in occupied Hong Kong during World War II. Her mother was the sixth concubine to a wealthy businessman who abandoned them and her younger brother when Wu was a child. She grew up in abject poverty, sleeping alone in a space behind a rice shop. Her mother was illiterate, but she urged her daughter to pursue an education and become independent of volatile men.

Wu graduated from a government school in Hong Kong and applied to 50 universities in the United States. She received a scholarship to attend Vassar College and arrived with $40 to her name.

Although she originally intended to become an artist, she was inspired to study physics after reading a biography of Marie Curie. She worked on experiments during consecutive summers at Brookhaven National Laboratory on Long Island, and she attended graduate school at Harvard University. She was the only woman in her cohort and was barred from entering the male dormitories to join the study groups that met there. She has labored since then to make a space for everyone in physics, mentoring more than 60 men and women through their doctorates.

*Quanta Magazine* joined Sau Lan Wu on a gray couch in sunny Cleveland in early June. She had just delivered an invited lecture about the discovery of gluons at a symposium to honor the 50th birthday of the Standard Model. The interview has been condensed and edited for clarity.

**You work on the largest experiments in the world, mentor dozens of students, and travel back and forth between Madison and Geneva. What is a normal day like for you? **

Very tiring! In principle, I am full-time at CERN, but I do go to Madison fairly often. So I do travel a lot.

**How do you manage it all? **

Well, I think the key is that I am totally devoted. My husband, Tai Tsun Wu, is also a professor, in theoretical physics at Harvard. Right now, he’s working even harder than me, which is hard to imagine. He’s doing a calculation about the Higgs boson decay that is very difficult. But I encourage him to work hard, because it’s good for your mental state when you are older. That’s why I work so hard, too.

**Of all the discoveries you were involved in, do you have a favorite? **

Discovering the gluon was a fantastic time. I was just a second- or third-year assistant professor. And I was so happy. That’s because I was the baby, the youngest of all the key members of the collaboration.

**The gluon was the first force-carrying particle discovered since the photon. The W and Z bosons, which carry the weak force, were discovered a few years later, and the researchers who found them won a Nobel Prize. Why was no prize awarded for the discovery of the gluon?**

Well, you are going to have to ask the Nobel committee that. I can tell you what I think, though. Only three people can win a Nobel Prize. And there were three other physicists on the experiment with me who were more senior than I was. They treated me very well. But I pushed the idea of searching for the gluon right away, and I did the calculations. I didn’t even talk to theorists. Although I married a theorist, I never really paid attention to what the theorists told me to do.

**How did you wind up being the one to do those calculations?**

If you want to be successful, you have to be fast. But you also have to be first. So I did the calculations to make sure that as soon as a new collider at DESY turned on in Hamburg, we could see the gluon and recognize its signal of three jets of particles. We were not so sure in those days that the signal for the gluon would be clear-cut, because the concept of jets had only been introduced a couple of years earlier, but this seemed to be the only way to discover gluons.

**You were also involved in discovering the Higgs boson, the particle in the Standard Model that gives many other particles their masses. How was that experiment different from the others that you were part of?**

I worked a lot more and a lot longer to discover the Higgs than I have on anything else. I worked for over 30 years, doing one experiment after another. I think I contributed a lot to that discovery. But the ATLAS collaboration at CERN is so large that you can’t even talk about your individual contribution. There are 3,000 people who built and worked on our experiment. How can anyone claim anything? In the old days, life was easier.

**Has it gotten any easier to be a woman in physics than when you started?**

Not for me. But for younger women, yes. There is a trend among funding agencies and institutions to encourage younger women, which I think is great. But for someone like me it is harder. I went through a very difficult time. And now that I am established others say: Why should we treat you any differently?

**Who were some of your mentors when you were a young researcher?**

Bjørn Wiik really helped me when I was looking for the gluon at DESY.

**How so?**** **

Well, when I started at the University of Wisconsin, I was looking for a new project. I was interested in doing electron-positron collisions, which could give the clearest indication of a gluon. So I went to talk to another professor at Wisconsin who did these kinds of experiments at SLAC, the lab at Stanford. But he was not interested in working with me.

So I tried to join a project at the new electron-positron collider at DESY. I wanted to join the JADE experiment . I had some friends working there, so I went to Germany and I was all set to join them. But then I heard that no one had told a big professor in the group about me, so I called him up. He said, “I am not sure if I can take you, and I am going on vacation for a month. I’ll phone you when I get back.” I was really sad because I was already in Germany at DESY.

But then I ran into Bjørn Wiik, who led a different experiment called TASSO, and he said, “What are you doing here?” I said, “I tried to join JADE, but they turned me down.” He said, “Come and talk to me.” He accepted me the very next day. And the thing is, JADE later broke their chamber, and they could not have observed the three-jet signal for gluons when we observed it first at TASSO. So I have learned that if something does not work out for you in life, something else will.

**You certainly turned that negative into a positive. **

Yes. The same thing happened when I left Hong Kong to attend college in the U.S. I applied to 50 universities after I went through a catalog at the American consulate. I wrote in every application, “I need a full scholarship and room and board,” because I had no money. Four universities replied. Three of them turned me down. Vassar was the only American college that accepted me. And it turns out, it was the best college of all the ones I applied to.

If you persist, something good is bound to happen. My philosophy is that you have to work hard and have good judgment. But you also have to have luck.

**I know this is an unfair question, because no one ever asks men, even though we should, but how can society inspire more women to study physics or consider it as a career? **

Well, I can only say something about my field, experimental high-energy physics. I think my field is very hard for women. I think partially it’s the problem of family.

My husband and I did not live together for 10 years, except during the summers. And I gave up having children. When I was considering having children, it was around the time when I was up for tenure and a grant. I feared I would lose both if I got pregnant. I was less worried about actually having children than I was about walking into my department or a meeting while pregnant. So it’s very, very hard for families.

**I think it still can be. **

Yeah, but for the younger generation it’s different. Nowadays, a department looks good if it supports women. I don’t mean that departments are deliberately doing that only to look better, but they no longer actively fight against women. It’s still hard, though. Especially in experimental high-energy physics. I think there is so much traveling that it makes having a family or a life difficult. Theory is much easier.

**You have done so much to help establish the Standard Model of particle physics. What do you like about it? What do you not like?**

It’s just amazing that the Standard Model works as well as it does. I like that every time we try to search for something that is not accounted for in the Standard Model, we do not find it, because the Standard Model says we shouldn’t.

But back in my day, there was so much that we had yet to discover and establish. The problem now is that everything fits together so beautifully and the Model is so well confirmed. That’s why I miss the time of the J/psi discovery. Nobody expected that, and nobody really had a clue what it was.

But maybe those days of surprise aren’t over.

**We know that the Standard Model is an incomplete description of nature. It doesn’t account for gravity, the masses of neutrinos, or dark matter — the invisible substance that seems to make up six-sevenths of the universe’s mass. Do you have a favorite idea for what lies beyond the Standard Model?**

Well, right now I am searching for the particles that make up dark matter. The only thing is, I am committed to working at the Large Hadron Collider at CERN. But a collider may or may not be the best place to look for dark matter. It’s out there in the galaxies, but we don’t see it here on Earth.

Still, I am going to try. If dark matter has any interactions with the known particles, it can be produced via collisions at the LHC. But weakly interacting dark matter would not leave a visible signature in our detector at ATLAS, so we have to intuit its existence from what we actually see. Right now, I am concentrating on finding hints of dark matter in the form of missing energy and momentum in a collision that produces a single Higgs boson.

**What else have you been working on?**

Our most important task is to understand the properties of the Higgs boson, which is a completely new kind of particle. The Higgs is more symmetric than any other particle we know about; it’s the first particle that we have discovered without any spin. My group and I were major contributors to the very recent measurement of Higgs bosons interacting with top quarks. That observation was extremely challenging. We examined five years of collision data, and my team worked intensively on advanced machine-learning techniques and statistics.

In addition to studying the Higgs and searching for dark matter, my group and I also contributed to the silicon pixel detector, to the trigger system , and to the computing system in the ATLAS detector. We are now improving these during the shutdown and upgrade of the LHC. We are also very excited about the near future, because we plan to start using quantum computing to do our data analysis.

**Do you have any advice for young physicists just starting their careers?**

Some of the young experimentalists today are a bit too conservative. In other words, they are afraid to do something that is not in the mainstream. They fear doing something risky and not getting a result. I don’t blame them. It’s the way the culture is. My advice to them is to figure out what the most important experiments are and then be persistent. Good experiments always take time.

**But not everyone gets to take that time.**

Right. Young students don’t always have the freedom to be very innovative, unless they can do it in a very short amount of time and be successful. They don’t always get to be patient and just explore. They need to be recognized by their collaborators. They need people to write them letters of recommendation.

The only thing that you can do is work hard. But I also tell my students, “Communicate. Don’t close yourselves off. Try to come up with good ideas on your own but also in groups. Try to innovate. Nothing will be easy. But it is all worth it to discover something new.”

*Correction July 18, 2018: Due to miscommunication, this article was slightly revised to more accurately reflect Wu’s view of the current state of particle physics.*

Take the social amoeba *Dictyostelium discoideum*, which has three: Each type can mate with members of the other two. *Coprinellus disseminatus*, a white-capped mushroom, has 143, each able to find a partner among the 142 others. The hairy, fan-shape fungus *Schizophyllum commune *boasts more than 23,000 mating types (though its more intricate reproductive strategy means that not every type can mate with every other).

Yet most species still have only two mating types. George Constable, a research fellow at the University of Bath, and Hanna Kokko, an evolutionary biologist at the University of Zurich, wanted to know why. In a paper published last month in *Nature Ecology & Evolution*, they developed a model that predicts how many mating types will emerge in a species based on just three fundamental ecological elements: the mutation rate (which introduces new types), the population size and — perhaps most surprisingly — the frequency of sex. Their work not only provides insights about the basic biology of these kinds of organisms, but could also contribute to our understanding of how the male and female sexes ultimately evolved.

Many scientists believe mating types evolved early in life’s history as a barrier against behaviors like inbreeding that might be harmful to a population or species. If an organism has sex with an incompatible mating type (including its own mating type), then the union generally produces no offspring.

That restriction aside, logic suggests that species should benefit from having as many mating types as possible. With two types, only half the population is eligible as a mate for any individual. With three, that rises to two-thirds — and so on as more mating types join the mix. Should a mutation lead to the appearance of a new type, it wouldn’t be stuck with the problem of finding a rare match for itself in the population; instead, it would be able to mate with everyone else, thereby producing offspring more quickly and growing its numbers.

“The intuitive expectation is that this should happen for larger and larger numbers of mating types, until you have thousands of them,” Constable said.

To date, the hypotheses about why the number of mating types only rarely soars to enormous heights revolve around considerations of stability. Maintaining just two types may be the better way to go: It allows for simpler, more efficient pheromone-signaling networks, and for an easier sorting system when it comes to passing on organelles from parent to offspring cells. But these theories don’t account for a slew of exceptions.

Then something occurred to Constable. “I realized that we’d been assuming that these species have sex all the time,” he said. That assumption made a huge difference in his predictions about how populations would evolve, because during periods without sex, mating type becomes a neutral trait: Chance events dictate the dominance of some types and the disappearance of many others.

According to the model, large populations that rely relatively more on sex to reproduce can sustain a greater number of mating types, while those having less sex cannot. Constable and Kokko wondered just how rare sexual reproduction would have to be to explain as few as two mating types. Very, very rare, as it turned out: just once every few thousand generations.

“At first, I was kind of disappointed,” Constable said. “That seemed really low.” But when he and Kokko turned to examples from nature, they found that their model’s predictions held up well. “That’s the beauty of it,” said Bart Nieuwenhuis, an evolutionary biologist at the University of Munich. Amoebas, fungi and other organisms that have two mating types tend to have sex very infrequently, opting most of the time for the faster, less energy-intensive process of asexual reproduction: Some species of yeast, for instance, have sex once in every 1,000 to 3,000 generations, when stressful environmental conditions make it advantageous for them to mix up their genes and improve their odds of evolving new beneficial traits.

Meanwhile, those species that do have hundreds or thousands of mating types, Constable said, are known as some of “the most sexual fungi in the fungal kingdom.” His model also seems to explain other observed sexual patterns, such as the ability of some species to switch their mating type or to reproduce with members of their own type.

“It takes a bit of a long-standing mystery and proposes a solution that is really quite simple, and that ties directly into the biology of these organisms in a clear way,” said Jussi Lehtonen, an evolutionary biologist at the University of Sydney.

In doing so, according to Kokko, it also highlighted that what we understand about fundamental biology, based on just a handful of model organisms (like mice, fruit flies or *E. coli*), fails to capture the real diversity of even the most basic functions that occur in nature. “Researchers a bit myopic when it comes to understanding diversity. Not all life obeys the most familiar rules,” Kokko wrote in an email. She hopes her research will inspire further empirical study of these other, less orthodox species. (Such work might also help scientists build on her model by adding species-specific mechanisms like pheromone signaling and organelle inheritance, which remain important parts of the story.)

The seemingly esoteric rules those organisms live by might even help us understand traits we do find familiar. “We can see the existence of two mating types as a trigger for the evolution of the male and female sexes” that descended from them, said Sylvain Billiard, a biologist at the University of Lille in France. Constable and Kokko’s model could potentially provide a sense of the conditions that were needed to lay the groundwork for that to happen. Nieuwenhuis speculated that, because two mating types dominate when the rate of sexual reproduction is low, situations may have arisen in which it was difficult to find a mate. Those conditions could have selected for specialized, smaller gametes capable of more easily reaching a partner — beginning the road toward today’s sexes.

Nieuwenhuis is trying to see some of this in his lab: He’s been working in fission yeast to create a third mating type able to reproduce with the two types that already exist. “But it’s very tricky,” he said, and he’s been unsuccessful so far.

Constable also thinks the work could have more direct applications. In pathogenic fungi that infect crops, one mating type is often particularly destructive, and the genes that determine that type may be linked to, say, its virulent resistance to antifungals. Understanding why these traits coexist might help with controlling or preventing blights.

Constable and Kokko’s finding, said Zena Hadjivasiliou, a postdoctoral fellow at the University of Geneva, “in some ways … starts with a quite simple and intuitive idea. But sometimes the nicest works come from these types of inspirations.”

]]>Many different complexity classes exist, though in most cases researchers haven’t been able to prove one class is categorically distinct from the others. Proving those types of categorical distinctions is among the hardest and most important open problems in the field. That’s why the new result I wrote about last month in *Quanta* was considered such a big deal: In a paper published at the end of May, two computer scientists proved (with a caveat) that the two complexity classes that represent quantum and classical computers really are different.

The differences between complexity classes can be subtle or stark, and keeping the classes straight is a challenge. For that reason, *Quanta* has put together this primer on seven of the most fundamental complexity classes. May you never confuse BPP and BQP again.

**Stands for:** Polynomial time

**Short version:** All the problems that are easy for a classical (meaning nonquantum) computer to solve.

**Precise version: **Algorithms in P must stop and give the right answer in at most *n ^{c }*time where

**Typical problems:**

• Is a number prime?

• What’s the shortest path between two points?

**What researchers want to know:** Is P the same thing as NP? If so, it would upend computer science and render most cryptography ineffective overnight. (Almost no one thinks this is the case.)

**Stands for:** Nondeterministic Polynomial time

**Short version:** All problems that can be quickly verified by a classical computer once a solution is given.

**Precise version:** A problem is in NP if, given a “yes” answer, there is a short proof that establishes the answer is correct. If the input is a string, *X*, and you need to decide if the answer is “yes,” then a short proof would be another string, *Y*, that can be used to verify in polynomial time that the answer is indeed “yes.” (*Y* is sometimes referred to as a “short witness” — all problems in NP have “short witnesses” that allow them to be verified quickly.)

**Typical problems:**

• The clique problem. Imagine a graph with edges and nodes — for example, a graph where nodes are individuals on Facebook and two nodes are connected by an edge if they’re “friends.” A clique is a subset of this graph where all the people are friends with all the others. One might ask of such a graph: Is there a clique of 20 people? 50 people? 100? Finding such a clique is an “NP-complete” problem, meaning that it has the highest complexity of any problem in NP. But if given a potential answer — a subset of 50 nodes that may or may not form a clique — it’s easy to check.

• The traveling salesman problem. Given a list of cities with distances between each pair of cities, is there a way to travel through all the cities in less than a certain number of miles? For example, can a traveling salesman pass through every U.S. state capital in less than 11,000 miles?

**What researchers want to know: **Does P = NP? Computer scientists are nowhere near a solution to this problem.

**Stands for:** Polynomial Hierarchy

**Short version:** PH is a generalization of NP — it contains all the problems you get if you start with a problem in NP and add additional layers of complexity.

**Precise version:** PH contains problems with some number of alternating “quantifiers” that make the problems more complex. Here’s an example of a problem with alternating quantifiers: Given *X*, does there exist *Y* such that for every *Z* there exists *W* such that *R* happens? The more quantifiers a problem contains, the more complex it is and the higher up it is in the polynomial hierarchy.

**Typical problem:**

• Determine if there exists a clique of size 50 but no clique of size 51.

**What researchers want to know: **Computer scientists have not been able to prove that PH is different from P. This problem is equivalent to the P versus NP problem because if (unexpectedly) P = NP, then all of PH collapses to P (that is, P = PH).

**Stands for:** Polynomial Space

**Short version:** PSPACE contains all the problems that can be solved with a reasonable amount of memory.

**Precise version:** In PSPACE you don’t care about time, you care only about the amount of memory required to run an algorithm. Computer scientists have proven that PSPACE contains PH, which contains NP, which contains P.

**Typical problem:**

• Every problem in P, NP and PH is in PSPACE.

**What researchers want to know: **Is PSPACE different from P?

**Stands for:** Bounded-error Quantum Polynomial time

**Short version:** All problems that are easy for a quantum computer to solve.

**Precise version:** All problems that can be solved in polynomial time by a quantum computer.

**Typical problems:**

• Identify the prime factors of an integer.

**What researchers want to know: **Computer scientists have proven that BQP is contained in PSPACE and that BQP contains P. They don’t know whether BQP is in NP, but they believe the two classes are incomparable: There are problems that are in NP and not BQP and vice versa.

**Stands for:** Exponential Time

**Short version:** All the problems that can be solved in an exponential amount of time by a classical computer.

**Precise version:** EXP contains all the previous classes — P, NP, PH, PSPACE and BQP. Researchers have proven that it’s different from P — they have found problems in EXP that are not in P.

**Typical problem:**

• Generalizations of games like chess and checkers are in EXP. If a chess board can be any size, it becomes a problem in EXP to determine which player has the advantage in a given board position.

**What researchers want to know: **Computer scientists would like to be able to prove that PSPACE does not contain EXP. They believe there are problems that are in EXP that are not in PSPACE, because sometimes in EXP you need a lot of memory to solve the problems. Computer scientists know how to separate EXP and P.

**Stands for:** Bounded-error Probabilistic Polynomial time

**Short version:** Problems that can be quickly solved by algorithms that include an element of randomness.

**Precise version:** BPP is exactly the same as P, but with the difference that the algorithm is allowed to include steps where its decision-making is randomized. Algorithms in BPP are required only to give the right answer with a probability close to 1.

**Typical problem:**

• You’re handed two different formulas that each produce a polynomial that has many variables. Do the formulas compute the exact same polynomial? This is called the polynomial identity testing problem.

**What researchers want to know: **Computer scientists would like to know whether BPP = P. If that is true, it would mean that every randomized algorithm can be de-randomized. They believe this is the case — that there is an efficient deterministic algorithm for every problem for which there exists an efficient randomized algorithm — but they have not been able to prove it.

The twin events could have been a cosmic coincidence. But when physicists looked through their archived data, they found several other neutrinos that appear to have come from the same direction. This supporting evidence has convinced them that they’ve achieved a cosmic first: tracing ultrahigh-energy neutrinos back to their astrophysical source.

That source appears to be a supermassive black hole at the center of a distant galaxy. Every time a black hole gobbles up a star, it spews out a stream of very high-energy radiation in the form of a laserlike jet. Astronomers say that it “flares.” They call these flaring black holes blazars and believe that energetic neutrinos are created as a by-product of the radiation.

The find, described in a series of two papers published in *Science*, has not only settled a long-standing debate in astronomy over the origin of high-energy neutrinos. This discovery is also the second example (after last year’s observation of colliding neutron stars) of a new scientific approach called multimessenger astronomy, in which astronomers use both light and another cosmic messenger (neutrinos in this case, gravitational waves in the other) to reveal the details of an astrophysical event. “This is like getting an entirely new way of looking at the universe,” said Roopesh Ojha, an astrophysicist at NASA Goddard Space Flight Center.

When the neutrino arrived, Albrecht Karle, a leader of the IceCube experiment, was in his office at the University of Wisconsin, Madison, preparing for his November trip to the South Pole. IceCube detects more than 50,000 neutrino candidates every year, but only about 10 of them are at the very high energies that indicate that they come from outside the Milky Way galaxy. When the detector spots a candidate high-energy neutrino, within minutes it sends an alert to members of the team and to observatories around the world.

The alert that popped up on Karle’s computer said that the candidate neutrino, dubbed IceCube-170922A, carried around 300 teraelectron-volts of energy, more than 40 times the energy of the protons produced by the Large Hadron Collider near Geneva. “I am normally not easy to get excited, but this one smells right,” Karle said to Elisa Bernardini, an astroparticle physicist at Humboldt University of Berlin.

Enter Fermi. When IceCube spotted the neutrino, the space-based Fermi Large Area Telescope happened to be scanning the area of the sky from which it appeared. It also recorded an unusually intense flare of gamma radiation. Ojha works with the Fermi telescope, and when IceCube’s alert landed in his inbox, he “straightaway knew that that was something interesting.” Sure enough, he found a match. There, a bit to the west of Bellatrix, a star in the constellation Orion, lives a blazar dubbed TXS 0506+056.

Next, Ojha checked a catalog of radio sources, while a colleague examined optical sources. They found a surge in gamma rays, radio waves and optical waves coming from the blazar. It was flaring.

All blazars are “inherently variable objects,” said Ojha, but over the previous three months, this one had moved from being the 51st brightest in one particular catalog into the top five. Along with Fermi, the flare was also seen by Swift, another space-based gamma-ray telescope.

On the Canary Islands, at the ground-based gamma-ray telescope Magic, the excitement was growing as well. After a few nights of bad weather, the flare came into focus — and the telescope registered gamma rays with energies exceeding 400 gigaelectron-volts. The surge in radiation was confirmed by several instruments, in “an exciting, frenetic series of email exchanges,” said Sara Buson, an astrophysicist at NASA Goddard Space Flight Center.

Just as in the case of the neutron star merger, astronomers all over the world leaped into action. In the days and weeks that followed IceCube’s original alert, more and more instruments swiveled toward the blazar. Eventually a total of 18 observatories would record radiation in various wavelengths in exactly the spot where the neutrino was thought to have come from.

Neutrinos, though infamously difficult to detect, offer a key advantage over observations with light: The universe is essentially opaque to electromagnetic radiation above a certain energy. High-energy neutrinos, on the other hand, zip across the cosmos unimpeded. When they are caught, the original information they carry remains intact, no matter how far they have had to travel. Scientists hope that with enough neutrinos, they’ll be able image their sources — including the most energetic events in the universe — just like we do with electromagnetic radiation.

Scientists believe that this discovery is a crucial step toward solving a number of physics puzzles, such as the origin of ultrahigh-energy cosmic rays. These particles, made of protons or heavier atomic nuclei, rain down on Earth from space and are believed to be born in the same processes as neutrinos. Researchers posit that they might come from a merger of two neutron stars or that perhaps they’re generated by magnetars — rapidly rotating neutron stars that produce strong magnetic fields. The finding could also help to determine the precise mass of the neutrino, spot other predicted types of neutrinos such as sterile neutrinos, and even possibly detect dark matter.

To do this, astronomers will have to find more neutrinos, at higher energies, and quickly link them back to their sources in the sky. Even this neutrino, energetic as it was, can’t compare to the most energetic neutrinos that IceCube has spotted, among them a trio of neutrinos with energies in the petaelectron-volt range, or 1,000 times that of a teraelectron-volt.

These neutrinos, with the Sesame Street–inspired names Ernie, Bert and Big Bird, haven’t yet been definitively traced back to their sources (although there is a suggestion that Big Bird might have originated in the flaring blazar PKS B1424-418). The makeup of these neutrinos is partly to blame for the uncertainty, because they all come from the electron neutrino family. Such neutrinos produce cascades of particles in the IceCube detector, which radiate out like a wide ice-cream cone. This makes it relatively easy to estimate the neutrino’s energy, but hard to track its direction. Muon neutrinos, on the other hand, produce a long, straight cascade of particles, which gives them a precise direction and an uncertain energy. (The neutrino observed last September was a muon neutrino.)

Over the past few years, said Karle, IceCube has detected neutrinos with probably even higher energy levels than the Sesame Street trio, although some of the data is still being analyzed. IceCube scientists have already analyzed four more years of data beyond the original data set from 2010 and 2011, when Ernie and Bert were detected. So far, they have found a total of 54 additional high-energy neutrino events. “This is a great time to be alive,” said Ojha.

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