At the time, Klein was 23 years old and living in her hometown of Budapest, Hungary. One day she brought a puzzle to two of her friends, Paul Erdős and George Szekeres: Given five points, and assuming no three fall exactly on a line, prove that it is always possible to form a convex quadrilateral — a four-sided shape that’s never indented (meaning that, as you travel around it, you make either all left turns or all right turns).

Erdős and Szekeres eventually found a way to show that Klein’s statement was true (she had worked out the proof before bringing it to them), and it got them thinking: If five points are enough to guarantee that you can always connect four to form this kind of quadrilateral, how many points are needed to guarantee that you can form this same kind of shape with five sides, or 11 sides, or any number of sides?

By 1935 Erdős and Szekeres had solved this problem for shapes with three, four and five sides. They knew it took three points to guarantee you could construct a convex triangle, five points to guarantee a convex quadrilateral, and nine points to guarantee a convex pentagon.

In the same paper in which they presented these solutions, Erdős and Szekeres proposed an exact formula for the number of points it would take to guarantee a convex polygon of any number of sides: 2^{(n–2)} + 1, where *n *is the number of sides. But their proposal was just that — a well-aimed conjecture. Erdős, as he did with many problems, offered a cash bounty of $500 to anyone who could prove the formula was correct.

The puzzle was given a memorable nickname, the “happy ending” problem (or “happy end” problem as originally dubbed by Erdős), for reasons that had nothing to do with math. Instead, it reflected the primary nonmathematical consequence of their discussion of points, lines and shapes: Esther Klein and George Szekeres fell in love and married on June 13, 1937.

Yet as the decades passed, mathematicians made virtually no progress in proving the conjecture. (The only other shape whose result is known is a hexagon, which requires at least 17 points, as proved by Szekeres and Lindsay Peters in 2006.) Now, in work recently published in the *Journal of the American Mathematical Society*, Andrew Suk of the University of Illinois, Chicago, provides nearly decisive evidence that the intuition that guided Erdős and Szekeres more than 80 years ago was correct.

At a very general level, the happy ending problem is about finding ways to add sides to a polygon. Say you have five points. You know that this is enough to guarantee that you can construct a convex quadrilateral by connecting four of those points. From there, you want to increase the number of sides of that polygon — from four sides to five, six and beyond. As you do this, you need to keep track of the number of points you need to add in order to guarantee that you can make the desired shape.

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Erdős and Szekeres were not able to prove their formula, but they had a clear sense of the path a mathematician might follow to get there. In the 1935 paper where they made the conjecture, they wrote about how convex polygons can be thought of as what are now called “cups” and “caps” glued together. A cup is made from a set of points in a “u” shape. It forms the bottom part of the polygon. A cap is a set of points in an “n” shape that forms the top.

The advantage of thinking about a polygon as a cup and cap glued together is that it’s easier to think about how you’d enlarge a cup or a cap than it is to figure out how to enlarge a polygon. If you have five points forming a convex cap, for instance, you can always enlarge it by adding a sixth point off one end or the other. But with a convex five-sided polygon, it’s less obvious how you’d add a sixth point while still preserving its convexity.

“They’re just a little easier to deal with,” said Ronald Graham, a mathematician at the University of California, San Diego, and a longtime friend of Erdős’s. “Say you have a cap. Depending on where you put one more point, it can still form a cap. You can kind of see where it’s going to be. If you have a convex polygon and you add one more point, you don’t have quite as much control about what’s going on.”

In their 1935 paper Erdős and Szekeres proved what’s called the cups-caps theorem, which tells you the minimum number of points you need before you’re guaranteed either a cup or a cap of a certain size. The cups-caps theorem also creates an upper bound on the happy ending problem for reasons that are intuitive enough: If you have a cup, you can always create a convex polygon simply by drawing a line between the cup’s endpoints. It’s not the most efficient way to construct a convex polygon, but it guarantees you can make one.

Using the cups-caps theorem, Erdős and Szekeres proved that if you have 4* ^{n}* points, you can always find a cup or a cap that gives you an

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Erdős and Szekeres understood that if they could find a more efficient way to combine a cup with a cap, they could prove that their conjectured value was correct. The problem they ran into was that you can’t always glue cups and caps together to form a convex polygon. For the gluing to work, the cap needs to be “strictly above” the cup, as mathematicians put it. There’s an exact way of defining what this condition means: A line drawn through any two points in the cap lies above all the points in the cup. When it’s not present — when the cup is not strictly above the cap — the shape you get when you glue them together is not convex.

This cups-caps impasse held for many years: Mathematicians viewed the gluing of cups and caps as the best way to approach the problem but couldn’t figure out how to do it. They wanted to find a large cup and a large cap that were in the right position with respect to each other, such that when they were glued together, you would get the biggest convex polygon you could be guaranteed (given the total number of points you started with). Yet in order to prove that their caps were always strictly above their cups (or their cups were strictly below their caps), they always ended up having to restrict the cup size and cap size to a degree that undermined the whole approach.

“They’ll be so small that what you gain by putting a cap and a cup together doesn’t help you much. You lose more than what you gain,” said Gábor Tardos, a mathematician at the Alfréd Rényi Institute of Mathematics of the Hungarian Academy of Sciences in Budapest. “That was the obstacle.”

It was, until Andrew Suk found a way over it.

Suk’s work focuses on an area of mathematics called Ramsey theory, which is about as old as the happy ending problem itself. Ramsey theory says, broadly, that within large disorganized sets — like a set of points dispersed randomly on a plane — you will always be able to find well-organized subsets. This basic idea was first discovered mathematically a few years before Klein posed her problem to Erdős and Szekeres, and it was rediscovered by the two men themselves in the course of studying the happy ending problem.

When applied to this problem, Ramsey theory states that no matter how you place your points on the plane, when the number of points grows sufficiently large, you’re guaranteed to get a well-organized subset of points that can be joined to form a convex *n*-gon. There’s absolutely no way to avoid it.

Suk uses an idea from Ramsey theory to divide up the points into subsets that are all in convex position with respect to each other. Ramsey theory tells him how many of these subsets he can create and roughly how many points have to be in each. Now each subset is contained within a triangular “spike” and all the spikes are arrayed around the outside of what you can picture as a convex hull. The graphic below shows what it might look like. (Note: During this step, and the subsequent steps, Suk doesn’t ever move points around; he just focuses on structured subsets of points that were present from the start.)

After organizing the original points into spikes, Suk begins to reason about subsets within each spike. His goal is to prove that within each spike he’s guaranteed either a cup or a cap of a certain size. And because the spikes are all in convex position with respect to each other, he knows that cups or caps within one spike can be combined with cups or caps within other spikes.

To hunt for cups and caps, Suk uses another part of Ramsey theory called Dilworth’s theorem to organize the points in each spike. Dilworth’s theorem implies that inside each spike there’s a set of some guaranteed minimum size in which the points are either parallel or perpendicular to the convex hull the spikes are organized around. (In this figure, the blue dots are perpendicular “chains” and the red dots are parallel “antichains.”)

From here, Suk builds cups and caps by going around the figure and combining points from different spikes. Based on another argument from Ramsey theory — the pigeonhole principle — Suk is able to show that the arrangement of chains and antichains around the convex hull must fall into one of two configurations.

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The first possibility is that he finds many red antichains. In this case, Suk argues that if one of the antichains contains an *n*-cup, then he’s finished, as he’s found the *n*-gon he’s looking for. Otherwise, the cups-caps theorem implies that there have to be mini-caps inside each of the antichains that can be combined to form one large, convex *n*-cap — again, giving him the shape he’s looking for.

The second scenario is thornier. In this case, he finds many consecutive spikes with blue perpendicular chains. Here, Suk moves left to right around the spikes. At each step he reasons about whether cups and caps can be combined to form a convex *n*-gon. If the answer is no at every step, he’s in the very worst-case scenario. But even there, Suk shows that as a consequence of the preceding futility, the last spike must contain an *n*-element cap all by itself.

“I keep applying this argument over and over such that by the time I get to the last region, I have to get it,” Suk said.

Suk uses tools from Ramsey theory to think about the kinds of organized substructures that have to emerge, and how large those substructures have to be, and how one kind of structure interacts with another. As a result, he’s able to assess whole categories of configurations in a small number of moves, cornering the points into their most difficult arrangements, and proving that even there, they have to yield the convex *n*-gon he’s looking for.

Suk’s work suggests that Erdős and Szekeres’ conjecture is almost certainly true, even if he stops short of proving it completely. Suk’s proof brings the previous upper bound on the problem — about 4* ^{n}* points — down to a value that’s almost equal to the formula proposed by Erdős and Szekeres when the number of points is large.

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“I think no one would have been shocked until now to see if the truth is close to 4* ^{n}*. Now we definitely know the conjecture is roughly true, which of course makes it more believable that it is actually true for every

Even when there’s a consensus about whether a long-standing conjecture is true or not, often it takes the advent of some complicated new mathematical machinery to settle it one way or the other. That was not the case here, where the basic path Suk followed, of combing cups and caps, was evident from the start.

“It’s very clever, very elementary, something Erdős and Szekeres could have done themselves,” Graham said. Pach adds, “I think Erdős and Szekeres would be extremely happy to see they were actually on the right track and these ideas they came up with can be made to work.”

Neither is alive today. Erdős passed away in 1996, though he lives on in the many cash prizes he endowed, which are now managed mostly by Graham. Szekeres outlived Erdős by nearly a decade. He and Klein were both in their mid-nineties when they moved into a nursing home in Adelaide, Australia, in 2005. They died there on August 28, 2005, within an hour of each other — 70 years after the birth of the happy ending problem, and approximately a decade before it was nearly settled.

]]>In 2008, the researcher Hilton Japyassú prompted 12 species of orb spiders collected from all over Brazil to go through this transition again. He waited until the spiders wove an ordinary web. Then he snipped its threads so that the silk drooped to where crickets wandered below. When a cricket got hooked, not all the orb spiders could fully pull it up, as a cobweb spider does. But some could, and all at least began to reel it in with their two front legs.

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Their ability to recapitulate the ancient spiders’ innovation got Japyassú, a biologist at the Federal University of Bahia in Brazil, thinking. When the spider was confronted with a problem to solve that it might not have seen before, how did it figure out what to do? “Where is this information?” he said. “Where is it? Is it in her head, or does this information emerge during the interaction with the altered web?”

In February, Japyassú and Kevin Laland, an evolutionary biologist at the University of Saint Andrews, proposed a bold answer to the question. They argued in a review paper, published in the journal *Animal Cognition,* that a spider’s web is at least an adjustable part of its sensory apparatus, and at most an extension of the spider’s cognitive system.

This would make the web a model example of extended cognition, an idea first proposed by the philosophers Andy Clark and David Chalmers in 1998 to apply to human thought. In accounts of extended cognition, processes like checking a grocery list or rearranging Scrabble tiles in a tray are close enough to memory-retrieval or problem-solving tasks that happen entirely inside the brain that proponents argue they are actually part of a single, larger, “extended” mind.

Among philosophers of mind, that idea has racked up citations, including supporters and critics. And by its very design, Japyassú’s paper, which aims to export extended cognition as a testable idea to the field of animal behavior, is already stirring up antibodies among scientists. “I got the impression that it was being very careful to check all the boxes for hot topics and controversial topics in animal cognition,” said Alex Jordan, a collective behaviorial scientist at the Max Planck Institute in Konstanz, Germany (who nonetheless supports the idea).

While many disagree with the paper’s interpretations, the study shouldn’t be confused for a piece of philosophy. Japyassú and Laland propose ways to test their ideas in concrete experiments that involve manipulating the spider’s web — tests that other researchers are excited about. “We can break that machine; we can snap strands; we can reduce the way that animal is able to perceive the system around it,” Jordan said. “And that generates some very direct and testable hypotheses.”

The suggestion that some of a spider’s “thoughts” happen in its web fits into a small but growing trend in discussions of animal cognition. Many animals interact with the world in certain complicated ways that don’t rely on their brains. In some cases, they don’t even use neurons. “We have this romantic notion that big brains are good, but most animals don’t work this way,” said Ken Cheng, who studies animal behavior and information processing at Macquarie University in Australia.

Parallel to the extended cognition that Japyassú sees in spiders, researchers have been gathering examples from elsewhere in the animal kingdom that seem to show a related concept, called embodied cognition: where cognitive tasks sprawl outside of the brain and into the body.

Perhaps the prime example is another eight-legged invertebrate. Octopuses are famously smart, but their central brain is only a small part of their nervous systems. Two-thirds of the roughly 500 million neurons in an octopus are found in its arms. That led Binyamin Hochner of the Hebrew University of Jerusalem to consider whether octopuses use embodied cognition to pass a piece of food held in their arms straight to their mouths.

For the octopus, with thousands of suckers studding symmetric arms, each of which can bend at any point, building a central mental representation of how to move seems like a computational nightmare. But experiments show that the octopus doesn’t do that. “The brain doesn’t have to know how to move this floppy arm,” Cheng said. Rather, the arm knows how to move the arm.

Readings of electric signals show that when a sucker finds a piece of food, it sends a wave of muscle activation inward up the arm. At the same time, the base of the arm sends another wave of clenched muscles outward, down the arm. Where the two signals meet each other, the arm makes an elbow — a joint in exactly the right place to reach the mouth.

Yet another related strategy, this one perhaps much more common and less controversial, is that the sensory systems of many animals are tuned in to the parts of the world that are relevant to their lives. Bees, for example, use ultraviolet vision to find flowers that have also evolved ultraviolet markings. That avoids the need to take in lots of data and parse it later. “If you do not have those receptors, that part of the world simply doesn’t exist,” said William Wcislo, a behaviorist at the Smithsonian Tropical Research Institute in Panama.

And then there are animals that appear to offload part of their mental apparatus to structures outside of the neural system entirely. Female crickets, for example, orient themselves toward the calls of the loudest males. They pick up the sound using ears on each of the knees of their two front legs. These ears are connected to one another through a tracheal tube. Sound waves come in to both ears and then pass through the tube before interfering with one another in each ear. The system is set up so that the ear closest to the source of the sound will vibrate most strongly.

In crickets, the information processing — the job of finding and identifying the direction that the loudest sound is coming from — appears to take place in the physical structures of the ears and tracheal tube, not inside the brain. Once these structures have finished processing the information, it gets passed to the neural system, which tells the legs to turn the cricket in the right direction.

Extended cognition may partly be an evolutionary response to an outsized challenge. According to a rule first observed by the Swiss naturalist Albrecht von Haller in 1762, smaller creatures almost always devote a larger portion of their body weight to their brains, which require more calories to fuel than other types of tissue.

Haller’s rule holds across the animal kingdom. It works for mammals from whales and elephants down to mice; for salamanders; and across the many species of ants, bees and nematodes. And in this latter range, as brains demand more and more resources from the tiny creatures that host them, scientists like Wcislo and his colleague William Eberhard, also at the Smithsonian, think new evolutionary tricks should arise.

In 2007, Eberhard compared data on the webs built by infant and adult spiders of the same species. The newborns, roughly a thousand times smaller than the adults in some cases, should be under much more pressure from Haller’s rule. As a result, they might be expected to slip up while performing a complex task. Perhaps the spiderlings would make more mistakes in attaching threads at the correct angles to build a geometrically precise web, among other measures. But their webs seemed “as precise as that of their larger relatives,” Eberhard said. “One of the questions is: How do they get away with that?”

Japyassú’s work offers a possible solution. Just as octopi appear to outsource information-processing tasks to their tentacles, or crickets to their tracheal tubes, perhaps spiders outsource information processing to objects outside of their bodies — their webs.

To test whether this is truly happening, Japyassú uses a framework suggested by the cognitive scientist David Kaplan. If spider and web are working together as a larger cognitive system, the two should be able to affect each other. Changes in the spider’s cognitive state will alter the web, and changes in the web will likewise ripple into the spider’s cognitive state.

Consider a spider at the center of its web, waiting. Many web-builders are near blind, and they interact with the world almost solely through vibrations. Sitting at the hub of their webs, spiders can pull on radial threads that lead to various outer sections, thereby adjusting how sensitive they are to prey that land in those particular areas.

As is true for a tin can telephone, a tighter string is better at passing along vibrations. Tensed regions, then, may show where the spider is paying attention. When insects land in tensed areas of the webs of the orb spider *Cyclosa octotuberculata*, a 2010 study found, the spider is more likely to notice and capture them. And when the experimenters in the same study tightened the threads artificially, it seemed to put the spiders on high alert — they rushed toward prey more quickly.

The same sort of effect works in the opposite direction, too. Let the orb spider *Octonoba sybotides* go hungry, changing its internal state, and it will tighten its radial threads so it can tune in to even small prey hitting the web. “She tenses the threads of the web so that she can filter information that is coming to her brain,” Japyassú said. “This is almost the same thing as if she was filtering things in her own brain.”

Another example of this sort of interplay between web and spider comes from the web-building process itself. According to decades of research from scientists like Eberhard, a spiderweb is easier to build than it looks. What seems like a baroque process involving thousands of steps actually requires only a short list of rules of thumb that spiders follow at each junction. But these rules can be hacked from inside or out.

When experimenters start cutting out pieces of a web as it’s being built, a spider makes different choices — as if the already-built portions of silk are reminders, chunks of external memory it needs to retrieve so it can keep things evenly spaced, Japyassú said. Similarly, what happens in a web once it is built can change what kind of web the spider builds next time. If one section of the web catches more prey, the spider may enlarge that part in the future.

And from the opposite direction, the state of a spider’s nervous system can famously affect its webs. Going back to the 1940s, researchers have exposed spiders to caffeine, amphetamines, LSD and other drugs, attracting plenty of media attention along the way. Unsurprisingly, these spiders make addled, irregular webs.

Even skeptics of the extended cognition idea agree that this back and forth between the web and spider is ripe ground for more investigation and debate on how to interpret what the spiders are doing to solve problems. “It introduces a biological setup to the philosophers,” said Fritz Vollrath, an arachnologist at the University of Oxford. “For that, I think it’s very valuable. We can start a discussion now.”

But many biologists doubt that this interplay adds up to a bigger cognitive system. The key issue for critics is a semantic — but crucial — distinction. Japyassú’s paper defines cognition in terms of acquiring, manipulating and storing information. That’s a set of criteria that a web can easily meet. But to many, that seems like a low bar. “I think we’re fundamentally losing a distinction between information and knowledge,” Wcislo said. Opponents argue that cognition involves not just passing along information, but also interpreting it into some sort of abstract, meaningful representation of the world, which the web — or a tray of Scrabble tiles — can’t quite manage by itself.

Further, Japyassú’s definition of cognition may even undersell the level of thought that spiders are capable of, say the spider behaviorists Fiona Cross and Robert Jackson, both of the University of Canterbury in New Zealand. Cross and Jackson study jumping spiders, which don’t have their own webs but will sometimes vibrate an existing web, luring another spider out to attack. Their work suggests that jumping spiders do appear to hold on to mental representations when it comes to planning routes and hunting specific prey. The spiders even seem to differentiate among “one,” “two” and “many” when confronted with a quantity of prey items that conflicts with the number they initially saw, according to a paper released in April.

“How an animal with such a small nervous system can do all this should keep us awake at night,” Cross and Jackson write in an email. “Instead of marveling at this remarkable use of representation, it seems that Japyassú and Laland are looking for an explanation that removes representation from the equation — in other words, it appears they may actually be removing cognition.”

Even leaving aside the problem of what cognition actually is, proving the simple version of the argument — that spiders outsource problem solving to their webs as an end run around Haller’s rule — is by itself an empirical challenge. You would need to show that the analytical power of the web saves calories a spider would have otherwise spent on the nervous tissue in a bigger brain, Eberhard said. That would require quantifying how much energy it takes to build and use a web compared with the cost of performing the same operations with brain tissue. Such a study “would be an interesting kind of data to collect,” Eberhard said.

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Whether this kind of engineered information-processing happens elsewhere in nature is likewise unclear. Laland is a high-profile advocate for the idea of niche construction, a term from evolutionary theory that encompasses burrows, beaver dams and nests of birds and termites.

Proponents argue that when animals build these artificial structures, natural selection starts to modify the structure and the animal in a reciprocal loop. For example: A beaver builds a dam, which changes the environment. The changes in the environment in turn affect which animals survive. And then the surviving animals further change the environment. Under this rubric, Japyassú thinks, this back-and-forth action makes all niche constructors at least candidates to outsource some of their problem solving to the structures they build, and thus possible practitioners of extended cognition.

Alternatively, more traditional theorists label these structures and spiderwebs alike as extended phenotypes, a term proposed by Richard Dawkins. Extended phenotypes are information from an animal’s genes that they express in the world. For example, bird nests are objects that are somehow encoded in the avian genome. And as with niche construction, natural selection affects the structure — different kinds of birds have evolved to build different kinds of nests, after all. But in the extended phenotype perspective, that selection ultimately just works inward, to tweak the controlling information in the animal’s genome.

It’s a subtle difference. But experts who subscribe to Dawkins’s extended phenotype idea, like Vollrath at Oxford, believe that webs are more like tools the spider uses. “The web is actually a computer, as it were,” he said. “It processes information and simplifies it.” In this view, webs evolved over time like an extension of the spider’s body and sensory system — not so much its mind. Vollrath’s lab will soon embark on a project to test just how webs help the spiders solve problems from the extended phenotype perspective, he said.

While Japyassú, Cheng and others continue to look for extensions of cognition outward into the world, critics say the only really strong case is the one with the most metaphysical baggage: us. “It is conceivable for cognition to be a property of a system with integrated nonbiological components,” Cross and Jackson write. “That seems to be where *Homo sapiens* is headed.”

*This article was reprinted on TheAtlantic.com.*

Yet in that same ruling, the court declined to strike down two Indiana maps under consideration, even though both “used every trick in the book,” according to a paper in the *University of Chicago Law Review*. And in the decades since then, the court has failed to throw out a single map as an unconstitutional partisan gerrymander.

“If you’re never going to declare a partisan gerrymander, what is it that’s unconstitutional?” said Wendy K. Tam Cho, a political scientist and statistician at the University of Illinois, Urbana-Champaign.

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The problem is that there is no such thing as a perfect map — every map will have some partisan effect. So how much is too much? In 2004, in a ruling that rejected nearly every available test for partisan gerrymandering, the Supreme Court called this an “unanswerable question.” Meanwhile, as the court wrestles with this issue, maps are growing increasingly biased, many experts say.

Even so, the current moment is perhaps the most auspicious one in decades for reining in partisan gerrymandering. New quantitative approaches — measures of how biased a map is, and algorithms that can create millions of alternative maps — could help set a concrete standard for how much gerrymandering is too much.

Last November, some of these new approaches helped convince a United States district court to invalidate the Wisconsin state assembly district map — the first time in more than 30 years that any federal court has struck down a map for being unconstitutionally partisan. That case is now bound for the Supreme Court.

“Will the Supreme Court say, ‘Here is a fairness standard that we’re willing to stand by?’” Cho said. “If it does, that’s a big statement by the court.”

So far, political and social scientists and lawyers have been leading the charge to bring quantitative measures of gerrymandering into the legal realm. But mathematicians may soon enter the fray. A workshop being held this summer at Tufts University on the “Geometry of Redistricting” will, among other things, train mathematicians to serve as expert witnesses in gerrymandering cases. The workshop has drawn more than 1,000 applicants.

“We have just been floored at the response that we’ve gotten,” said Moon Duchin, a mathematician at Tufts who is one of the workshop’s organizers.

Gerrymanderers rig maps by “packing” and “cracking” their opponents. In packing, you cram many of the opposing party’s supporters into a handful of districts, where they’ll win by a much larger margin than they need. In cracking, you spread your opponent’s remaining supporters across many districts, where they won’t muster enough votes to win.

For instance, suppose you’re drawing a 10-district map for a state with 1,000 residents, who are divided evenly between Party A and Party B. You could create one district that Party A will win, 95 to 5, and nine districts that it will lose, 45 to 55. Even though the parties have equal support, Party B will win 90 percent of the seats.

Such gerrymanders are sometimes easy to spot: To pick up the right combination of voters, cartographers may design districts that meander bizarrely. This was the case with the “salamander”-shaped district signed into law in 1812 by Massachusetts governor Elbridge Gerry — the incident that gave the practice its name. In an assortment of racial gerrymandering cases, the Supreme Court has “stated repeatedly … that crazy-looking shapes are an indicator of bad intent,” Duchin said.

Yet it’s one thing to say bizarre-looking districts are suspect, and another thing to say precisely what bizarre-looking means. Many states require that districts should be reasonably “compact” wherever possible, but there’s no one mathematical measure of compactness that fully captures what these shapes should look like. Instead, there are a variety of measures; some focus on a shape’s perimeter, others on how close the shape’s area is to that of the smallest circle around it, and still others on things like the average distance between residents.

The Supreme Court justices have “thrown up their hands,” Duchin said. “They just don’t know how to decide what shapes are too bad.”

The compactness problem will be a primary focus of the Tufts workshop. The goal is not to come up with a single compactness measure, but to bring order to the jostling crowd of contenders. The existing literature on compactness by nonmathematicians is filled with elementary errors and oversights, Duchin said, such as comparing two measures statistically without realizing that they are essentially the same measure in disguise.

Mathematicians may be able to help, but to truly make a difference, they will have to go beyond the simple models they’ve used in past papers and consider the full complexity of real-world constraints, Duchin said. The workshop’s organizers “are absolutely, fundamentally motivated by being useful to this problem,” she said. Because of the flood of interest, plans are afoot for several satellite workshops, to be held across the country over the coming year.

Ultimately, the workshop organizers hope to develop a deep bench of mathematicians with expertise in gerrymandering, to “get persuasive, well-armed mathematicians into these court conversations,” Duchin said.

A compactness rule would limit the range of tactics available for drawing unfair maps, but it would be far from a panacea. For starters, there are a lot of legitimate reasons why some districts are not compact: In many states, district maps are supposed to try to preserve natural boundaries such as rivers and county lines as well as “communities of interest,” and they must also comply with the Voting Rights Act’s protections for racial minorities. These requirements can lead to strange-looking districts — and can give cartographers latitude to gerrymander under the cover of satisfying these other constraints.

More fundamentally, drawing compact districts gives no guarantee that the resulting map will be fair. On the contrary, a 2013 study suggests that even when districts are required to be compact, drawing biased maps is often easy, and sometimes almost unavoidable.

The study’s authors — political scientists Jowei Chen of the University of Michigan and Jonathan Rodden of Stanford University — examined the 2000 presidential race in Florida, where George W. Bush and Al Gore received an almost identical number of votes. Despite this perfect partisan balance, in the round of redistricting after the 2000 census, the Republican-controlled Florida legislature created a congressional district map in which Bush voters outnumbered Gore voters in 68 percent of the districts — a seemingly classic case of gerrymandering.

Yet when Chen and Rodden drew hundreds of random district maps using a nonpartisan computer algorithm, they found that their maps were biased in favor of Republicans too, sometimes as much as the official map. Democratic voters in the early 2000s, they found, were clustering into highly homogeneous neighborhoods in big cities like Miami and spreading out their remaining support in suburbs and small towns that got swallowed up inside Republican-leaning districts. They were packing and cracking themselves.

This kind of “unintentional gerrymandering” creates problems for Democrats in many of the large, urbanized states, Chen and Rodden found, although some states — such as New Jersey, in which Democratic voters are evenly spread through a large urban corridor — have population distributions that favor Democrats.

Chen and Rodden’s work indicates that biased maps can often arise even in the absence of partisan intent, and that drawing fair maps under such circumstances requires considerable care. Maps can be drawn that break up the tight city clusters, as in Illinois, where the Democratic-controlled legislature has created districts that unite segments of Chicago with suburbs and nearby rural areas.

Nevertheless, Chen and Rodden write, Democratic cartographers have a tougher job than Republican ones, who “can do strikingly well by literally choosing precincts at random.”

Since drawing compact districts is not a cure-all, solving the gerrymandering problem also requires ways to measure how biased a given map is. In a 2006 ruling, the Supreme Court offered tantalizing hints about what kind of measure it might look kindly on: one that captures the notion of “partisan symmetry,” which requires that each party have an equal opportunity to convert its votes into seats.

The court’s interest in partisan symmetry, coming after its rejection of so many other possible gerrymandering principles, represents “the most promising development in this area in decades,” wrote two researchers — Nicholas Stephanopoulos, a law professor at the University of Chicago, and Eric McGhee, a research fellow at the Public Policy Institute of California — in a 2015 paper.

In that paper, they proposed a simple measure of partisan symmetry, called the “efficiency gap,” which tries to capture just what it is that gerrymandering does. At its core, gerrymandering is about wasting your opponent’s votes: packing them where they aren’t needed and spreading them where they can’t win. So the efficiency gap calculates the difference between each party’s wasted votes, as a percentage of the total vote — where a vote is considered wasted if it is in a losing district or if it exceeds the 50 percent threshold needed in a winning district.

For instance, in our 10-district plan above, Party A wastes 45 votes in the one district it wins, and 45 votes each in the nine districts it loses, for a total of 450 wasted votes. Party B wastes only 5 votes in the district it loses, and 5 votes in each of the districts it wins, for a total of 50. That makes a difference of 400, or 40 percent of all voters. This percentage has a natural interpretation: It is the percentage of seats Party B has won beyond what it would receive in a balanced plan with an efficiency gap of zero.

Stephanopoulos and McGhee have calculated the efficiency gaps for nearly all the congressional and state legislative elections between 1972 and 2012. “The efficiency gaps of today’s most egregious plans dwarf those of their predecessors in earlier cycles,” they wrote.

The efficiency gap played a key role in the Wisconsin case, where the map in question, according to expert testimony by the political scientist Simon Jackman, had an efficiency gap of 13 percent in 2012 and 10 percent in 2014. By comparison, the average efficiency gap among state legislatures in 2012 was just over 6 percent, Stephanopoulos and McGhee have calculated.

The two have proposed the efficiency gap as the centerpiece of a simple standard the Supreme Court could adopt for partisan gerrymandering cases. To be considered an unconstitutional gerrymander, they suggest, a district plan must first be shown to exceed some chosen efficiency gap threshold, to be determined by the court. Second, since efficiency gaps tend to fluctuate over the decade that a district map is in force, the plaintiffs must show that the efficiency gap is likely to favor the same party over the entire decade, even if voter preferences shift about somewhat.

If these two requirements are met, Stephanopoulos and McGhee propose, the burden then falls to the state to explain why it created such a biased plan; perhaps, the state could argue, other considerations such as compactness and preservation of boundaries tied its hands. The plaintiffs could then rebut that claim by producing a less biased plan that performed as well as the existing map on measures like compactness.

This approach, the pair wrote, “would neatly slice the Gordian knot the Court has tied for itself,” by explicitly laying down just how much partisan effect is too much.

The efficiency gap can help to identify plans with strong partisan bias, but it cannot say whether that bias was created intentionally. To disentangle the threads of intentional and unintentional gerrymandering, last year Cho — along with her colleagues at Urbana-Champaign, senior research programmer Yan Liu and geographer Shaowen Wang — unveiled a simulation algorithm that generates a large number of maps to compare to any given districting map, to determine whether it is an outlier.

There’s an almost unfathomably large number of possible maps out there, far too many for any algorithm to fully enumerate. But by spreading their algorithm’s tasks across a massive number of processors, Cho’s team found a way to create millions or even billions of what they call “reasonably imperfect” maps — ones that perform at least as well as the original map on whatever nonpartisan measures (such as compactness) a court might be interested in. “As long as a particular facet can be quantified, we can incorporate it into our algorithm,” Cho and Liu wrote in a second paper.

In that paper, Cho and Liu used their algorithm to draw 250 million imperfect but reasonable congressional district maps for Maryland, whose existing plan is being challenged in court. Nearly all their maps, they found, are biased in favor of Democrats. But the official plan is even more biased, favoring Democrats more strongly than 99.79 percent of the algorithm’s maps — a result extremely unlikely to occur in the absence of an intentional gerrymander.

In a similar vein, Chen and Rodden have used simulations (though with many fewer maps) to suggest that Florida’s 2012 congressional plan was almost surely intentionally gerrymandered. Their expert testimony contributed to the Florida Supreme Court’s decision in 2015 to strike down eight of the plan’s 27 districts.

“We didn’t have this level of sophistication in simulation available a decade ago, which was the last major case on this topic before the [U.S. Supreme] Court,” said Bernard Grofman, a political scientist at the University of California, Irvine.

The Florida ruling was based on the state constitution, so its implications for other states are limited. But the Wisconsin case has “potential incredible precedent value,” Grofman said.

Grofman has developed a five-pronged gerrymandering test that distills the key elements of the Wisconsin case. Three prongs are similar to those Stephanopoulos and McGhee have proposed: evidence of partisan bias, indications that the bias would likely endure for the whole decade, and the existence of at least one replacement plan that would remedy the existing plan’s bias. To these, Grofman adds two more requirements: simulations showing that the plan is an extreme outlier, suggesting that the gerrymander was intentional, and evidence that the people who made the map knew they were drawing a much more biased plan than necessary.

If the Supreme Court does adopt a gerrymandering standard, it remains to be seen whether it will require evidence of intent, as Grofman’s standard does, or instead focus on outcomes, as Stephanopoulos and McGhee’s standard does.

“Do we believe that districts should come as close as possible to fair representation of the parties?” Rodden said. “If so, we shouldn’t really care about whether [gerrymandering is] intentional or unintentional.” But, he added, “I don’t know where the courts will end up coming down. I don’t think anyone knows.”

The choice has major ramifications. Last year, Chen and David Cottrell, a quantitative social scientist at Dartmouth University, used simulations to measure the extent of intentional gerrymandering in congressional district maps across most of the 50 states; they uncovered a fair bit, but they also found that on the national level, it mostly canceled out. Banning only intentional gerrymandering, they concluded, would likely have little effect on the partisan balance of the U.S. House of Representatives (although it could have a significant effect on individual state legislatures).

Banning unintentional gerrymandering as well would lead to a more radical redrawing of district maps, one that “could potentially make a very big change to the membership of the House,” McGhee said.

That decision is up to the court. But there’s plenty of work left for gerrymandering researchers, from understanding the limitations of their measures (many of which produce odd results in lopsided elections, for instance) to studying the trade-offs between ensuring partisan symmetry and, say, protecting the voting power of minorities or drawing compact districts. Collaboration between political and social scientists, mathematicians, and computer scientists is the ideal way forward, Rodden and McGhee both say.

“We should be encouraging cross-pollination and bringing in outside ideas, and then debating those ideas robustly,” McGhee said.

*This article was reprinted on Wired.com.*

Known as the Gaussian correlation inequality (GCI), the conjecture originated in the 1950s, was posed in its most elegant form in 1972 and has held mathematicians in its thrall ever since. “I know of people who worked on it for 40 years,” said Donald Richards, a statistician at Pennsylvania State University. “I myself worked on it for 30 years.”

Royen hadn’t given the Gaussian correlation inequality much thought before the “raw idea” for how to prove it came to him over the bathroom sink. Formerly an employee of a pharmaceutical company, he had moved on to a small technical university in Bingen, Germany, in 1985 in order to have more time to improve the statistical formulas that he and other industry statisticians used to make sense of drug-trial data. In July 2014, still at work on his formulas as a 67-year-old retiree, Royen found that the GCI could be extended into a statement about statistical distributions he had long specialized in. On the morning of the 17th, he saw how to calculate a key derivative for this extended GCI that unlocked the proof. “The evening of this day, my first draft of the proof was written,” he said.

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Not knowing LaTeX, the word processer of choice in mathematics, he typed up his calculations in Microsoft Word, and the following month he posted his paper to the academic preprint site arxiv.org. He also sent it to Richards, who had briefly circulated his own failed attempt at a proof of the GCI a year and a half earlier. “I got this article by email from him,” Richards said. “And when I looked at it I knew instantly that it was solved.”

Upon seeing the proof, “I really kicked myself,” Richards said. Over the decades, he and other experts had been attacking the GCI with increasingly sophisticated mathematical methods, certain that bold new ideas in convex geometry, probability theory or analysis would be needed to prove it. Some mathematicians, after years of toiling in vain, had come to suspect the inequality was actually false. In the end, though, Royen’s proof was short and simple, filling just a few pages and using only classic techniques. Richards was shocked that he and everyone else had missed it. “But on the other hand I have to also tell you that when I saw it, it was with relief,” he said. “I remember thinking to myself that I was glad to have seen it before I died.” He laughed. “Really, I was so glad I saw it.”

Richards notified a few colleagues and even helped Royen retype his paper in LaTeX to make it appear more professional. But other experts whom Richards and Royen contacted seemed dismissive of his dramatic claim. False proofs of the GCI had been floated repeatedly over the decades, including two that had appeared on arxiv.org since 2010. Bo’az Klartag of the Weizmann Institute of Science and Tel Aviv University recalls receiving the batch of three purported proofs, including Royen’s, in an email from a colleague in 2015. When he checked one of them and found a mistake, he set the others aside for lack of time. For this reason and others, Royen’s achievement went unrecognized.

Proofs of obscure provenance are sometimes overlooked at first, but usually not for long: A major paper like Royen’s would normally get submitted and published somewhere like the *Annals of Statistics*, experts said, and then everybody would hear about it. But Royen, not having a career to advance, chose to skip the slow and often demanding peer-review process typical of top journals. He opted instead for quick publication in the *Far East Journal of Theoretical Statistics*, a periodical based in Allahabad, India, that was largely unknown to experts and which, on its website, rather suspiciously listed Royen as an editor. (He had agreed to join the editorial board the year before.)

With this red flag emblazoned on it, the proof continued to be ignored. Finally, in December 2015, the Polish mathematician Rafał Latała and his student Dariusz Matlak put out a paper advertising Royen’s proof, reorganizing it in a way some people found easier to follow. Word is now getting around. Tilmann Gneiting, a statistician at the Heidelberg Institute for Theoretical Studies, just 65 miles from Bingen, said he was shocked to learn in July 2016, two years after the fact, that the GCI had been proved. The statistician Alan Izenman, of Temple University in Philadelphia, still hadn’t heard about the proof when asked for comment last month.

No one is quite sure how, in the 21st century, news of Royen’s proof managed to travel so slowly. “It was clearly a lack of communication in an age where it’s very easy to communicate,” Klartag said.

“But anyway, at least we found it,” he added — and “it’s beautiful.”

In its most famous form, formulated in 1972, the GCI links probability and geometry: It places a lower bound on a player’s odds in a game of darts, including hypothetical dart games in higher dimensions.

Imagine two convex polygons, such as a rectangle and a circle, centered on a point that serves as the target. Darts thrown at the target will land in a bell curve or “Gaussian distribution” of positions around the center point. The Gaussian correlation inequality says that the probability that a dart will land inside both the rectangle and the circle is always as high as or higher than the individual probability of its landing inside the rectangle multiplied by the individual probability of its landing in the circle. In plainer terms, because the two shapes overlap, striking one increases your chances of also striking the other. The same inequality was thought to hold for any two convex symmetrical shapes with any number of dimensions centered on a point.

Special cases of the GCI have been proved — in 1977, for instance, Loren Pitt of the University of Virginia established it as true for two-dimensional convex shapes — but the general case eluded all mathematicians who tried to prove it. Pitt had been trying since 1973, when he first heard about the inequality over lunch with colleagues at a meeting in Albuquerque, New Mexico. “Being an arrogant young mathematician … I was shocked that grown men who were putting themselves off as respectable math and science people didn’t know the answer to this,” he said. He locked himself in his motel room and was sure he would prove or disprove the conjecture before coming out. “Fifty years or so later I still didn’t know the answer,” he said.

Despite hundreds of pages of calculations leading nowhere, Pitt and other mathematicians felt certain — and took his 2-D proof as evidence — that the convex geometry framing of the GCI would lead to the general proof. “I had developed a conceptual way of thinking about this that perhaps I was overly wedded to,” Pitt said. “And what Royen did was kind of diametrically opposed to what I had in mind.”

Royen’s proof harkened back to his roots in the pharmaceutical industry, and to the obscure origin of the Gaussian correlation inequality itself. Before it was a statement about convex symmetrical shapes, the GCI was conjectured in 1959 by the American statistician Olive Dunn as a formula for calculating “simultaneous confidence intervals,” or ranges that multiple variables are all estimated to fall in.

Suppose you want to estimate the weight and height ranges that 95 percent of a given population fall in, based on a sample of measurements. If you plot people’s weights and heights on an *x*–*y* plot, the weights will form a Gaussian bell-curve distribution along the *x*-axis, and heights will form a bell curve along the *y*-axis. Together, the weights and heights follow a two-dimensional bell curve. You can then ask, what are the weight and height ranges — call them –*w* < *x *< *w* and –*h* < *y* <* h *— such that 95 percent of the population will fall inside the rectangle formed by these ranges?

If weight and height were independent, you could just calculate the individual odds of a given weight falling inside –*w* < *x *< *w* and a given height falling inside –*h* < *y* <* h*, then multiply them to get the odds that both conditions are satisfied. But weight and height are correlated. As with darts and overlapping shapes, if someone’s weight lands in the normal range, that person is more likely to have a normal height. Dunn, generalizing an inequality posed three years earlier, conjectured the following: The probability that both Gaussian random variables will simultaneously fall inside the rectangular region is always greater than or equal to the product of the individual probabilities of each variable falling in its own specified range. (This can be generalized to any number of variables.) If the variables are independent, then the joint probability equals the product of the individual probabilities. But any correlation between the variables causes the joint probability to increase.

Royen found that he could generalize the GCI to apply not just to Gaussian distributions of random variables but to more general statistical spreads related to the squares of Gaussian distributions, called gamma distributions, which are used in certain statistical tests. “In mathematics, it occurs frequently that a seemingly difficult special problem can be solved by answering a more general question,” he said.

Royen represented the amount of correlation between variables in his generalized GCI by a factor we might call *C*, and he defined a new function whose value depends on *C*. When* C* = 0 (corresponding to independent variables like weight and eye color), the function equals the product of the separate probabilities. When you crank up the correlation to the maximum, *C* = 1, the function equals the joint probability. To prove that the latter is bigger than the former and the GCI is true, Royen needed to show that his function always increases as *C* increases. And it does so if its derivative, or rate of change, with respect to *C* is always positive.

His familiarity with gamma distributions sparked his bathroom-sink epiphany. He knew he could apply a classic trick to transform his function into a simpler function. Suddenly, he recognized that the derivative of this transformed function was equivalent to the transform of the derivative of the original function. He could easily show that the latter derivative was always positive, proving the GCI. “He had formulas that enabled him to pull off his magic,” Pitt said. “And I didn’t have the formulas.”

Any graduate student in statistics could follow the arguments, experts say. Royen said he hopes the “surprisingly simple proof … might encourage young students to use their own creativity to find new mathematical theorems,” since “a very high theoretical level is not always required.”

Some researchers, however, still want a geometric proof of the GCI, which would help explain strange new facts in convex geometry that are only de facto implied by Royen’s analytic proof. In particular, Pitt said, the GCI defines an interesting relationship between vectors on the surfaces of overlapping convex shapes, which could blossom into a new subdomain of convex geometry. “At least now we know it’s true,” he said of the vector relationship. But “if someone could see their way through this geometry we’d understand a class of problems in a way that we just don’t today.”

Beyond the GCI’s geometric implications, Richards said a variation on the inequality could help statisticians better predict the ranges in which variables like stock prices fluctuate over time. In probability theory, the GCI proof now permits exact calculations of rates that arise in “small-ball” probabilities, which are related to the random paths of particles moving in a fluid. Richards says he has conjectured a few inequalities that extend the GCI, and which he might now try to prove using Royen’s approach.

Royen’s main interest is in improving the practical computation of the formulas used in many statistical tests — for instance, for determining whether a drug causes fatigue based on measurements of several variables, such as patients’ reaction time and body sway. He said that his extended GCI does indeed sharpen these tools of his old trade, and that some of his other recent work related to the GCI has offered further improvements. As for the proof’s muted reception, Royen wasn’t particularly disappointed or surprised. “I am used to being frequently ignored by scientists from [top-tier] German universities,” he wrote in an email. “I am not so talented for ‘networking’ and many contacts. I do not need these things for the quality of my life.”

The “feeling of deep joy and gratitude” that comes from finding an important proof has been reward enough. “It is like a kind of grace,” he said. “We can work for a long time on a problem and suddenly an angel — [which] stands here poetically for the mysteries of our neurons — brings a good idea.”

*This article was reprinted on Wired.com.*

Modern astrophysicists have their own story. The coda, at least, is relatively clear: About four billion years ago, during a period called the “late veneer,” meteorites flecked with small amounts of precious metals — gold included — hammered the nascent Earth. But the more fundamental question of where gold was forged in the cosmos is still contentious.

For decades, the prevailing account has been that supernova explosions make gold, along with dozens of other heavy elements on the bottom few rows of the periodic table. But as computer models of supernovas have improved, they suggest that most of these explosions do just about as well at making gold as history’s alchemists. Perhaps a new kind of event — one that has traditionally been difficult, if not impossible, to study — is responsible.

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In the past few years, a debate has erupted. Many astronomers now believe that the space-quaking merger of two neutron stars can forge the universe’s supply of heavy elements. Others hold that even if garden-variety supernovas can’t do the trick, more exotic examples might still be able to. To settle the argument, astrophysicists are searching for clues everywhere, from alchemical computer simulations to gamma-ray telescopes to the manganese crust of the deep ocean. And the race is on to make an observation that would seal the deal — catching one of the cosmos’s rarest mints with its assembly line still running.

In 1957, the physicists Margaret and Geoffrey Burbidge, William Fowler and Fred Hoyle laid out a set of recipes for how the lives and deaths of stars could fill in almost every slot in the periodic table. That implied that humans, or at least the elements making up our bodies, were once stardust. So was gold — somehow.

“The problem itself is rather old, and now for a long time has been the last stardust secret,” said Anna Frebel, an astronomer at the Massachusetts Institute of Technology.

The Big Bang left behind hydrogen, helium and lithium. Stars then fused these elements into progressively heavier elements. But the process stops at iron, which is among the most stable elements. Nuclei bigger than iron are so positively charged, and so difficult to bring together, that fusion no longer returns more energy than you have to put in.

To make heavy elements more reliably, you can bombard iron nuclei with charge-free neutrons. The new neutrons often make the nucleus unstable. In this case, a neutron will decay into a proton (popping out both an electron and an antineutrino). The net increase of a proton leads to a new, heavier element.

When additional neutrons are thrown into a nucleus more slowly than it can decay, the process is called slow neutron capture, or the *s* process. This makes elements such as strontium, barium and lead. But when neutrons land on a nucleus faster than they decay, rapid neutron capture — the *r *process — occurs, beefing up nuclei to form heavy elements including uranium and gold.

In order to coax out the *r-*process elements, the Burbidges and their colleagues recognized, you would need a few things. First, you have to have a relatively pure, unadulterated source of neutrons. You also need heavy “seed” nuclei (such as iron) to capture those neutrons. You need to bring them together in a hot, dense (but not too dense) environment. And you want all this to happen during an explosive event that will scatter the products out into space.

To many astronomers, those requirements implicate one specific kind of object: a supernova.

A supernova erupts when a massive star, having fused its core into progressively heavier elements, reaches iron. Then fusion stops paying off, and the star’s atmosphere crashes down. A sun’s worth of mass collapses into a sphere only about a dozen kilometers in radius. Then, when the core reaches the density of nuclear matter, it holds firm. Energy rebounds outward, ripping apart the star in a supernova explosion visible from billions of light years away.

A supernova seems to tick the necessary boxes. During the star’s collapse, protons and electrons in the core are forced together, making neutrons and converting the core into an infant neutron star. Iron is abundant. So is heat. And the glowing ejecta keep expanding out into space for millennia, dispersing the products.

By the 1990s, a specific picture had begun to emerge in computational models. Half a second after the core of a massive star collapses, a gale of neutrinos streams out, continuing for up to a minute. Some of that wind would blow off iron nuclei that could serve as seeds, along with lots and lots of neutrons.

“That was the hope,” said Thomas Janka of the Max Planck Institute for Astrophysics in Garching, Germany. “This was, I would say, the most interesting and the most promising site for forming the *r-*process elements for almost 20 years.” And the explanation still has its adherents. “If you open a textbook, it will tell you that the *r *process is made by supernova explosions,” said Enrico Ramirez-Ruiz, an astrophysicist at the University of California, Santa Cruz.

But as supernova models got more and more sophisticated, the situation got worse, not better. Temperatures in the neutrino-driven wind didn’t seem to be high enough. The wind might also be too slow, allowing seed nuclei to form so abundantly that they wouldn’t find enough neutrons to build up heavy elements all the way up to uranium. And the neutrinos could also convert neutrons back into protons — meaning there might not even be a lot of neutrons to work with.

That left theorists circling back to one of the strongest points of the supernova model. Supernovas make neutron stars, which seem indispensable to the process.

“They are fantastic for this type of nucleosynthesis,” said Stephan Rosswog at Stockholm University. “You start with this gigantic amount of neutrons that you don’t have anywhere else in the universe.” But a neutron star also has a strong gravitational field, he said. “The question is just, well, how can you convince the neutron star to eject something?”

One way to crack open a neutron star would be to use the same explosion that birthed it. That didn’t seem to work. But what if you came back later, and tore one open again?

In 1974, radio astronomers found the first binary neutron star system. With each orbit, the pair were losing energy, implying that one day they would collide. The same year, the astrophysicists James Lattimer and David Schramm modeled what would happen in such a situation — not specifically the clash of two neutron stars, since that was too complicated to calculate at the time, but the similar merger of a neutron star and a black hole.

While supernova explosions can briefly outshine the entire galaxies that host them, neutron stars are extremely difficult to see. The supernova that produced the Crab nebula was observed by many different cultures in the year 1054; the neutron star it left behind wasn’t detected until 1968. A merger of two neutron stars would be still more difficult to find and understand. But although nobody had ever seen one, this kind of exotic event could be responsible for the *r-*process elements, Lattimer and Schramm said.

Picture two neutron stars approaching their final embrace. In the last few orbits around each other before glomming together into a bigger neutron star or a black hole, the pair are wracked by enormous gravitational tides. The collision ejects enormous amounts of material.

“Kind of like you squeeze a tube of toothpaste, stuff comes flying out the end,” said Brian Metzger, a theoretical astrophysicist at Columbia University. Behind each neutron star stretches a tail, with perhaps 10 neutrons to every proton, all heated to billions of degrees. Heavy nuclei form in about a second. Because they have so many extra neutrons they are unstable, radioactive. They glow, eventually decaying to things like gold and platinum.

At least, this is how it works in simulations.

Neutron star mergers and supernovas are both capable of making making *r-*process elements. But there’s a big difference in just how much each of those options can make. Supernovas produce perhaps our moon’s worth of gold. Neutron star mergers, by contrast, make about a Jupiter-size mass of gold — thousands of times more than in a supernova — but they happen far less frequently. This allows astronomers to search for the distribution of *r*-process elements as a way to track their origins.

“Think of *r-*process [elements] as chocolate,” Ramirez-Ruiz said. A universe enriched in the *r-*process elements predominantly by supernovas would be like a cookie with a thin, evenly spread glaze of chocolate. By contrast, “neutron star mergers are like chocolate chip cookies,” he said. “All of the chocolate, or the *r* process, is concentrated.”

One way to assess the distribution and rate of *r-*process events is to look for their byproducts on Earth. Long after supernovas light up the Milky Way, the nuclei they make can coalesce onto interstellar dust grains, slip past the solar and terrestrial magnetic fields, and fall to Earth, where they should be preserved in the deep ocean. A 2016 paper in *Nature* that looked at radioactive iron-60 in the deep-sea crust found traces of multiple nearby supernovas in the past 10 million years. Yet those supernovas did not appear to correspond with *r-*process elements. When the same team looked in deep-sea crust samples for plutonium 244, an unstable *r-*process product that decays over time, they found very little. “Whatever site is creating these heaviest elements is not a very frequent one in our galaxy,” Metzger said.

Not everyone agrees with that conclusion. Another team, led by Shawn Bishop at the Munich Technical University, still hopes to find radioactive plutonium on Earth from recent supernovas. In work now underway, his team is searching for hints of *r-*process elements in sediments that contain microfossils: the tiny remnants of bacteria that take in metals from their environment to make magnetic crystals.

Astronomers can also look for evidence of a chocolate chip-cookie universe farther afield. The *r-*process element europium has one strong spectral line, allowing astronomers to look for it in the atmospheres of stars. Among the old stars that are found in the halo of the Milky Way, observed *r-*process signatures have been hit or miss. “We can find two stars that have very similar, say, iron content,” Ramirez-Ruiz said. “But their europium content, which is the signature for the *r* process, can change by two orders of magnitude.” Because of this, the universe is looking more chocolate chip than chocolate glaze, argues Ramirez-Ruiz.

Astronomers have found an even cleaner example. Many dwarf galaxies experience just one brief burst of activity before settling down. That gives them a narrow window for an *r-*process event to occur — or not. And up until 2016, not one star in any dwarf galaxy seemed to be enriched in *r-*process elements.

That’s what made the phone call MIT’s Frebel received one night so surprising. Her graduate student Alex Ji had been observing stars in a dwarf galaxy called Reticulum II. “He called me at two in the morning and said ‘Anna, I think there’s a problem with the spectrograph.’” One star in particular appeared to have a strong europium line. “I made this joke. I said, ‘Well, Alex, maybe you found an *r-*process galaxy,’” Frebel said. He actually had, though. Reticulum II has seven stars enriched in the *r-*process elements, all implicating a single, otherwise uncommon event.

To advocates of the neutron star merger model, all of this fits nicely. Neutron star mergers are naturally rare. Unlike a single massive star collapsing and going supernova, they require two neutron stars to form, to be in a binary orbit, and to merge perhaps a hundred million years later. But critics also point out that they might be too rare.

In our galaxy, neutron star mergers could happen as rarely as once every hundred million years, or as often as once every 10 thousand years — rates that differ by a factor of 10,000. “The thing that shook me is: The people who were saying neutron star mergers are going to explain the *r* process were also taking this highest rate,” said Christopher Fryer, an astrophysicist at Los Alamos National Laboratory.

When Fryer and colleagues used more moderate guesses about how often neutron star mergers occur and how much *r-*process material they yield, they found that neutron star mergers can explain only 1 percent of the *r-*process elements observed in the universe. And if the true rate lies at the lowest end, they could contribute a hundred times less again. “More people are going back to ‘Huh, what other sources of *r *process can we have?’” Fryer said.

That’s where supernovas may see their stock rise again. If perhaps 1 percent of core collapse supernovas behave differently than the standard simulations predict, they might also be able to make considerable amounts of *r-*process elements in a chocolate chip pattern. One way to salvage a supernova explosion is if a star detonates with massive, magnetically powered jets instead of neutrinos, argues Nobuya Nishimura, an astrophysicist at Keele University in England, and his colleagues in a recent paper. That would create a rapid explosion of neutron-rich matter, allowing seed nuclei to grow into at least some of the *r-*process elements. “It’s not like you can have a tea party there,” Fryer said. “You just need to stay [in that region] for 100 milliseconds.”

The answer, many astronomers believe, will end up being some kind of compromise. That shift may already be happening. “*R *process is really not *r *process anymore now,” Frebel said. Maybe it can be broken in half, with the “weak” *r-*process elements lighter than barium coming from supernovas, and the heavier ones like gold coming from neutron star collisions.

And there’s one more dark horse still lurking out there: the merger of a neutron star and a black hole, which Lattimer and Schramm had originally considered. The neutron star in the pair would still eject material, just as before. But the rate of those events is even fuzzier. “Maybe even they are the dominant ones producing *r-*process elements,” Janka said. “We don’t know. We need better data.”

That data may already be on its way. The last few orbits of a neutron star merger or a merger between a neutron star and a black hole warp and drag space-time so much that gravitational waves roar out of the system. LIGO (the Laser Interferometer Gravitational-Wave Observatory), which has already succeeded at “hearing” such a crescendo between merging black holes, is approaching a sensitivity that should let it start picking up neutron star mergers in distant galaxies. The longer it doesn’t, the less often it seems these events occur. Once LIGO reaches its full design sensitivity, a nondetection could spell doom for neutron star merger models. “If they still have not found something, there will be a moment in which Enrico [Ramirez-Ruiz] and Brian [Metzger], etcetera, should wonder, and get back to the board,” said Selma de Mink, an astrophysicist at the University of Amsterdam.

The dream, though, is to go beyond making inferences about *r-*process events and see one actually in action. Two teams may have already done so. In 2013, the Swift satellite picked up a short gamma-ray burst: a type of event also attributed to colliding neutron stars. Other telescopes zoomed in on the aftermath.

In simulations, an observational signature called a kilonova follows neutron-star mergers. The radioactive nuclei made through the *r *process spread and glow, causing the system to ramp up in brightness for about a week before starting to fade. And these elements are so opaque that only red light can penetrate out. The 2013 event matched both predictions, but it was so far away that it was hard to fully interpret. “It’s not compelling but it’s suggestive,” Metzger said.

Many of the astronomers who made that discovery are now part of teams hoping to find a closer, more definitive kilonova. That entails pouncing on a LIGO signal from merging neutron stars and quickly finding its source in the sky with more traditional telescopes — perhaps even measuring its light spectrum using something like the upcoming James Webb Space Telescope. In doing so, it may be possible to see a cloud of newborn *r-*process elements — or to infer something from their absence. “The world of gamma-ray bursts has trained us very well,” said Wen-fai Fong of the University of Arizona. “It is definitely like a race. How quickly can you react?”

*This article was reprinted on TheAtlantic.com.*

A new study, out today, suggests that the shift to lungs and limbs doesn’t tell the full story of these creatures’ transformation. As they emerged from the sea, they gained something perhaps more precious than oxygenated air: information. In air, eyes can see much farther than they can under water. The increased visual range provided an “informational zip line” that alerted the ancient animals to bountiful food sources near the shore, according to Malcolm MacIver, a neuroscientist and engineer at Northwestern University.

This zip line, MacIver maintains, drove the selection of rudimentary limbs, which allowed animals to make their first brief forays onto land. Furthermore, it may have had significant implications for the emergence of more advanced cognition and complex planning. “It’s hard to look past limbs and think that maybe information, which doesn’t fossilize well, is really what brought us onto land,” MacIver said.

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MacIver and Lars Schmitz, a paleontologist at the Claremont Colleges, have created mathematical models that explore how the increase in information available to air-dwelling creatures would have manifested itself, over the eons, in an increase in eye size. They describe the experimental evidence they have amassed to support what they call the “buena vista” hypothesis in the *Proceedings of the National Academy of Sciences*.

MacIver’s work is already earning praise from experts in the field for its innovative and thorough approach. While paleontologists have long speculated about eye size in fossils and what that can tell us about an animal’s vision, “this takes it a step further,” said John Hutchinson of the Royal Veterinary College in the U.K. “It isn’t just telling stories based on qualitative observations; it’s testing assumptions and tracking big changes quantitatively over macro-evolutionary time.”

MacIver first came up with his hypothesis in 2007 while studying the black ghost knifefish of South America — an electric fish that hunts at night by generating electrical currents in the water to sense its environment. MacIver compares the effect to a kind of radar system. Being something of a polymath, with interests and experience in robotics and mathematics in addition to biology, neuroscience and paleontology, MacIver built a robotic version of the knifefish, complete with an electrosensory system, to study its exotic sensing abilities and its unusually agile movement.

When MacIver compared the volume of space in which the knifefish can potentially detect water fleas, one of its favorite prey, with that of a fish that relies on vision to hunt the same prey, he found they were roughly the same. This was surprising. Because the knifefish must generate electricity to perceive the world — something that requires a lot of energy — he expected it would have a smaller sensory volume for prey compared to that of a vision-centric fish. At first he thought he had made a simple calculation error. But he soon discovered that the critical factor accounting for the unexpectedly small visual sensory space was the amount that water absorbs and scatters light. In fresh shallow water, for example, the “attenuation length” that light can travel before it is scattered or absorbed ranges from 10 centimeters to two meters. In air, light can travel between 25 to 100 kilometers, depending on how much moisture is in the air.

Because of this, aquatic creatures rarely gain much evolutionary benefit from an increase in eye size, and they have much to lose. Eyes are costly in evolutionary terms because they require so much energy to maintain; photoreceptor cells and neurons in the visual areas of the brain need a lot of oxygen to function. Therefore, any increase in eye size had better yield significant benefits to justify that extra energy. MacIver likens increasing eye size in the water to switching on high beams in the fog in an attempt to see farther ahead.

But once you take eyes out of the water and into air, a larger eye size leads to a proportionate increase in how far you can see.

MacIver concluded that eye size would have increased significantly during the water-to-land transition. When he mentioned his insight to the evolutionary biologist Neil Shubin — a member of the team that discovered *Tiktaalik roseae*, an important transitional fossil from 375 million years ago that had lungs and gills — MacIver was encouraged to learn that paleontologists had noticed an increase in eye size in the fossil record. They just hadn’t ascribed much significance to the change. MacIver decided to investigate for himself.

MacIver had an intriguing hypothesis, but he needed evidence. He teamed up with Schmitz, who had expertise in interpreting the eye sockets of four-legged “tetrapod” fossils (of which *Tiktaalik *was one), and the two scientists pondered how best to test MacIver’s idea.

MacIver and Schmitz first made a careful review of the fossil record to track changes in the size of eye sockets, which would indicate corresponding changes in eyes, since they are proportional to socket size. The pair collected 59 early tetrapod skulls spanning the water-to-land transition period that were sufficiently intact to allow them to measure both the eye orbit and the length of the skull. Then they fed those data into a computer model to simulate how eye socket size changed over many generations, so as to gain a sense of the evolutionary genetic drift of that trait.

They found that there was indeed a marked increase in eye size — a tripling, in fact — during the transitional period. The average eye socket size before transition was 13 millimeters, compared to 36 millimeters after. Furthermore, in those creatures that went from water to land and back to the water — like the Mexican cave fish *Astyanax mexicanus* — the mean orbit size shrank back to 14 millimeters, nearly the same as it had been before.

There was just one problem with these results. Originally, MacIver had assumed that the increase occurred after animals became fully terrestrial, since the evolutionary benefits of being able to see farther on land would have led to the increase in eye socket size. But the shift occurred before the water-to-land transition was complete, even before creatures developed rudimentary digits on their fishlike appendages. So how could being on land have driven the gradual increase in eye socket size.

In that case, “it looks like hunting like a crocodile was the gateway drug to terrestriality,” MacIver said. “Just as data comes before action, coming up on land was likely about how the huge gain in visual performance from poking eyes above the water to see an unexploited source of prey gradually selected for limbs.”

This insight is consistent with the work of Jennifer Clack, a paleontologist at the University of Cambridge, on a fossil known as *Pederpes finneyae*, which had the oldest known foot for walking on land, yet was not a truly terrestrial creature. While early tetrapods were primarily aquatic, and later tetrapods were clearly terrestrial, paleontologists believe this creature likely spent time in water and on land.

After determining how much eye sizes increased, MacIver set out to calculate how much farther the animals could see with bigger eyes. He adapted an existing ecological model that takes into account not just the anatomy of the eye, but other factors such as the surrounding environment. In water, a larger eye only increases the visual range from just over six meters to nearly seven meters. But increase the eye size in air, and the improvement in range goes from 200 meters to 600 meters.

MacIver and Schmitz ran the same simulation under many different conditions: daylight, a moonless night, starlight, clear water and murky water. “It doesn’t matter,” MacIver said. “In all cases, the increment [in air] is huge. Even if they were hunting in broad daylight in the water and only came out on moonless nights, it’s still advantageous for them, vision-wise.”

Using quantitative tools to help explain patterns in the fossil record is something of a novel approach to the problem, but a growing number of paleontologists and evolutionary biologists, like Schmitz, are embracing these methods.

“So much of paleontology is looking at fossils and then making up narratives on how the fossils might have fit into a particular environment,” said John Long, a paleobiologist at Flinders University in Australia who studies how fish evolved into tetrapods. “This paper has very good hard experimental data, testing vision in different environments. And that data does fit the patterns that we see in these fish.”

Schmitz identified two key developments in the quantitative approach over the past decade. First, more scientists have been adapting methods from modern comparative biology to fossil record analysis, studying how animals are related to each other. Second, there is a lot of interest in modeling the biomechanics of ancient creatures in a way that is actually testable — to determine how fast dinosaurs could run, for instance. Such a model-based approach to interpreting fossils can be applied not only to biomechanics but to sensory function — in this case, it explained how coming out of the water affected the vision of the early tetrapods.

“Both approaches bring something unique, so they should go hand in hand,” Schmitz said. “If I had done the [eye socket size] analysis just by itself, I would be lacking what it could actually mean. Eyes do get bigger, but why?” Sensory modeling can answer this kind of question in a quantitative, rather than qualitative, way.

Schmitz plans to examine other water-to-land transitions in the fossil record — not just that of the early tetrapods — to see if he can find a corresponding increase in eye size. “If you look at other transitions between water and land, and land back to water, you see similar patterns that would potentially corroborate this hypothesis,” he said. For example, the fossil record for marine reptiles, which rely heavily on vision, should also show evidence for an increase in eye socket size as they moved from water to land.

MacIver’s background as a neuroscientist inevitably led him to ponder how all this might have influenced the behavior and cognition of tetrapods during the water-to-land transition. For instance, if you live and hunt in the water, your limited vision range — roughly one body length ahead — means you operate primarily in what MacIver terms the “reactive mode”: You have just a few milliseconds (equivalent to a few cycle times of a neuron in the brain) to react. “Everything is coming at you in a just-in-time fashion,” he said. “You can either eat or be eaten, and you’d better make that decision quickly.”

But for a land-based animal, being able to see farther means you have much more time to assess the situation and strategize to choose the best course of action, whether you are predator or prey. According to MacIver, it’s likely the first land animals started out hunting for land-based prey reactively, but over time, those that could move beyond reactive mode and think strategically would have had a greater evolutionary advantage. “Now you need to contemplate multiple futures and quickly decide between them,” MacIver said. “That’s mental time travel, or prospective cognition, and it’s a really important feature of our own cognitive abilities.”

That said, other senses also likely played a role in the development of more advanced cognition. “It’s extremely interesting, but I don’t think the ability to plan suddenly arose only with vision,” said Barbara Finlay, an evolutionary neuroscientist at Cornell University. As an example, she pointed to how salmon rely on olfactory pathways to migrate upstream.

Hutchinson agrees that it would be useful to consider how the many sensory changes over that critical transition period fit together, rather than studying vision alone. For instance, “we know smell and taste were originally coupled in the aquatic environment and then became separated,” he said. “Whereas hearing changed a lot from the aquatic to the terrestrial environment with the evolution of a proper external ear and other features.”

The work has implications for the future evolution of human cognition. Perhaps one day we will be able to take the next evolutionary leap by overcoming what MacIver jokingly calls the “paleoneurobiology of human stupidity.” Human beings can grasp the ramifications of short-term threats, but long-term planning — such as mitigating the effects of climate change — is more difficult for us to process. “Maybe some of our limitations in strategic thinking come back to the way in which different environments favor the ability to plan,” he said. “We can’t think on geologic time scales.” He hopes this kind of work with the fossil record can help identify our own cognitive blind spots. “If we can do that, we can think about ways of getting around those blind spots.”

]]>His method had been hiding in plain sight. Fermat’s Last Theorem, which states that there are no positive integer solutions to equations of the form *a ^{n }*+

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Adding (or “adjoining”) new values to the old numbers is not hard to do — there’s a straightforward mathematical recipe for how to incorporate the square root of 5 as a normal number between 2 and 3, for example, after which you can carry on with the business of arithmetic as usual. All you do is write each value in the new number system as *a *+ *b*√5, where *a* and *b *are integers. This might seem like an ungainly way to write a number, but it serves as a coherent basis for creating a new number system that functions just like the old one. Mathematicians call this new system a number “ring”; they can create an infinite variety of them, depending on the new values they choose to incorporate.

Of course, it’s hard to tinker with something as intricate as a number system without producing unintended consequences. When Lamé started adjoining these funny numbers, everything looked great at first. But then other mathematicians pointed out that this newly gained flexibility came at a high cost: The new number system lacked unique prime factorization, by which a number — for example, 12 — can be expressed uniquely as a product of primes: 2 x 2 x 3. This violated a bedrock principle of conventional arithmetic.

Unique prime factorization ensures that each number in a number system can be built up from prime numbers in exactly one way. In a number ring that includes a √-5 (in practice, mathematicians often employ number systems that use the square roots of negative numbers), duplicity creeps in: 6 is both 2 x 3 and also (1 + √-5) x (1 – √-5). All four of those factors are prime in the new number ring, giving 6 a dual existence that just won’t do when you’re trying to nail things down mathematically.

“In algebra classes when we first teach that unique prime factorization sometimes doesn’t hold, students gasp, they say, ‘Oh my, what happened?’ We always take for granted that everything can be uniquely decomposed into primes,” said Manjul Bhargava, a professor at Princeton University and winner of the Fields Medal, math’s highest honor.

Unique prime factorization is a way of constructing a number system from fundamental building blocks. Without it, proofs can turn leaky. Mixing roots with the regular numbers failed as an attack on Fermat’s Last Theorem, but as often happens in math, the way in which it failed was provocative. It launched an area of inquiry unto itself called algebraic number theory.

Today mathematicians are actively engaged in the study of “class numbers” of number systems. In their crudest form, they’re a rating of how badly a number system fails the test of unique prime factorization, depending on which roots get mixed in: A number system that gets a “1” has unique prime factorization; a system that gets a “2” misses unique prime factorization by a little; a system that gets a “7” misses it by a lot more.

On their face, you’d expect class numbers to be randomly distributed — that class number 5 occurs with the same frequency as class number 6, or that half of all class numbers are even. That’s not the case, though, and current research in the subject aims to understand why. Today mathematicians are circling in on the structure that underlies class numbers and inching closer to establishing the truth about long-conjectured values. It’s an effort that has generated insights about how numbers behave that go far beyond a proof of any one problem.

Around the same time Lamé gave his failed proof, the German mathematician Ernst Kummer developed a way to fix the loss of prime factorization with what he called “ideal numbers.” They’re not numbers in any conventional sense. Rather, they’re sprawling constructions in set theory that perform a number-like function.

For example, the simplest ideal is the infinite set of all multiples of a given integer — 5, 10, 15, 20 and so on. Ideals can be added into an already expanded number ring to restore unique factorization. They allow mathematicians to reconcile competing prime factorizations into a single set of prime factors.

Ideals can be categorized into various classes. The number of different classes of ideals you need to add to a number ring in order to restore unique factorization is the ring’s “class number.”

The study of class numbers goes at least as far back as Carl Friedrich Gauss in the early 19th century. In a sign of how hard it’s been to make progress in this area, many of his results are still state of the art. Among his contributions, Gauss conjectured that there are infinitely many positive square roots that can be adjoined to the whole numbers without losing unique factorization — a proof of which remains the most sought-after result in class numbers (and is rumored to have frustrated Kurt Gödel enough to make him give up number theory for logic). Gauss also conjectured that there are only nine negative square roots that preserve prime factorization. √-163 is the very last one.

Today, research on class numbers remains inspired by Gauss, but much of it takes place in a context established in the late 1970s by the mathematicians Henri Cohen, emeritus professor of mathematics at the University of Bordeaux, and Hendrik Lenstra, who recently retired from Leiden University in the Netherlands. Together they formulated the Cohen-Lenstra heuristics, which are a series of predictions about how frequently particular kinds of class numbers should appear. For example, the heuristics predict that 43 percent of class numbers are divisible by 3 in situations where you’re adjoining square roots of negative numbers.

“That’s interesting because it tells you this way in which class numbers are behaving unexpectedly. If you go and look at a list of telephone numbers or something, then generally speaking one in three of them should be divisible by 3,” said Akshay Venkatesh, a mathematician at Stanford University.

Gauss had to compute class numbers by hand. By the time Cohen and Lenstra made their predictions, computers made it possible to rapidly calculate class numbers for billions of different number rings. As a result, there is good experimental evidence to support the Cohen-Lenstra heuristics. However, knowing something with confidence is entirely different from proving it.

“Probably in other sciences this is where you’d be done. However, in math that’s just the beginning. Now we want to know for sure,” said Melanie Wood, a mathematician at the University of Wisconsin-Madison.

The fact that class numbers are not distributed randomly suggests something interesting is going on beneath the surface. A class number, remember, tells us something about a given number ring: the number of classes of ideals required to restore unique factorization. Those ideal classes form the “class group” of that number ring. Groups have all sorts of interesting structural properties that are not evident just from knowing the number of elements they contain, in the same way that knowing the number of people in a family doesn’t tell you much about how those people are related.

In order to understand why class numbers are distributed as they are, mathematicians need to study the structure of the class groups that give rise to the class numbers. In particular, they’re interested in the amount of symmetry in one group versus another, with the understanding that groups that have more symmetry will occur proportionally less often than groups with less symmetry.

To see the relationship between the amount of symmetry something has and the frequency with which it occurs, consider a geometric example. Start with three points arranged to make a triangle. (These points are analogous to elements of a group, but they’re not a group in any real mathematical sense.) Now think of all possible ways of connecting those points with lines, which are a stand-in for mathematical relationships. There are eight possible configurations:

- One with three lines that make a triangle.
- Three with two lines that make an ‘‘open jaw’’ shape.
- Three with one line that connects two points.
- One with no lines.

The triangle has six symmetries and appears once. The open jaw shape has two symmetries and appears three times. Or, put another way, the triangle has three times as much symmetry as the open jaw and appears one-third as often. This relationship — the more symmetry something has, the less often it occurs — holds throughout mathematics. It’s true because the less symmetry something has, the more ways it can appear. Consider that there are an infinite number of two-dimensional shapes with no symmetry, but only one shape that has infinite lines of symmetry — the circle.

“It’s not just a rough [correlation], it’s exact and precise: If one thing has three times as many symmetries, it appears one-third as often,” said Wood.

The same relationship between how symmetric something is and how frequently it occurs holds for the way groups are constructed. In the example above, relationships are defined by lines drawn between points. In a group, relationships are established by the way the elements of the group can be added together.

To be a group, those additive relationships have to satisfy certain axioms. The elements of class groups must obey the associative and commutative properties of addition, and must include a zero element, such that zero plus any other element leaves the element unchanged. The integers are in a sense the original group because they satisfy all these axioms. But certain finite sets (like class groups) also satisfy these axioms, creating, in essence, miniature number systems.

Knowing that a group has, say, four elements doesn’t tell you everything about how those four elements are related to one another. Consider two groups — call them Group 1 and Group 2 — each with four elements*. *What’s different about the two groups is the additive relationships between those elements. The tables below show what happens when you add an element to another element in each group.

In this setting, a “symmetry” of the group occurs wherever it’s possible to rearrange elements of the group in a way that preserves the addition structure of the group. For Group 2, there are two such symmetries: the “identity” symmetry (in which you don’t change the places of any elements), and the symmetry that swaps *x* with *z*. (Because *x *+ *x* = *y* and *z *+ *z *= *y*, *x* and *z* are interchangeable.)

Group 1 has more symmetries. The elements *a*, *b*, and* c* are all interchangeable, since *a *+ *a *= 0, *b *+ *b* = 0, and *c* + *c* = 0. Given that, every way of rearranging these three elements is a symmetry (or “automorphism”) of the group. If you work through all the combinations you see there are six symmetries in all. Putting this together, Group 1 has three times as many symmetries as Group 2. You’d therefore expect to find Group 2 three times as often as you would Group 1, in keeping with the rule that arrangements occur in inverse proportion to their number of symmetries. This law is as true for groups with four simple elements like Group 1 and Group 2 as it is for other, more complicated, groups of ideals.

When mathematicians are confronted with a class number, they want to know the structure of the underlying group it represents. If they can establish the structure of the underlying group, and establish how frequently groups of a given structure arise, they can bring that information back to the surface and use it to understand how often a given class number should occur.

If you start to examine the group structure and its symmetries, then “suddenly it gives you what the distribution of class numbers should be on the nose,” said Bhargava.

The two groups above are (relatively) easy to parse. Groups of ideals are much harder to pin down; it’s not easy to sketch out their addition tables. Instead, mathematicians have ways of probing the groups, testing their structure, even when they can’t see the whole thing completely. In particular, they test how far each element in the group is from zero.

Recall that every group has a zero element that, when added to any other number, leaves that number unchanged. To investigate the structure of class groups, mathematicians try to get a feel for the number of elements in a given class group that have what they call “*n*-torsion,” which means that when you add *n* copies of the element, you wind up at the zero of the group. An element is 2-torsion, for example, if *x *+ *x *= 0, 3-torsion if *x *+* x *+ *x *= 0, 4-torsion if *x *+* x *+ *x *+* x *= 0 and so on.

One way to make clear the difference between the two groups above is to consider how many of their elements are 2-torsion. In Group 1, all four elements are 2-torsion, which is evident by the line of zeroes on the diagonal: 0 + 0 = 0, *a *+* a *= 0, *b* +* b* = 0, *c* + *c* = 0. In Group 2, only 0 and *y* are 2-torsion. The amount of different types of torsion in the group is an exact reflection of the group’s overall structure.

“If the number of *n*-torsion elements in two groups is the same for all *n* then they’re the same group. Investigating how many *n*-torsion elements there are is a simple strategy that probes the group and is enough to recover the group if you understand everything about torsion,” said Bhargava.

A lot of the work on the Cohen-Lenstra heuristics today has to do with establishing how many elements in a class group have different types of torsion. The Cohen-Lenstra predictions with respect to torsion are quite easy to state. For example, if you’re adjoining the square roots of negative numbers, how many ideals in their class group should have 3-torsion? Cohen-Lenstra predict that there should be on average two 3-torsion elements per number ring. How many should have 5-torsion? 7-torsion? 11-torsion? The answer again, for each prime, is two.

This constancy is striking because from a naïve perspective, you’d expect the number of elements with a given torsion to grow as the size of the class group grows. Yet even as the sizes of the class groups vary, the Cohen-Lenstra heuristics predict that the number of elements with, say, 3-torsion, will on average remain constant.

“It’s interesting that this prediction is independent of the prime number,” said Bhargava. “It’s an amazing prediction.”

It’s an amazing prediction that’s been borne out statistically in countless computer runs, yet remains hard to prove.

The Cohen-Lenstra heuristics, further extended by Cohen and Jacques Martinet in 1987, have been around for more than 40 years. Yet you could summarize progress on them on a Post-it. Only two cases have ever been proved: one in 1971, by Harold Davenport and Hans Heilbronn, and another in 2005 by Bhargava. Otherwise, “almost nothing has been proven,” said Bhargava.

With proofs of the heuristics being hard to come by, mathematicians have adopted more modest goals. They’d like to prove that the average number of *n*-torsion elements for a given prime is as expected, but short of that, they’ll settle for at least putting a ceiling on the number. This is called establishing an upper bound, and mathematicians have been making gradual progress in this regard.

When you’re adjoining the square root of a negative number to your number system, the class number grows in proportion to the size of the square root. If you’re adjoining the square root of –13, you can expect the class group to be, at most, about square root of 13 elements in size. Another way of writing the square root for any number *n* is *n*^{0.5}, and that number — the 0.5 in the exponent — is the place mathematicians start when trying to fix an upper bound. If the whole class group contains *n*^{0.5} elements, then you know from the start that there can’t be more than *n*^{0.5} elements with say, 3-torsion, because that would be every element. For that reason, *n*^{0.5} is considered the trivial bound on *n*-torsion in the class group.

Mathematicians typically use one of several general approaches to lowering these bounds. One is an approach called a “sieve,” which you can analogize as “panning” for *n*-torsion elements the way a prospector pans for gold. The two other methods involve complicated transformations through which elements with *n*-torsion can be counted as lattice points in a region or on a curve.

One of the first to break the trivial bound was Lillian Pierce, a mathematician at Duke University, when, in 2006, she proved that the number of 3-torsion elements in a particular number ring is at most *n*^{0.49}. It was a small improvement over the trivial bound, but it started a trail that other mathematicians followed. Independently and around the same time, Venkatesh and Harald Helfgott of the University of Göttingen lowered the bound to *n*^{0.44}, and the next year Venkatesh and Jordan Ellenberg of the University of Wisconsin-Madison brought the bound down even further, to *n*^{0.33}. These are not expected to be the optimal bounds, but they do move the field forward. “From my point of view it’s much more important to prove anything at all in the first place,” said Venkatesh.

The most recent result in this area comes from Bhargava and five coauthors, Arul Shankar, Takashi Taniguchi, Frank Thorne, Jacob Tsimerman, and Yongqiang Zhao. In January, they posted a paper to the scientific preprint site arxiv.org that lowered the bound for 2-torsion in cubic and quartic number rings to *n*^{0.28}. In that same paper they also proved that they can break the trivial bound for 2-torsion for number rings in any degree.

“It is just a small savings, but it’s chipped away at the trivial bound for the first time in infinitely many cases,” said Pierce.

Even that small savings has already paid mathematical dividends. The methods Bhargava and his collaborators used have proved useful for bounding the number of solutions to a specific class of polynomial equation called elliptic curves, which is consistent with the way that class numbers seem to be situated at the intersection of many different mathematical fields. And, while there’s a long way to go before this happens, progress on class numbers could end up redeeming the original purpose of the number rings they describe.

“A proof of Fermat’s Last Theorem has never been obtained just by studying these class numbers,” said Bhargava. “If we fully understood how class groups behave in general, it seems conceivable a proof of that kind could work for FLT and for many other equations. It’s hard to say because we still have a long way to go.”

*Correction: On March 6 this article was updated to clarify that the integers, not the whole numbers, are in a sense the original group.*

*This article was reprinted on ScientificAmerican.com.*

But for all these meticulous observations, the molecular machinery behind our nightly departure from consciousness remains one of the deepest mysteries in modern biology. “On a mechanistic level,” said Amita Sehgal, a chronobiologist at the University of Pennsylvania who has pioneered the study of sleep using fruit flies, “we don’t know what’s happening to make us sleepy.” Biologists have mutated thousands of lab organisms like flies, picked out those with sleep abnormalities, and studied their genes, as well as those of people with sleep disorders. The idea is that if we can identify which genes are disrupted — what is making people fall asleep at their desks, or what makes flies too snoozy to mate — we’ll be closer to understanding why we get sleepy and what, on the molecular level, sleep consists of.

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Now, a team of researchers in Japan and Texas have unveiled the first results of a colossal experiment that takes this approach with mice, on a scale never seen before: Over the course of several years, the team mutated the genes of thousands of mice, hooked them up to brain wave monitors and watched them sleep. In a recent paper in *Nature*, the researchers revealed two genes that seem to be involved in sleep’s biological machinery. The first was found in mice that need much more sleep than normal. The other comes from mice that spend less time in dreaming, or REM, sleep. The results suggest new leads for sleep researchers to chase, and they contribute to a rising tide of research dissecting the molecular underpinnings of sleep.

The idea for the experiment came in 2007 or so, said Masashi Yanagisawa, a biologist who at the time was at the University of Texas Southwestern Medical Center. He had stumbled into the sleep field about a decade before, when he and his colleagues, on a hunt for interesting neurotransmitters, identified a protein that turned out to be involved with narcolepsy. Sufferers of the condition can go from chatting and laughing to REM sleeping in seconds. Studies with mice lacking the protein and a colony of narcoleptic dogs at Stanford, as well as human research, soon established the protein’s role. Called orexin, it helps keep people awake — but that was far from the whole story, as researchers uncovering the circuits of neurons behind alertness and sedation knew. Yanagisawa grew more and more interested in cracking open the black box of sleep.

Biologists had recently begun screening thousands of mutant flies in search of sleep genes. In flies, which are in constant motion when they’re awake, researchers define “sleep” as periods of inactivity. But the gold standard for studying human sleep is electroencephalography, or EEG, which lets observers track the stages of sleep by measuring brain activity. It would be interesting to try it with mice, Yanagisawa thought, which, while wildly more expensive than flies — they take much longer to breed and take up far more space and resources — can be fitted with implants to allow EEG. And mice are evolutionarily closer to humans as well. He conferred with Joseph Takahashi, a colleague at UT Southwestern who specialized in immense mouse studies.

Each week, they decided, they could mutate maybe 30 or 40 mice using a drug, implant brain-activity monitors in their heads, and see how long they slept and how long they spent in each phase of sleep. The study generated data like those from human sleep studies. Mice with abnormalities could be picked out from their sleeping brain activity, and their genomes sifted through to find the genes responsible.

The pilot phase began with just a few hundred animals. “Each mouse could be a treasure,” Yanagisawa said. “You just have to find out which one.” Early on they were encouraged by the discovery of one mouse with sleep abnormalities. And in 2009, Yanagisawa received an enormous grant from the Japanese government to study sleep. The grant was so large that it enabled him to set up a facility at Tsukuba University in Japan to continue the screening on a grand scale. “Now we do up to 100 mice a week,” Yanagisawa said.

The *Nature* paper reports the results from the first 8,000 mice. For years now, Sehgal and many other sleep researchers say, they’ve been hearing updates on the team’s progress at meetings. Yet the identity of the genes has always been a closely kept secret. Sehgal recounted a meeting at which, during an informal presentation session, a researcher from the Yanagisawa lab teasingly told the audience, “Ask me anything you want!”; the crowd, a few drinks in to the evening, roared back, “What are the genes?”

“So I was excited to see the paper,” Sehgal said with a laugh, “and finally find out what these genes were.”

The first gene the team describes makes a protein called SIK3. Mice with a mutated *Sik3* gene spend little time awake — each day they spent more than 30 percent more time asleep than other mice. SIK3 is an enzyme that uses up the cell’s batteries — energy-storage molecules called ATP — to activate other proteins. When the researchers investigated the nature of the mutation, they found that it had removed SIK3’s on-off switch. They believe the mutation jams SIK3 in the on mode.

The other gene came from a mouse that spent less than the usual amount of time in REM sleep, bobbing randomly in and out of a state that usually lasts for a predictable period of time. The mutation in this case was in a protein called NALCN, which acts as a channel in the cell walls of neurons. The protein lets ions burst through as the cells fire. The mutation in this particular mouse may be causing neuronal activity that brings the animal out of REM sleep too soon.

Identifying genes related to sleep doesn’t lead to answers as quickly as one might hope, however. Sleep is so crucial to survival that the ancestors of every sleeping creature alive today risked death to do it daily. That means that there are likely many ways to ensure that it happens, a redundancy that makes it difficult to dissect in experiments. A mutant fly strain that starts out with a severe sleep abnormality will evolve over the generations to sleep more and more normally as time goes on, Sehgal said. That can be frustrating for researchers, though it provides evidence of sleep’s robustness. And most of the genes linked with sleep in the last decade or so of studies likely have many purposes. “There is not something called a ‘sleep gene,’” said Divya Sitaraman, a sleep researcher at the University of San Diego. “There are these genes that have very basic functions that also affect sleep.”

This new paper provides a great example. SIK3, as it happens, doesn’t show up in just the brain. Over the years researchers have found it being used all over the body. It has been linked to metabolic problems, which can make organisms sleepy, said Tom Scammell, a doctor and sleep researcher at Harvard Medical School. A medical work-up of the mouse might help determine whether its drowsiness is because SIK3 affects sleep directly or because of damage to the liver or kidneys, he said.

Still, as work in animal models has continued to churn out these sleep-related genes, the collection of genes linked to sleep in humans has been kicked up a notch. Some researchers have used vast sets of data — thousands and thousands of people, sorted according to how much time they spend sleeping or whether they get up early or late — to see whether these can be linked to specific genetic variations. At least three such studies have been published in the last year.

So far, not many of the associations of genes to sleep have been found by more than one study, said Richa Saxena, a geneticist at Harvard Medical School, Massachusetts General Hospital and the Broad Institute who is an author of one of those studies. That’s likely the redundancy of sleep raising its head again — almost no genes are so strongly connected to sleep that their influence can rise above the noise. But she is hopeful that these statistical associations, paired with the results of screening studies, can bring knowledge of the mechanisms of waking and sleep into reach. For instance, the NALCN channel complex includes a protein that Saxena’s work has linked with sleep.

Finding those connections will make all the difference, said Dragana Rogulja, a fruit fly sleep researcher at Harvard Medical School. She was pleased to learn recently that a sleep-related protein she’d discovered showed up in the same cells as another sleep-related protein identified by another lab. Even though the significance of the link isn’t clear, it’s a step forward. “I was pretty happy about that,” she said, “because most of this stuff is not really connected yet.

“But this is normal in a field that is relatively new,” she said. Yes, sleep research is an old field, she said, “but it was pretty descriptive. I think it’s a really exciting time in sleep research right now, because we’re able to move from that very descriptive, poetic style to a mechanistic one.”

That transformation will continue as researchers delve more deeply into the private life of each protein uncovered by genetic screens and by human studies. Yanagisawa and collaborators are in the thick of characterizing both proteins — SIK3 and NALCN — working on the questions of where they are active and what they interact with. The group continues to mutate mice and watch them sleep; they expect to publish further findings soon. “I feel like these last several years have been the funnest part of my career,” Yanagisawa said. “It’s really exciting.”

]]>“Technically, this experiment is truly impressive,” said Nicolas Gisin, a quantum physicist at the University of Geneva who has studied this loophole around entanglement.

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According to standard quantum theory, particles have no definite states, only relative probabilities of being one thing or another — at least, until they are measured, when they seem to suddenly roll the dice and jump into formation. Stranger still, when two particles interact, they can become “entangled,” shedding their individual probabilities and becoming components of a more complicated probability function that describes both particles together. This function might specify that two entangled photons are polarized in perpendicular directions, with some probability that photon A is vertically polarized and photon B is horizontally polarized, and some chance of the opposite. The two photons can travel light-years apart, but they remain linked: Measure photon A to be vertically polarized, and photon B instantaneously becomes horizontally polarized, even though B’s state was unspecified a moment earlier and no signal has had time to travel between them. This is the “spooky action” that Einstein was famously skeptical about in his arguments against the completeness of quantum mechanics in the 1930s and ’40s.

In 1964, the Northern Irish physicist John Bell found a way to put this paradoxical notion to the test. He showed that if particles have definite states even when no one is looking (a concept known as “realism”) and if indeed no signal travels faster than light (“locality”), then there is an upper limit to the amount of correlation that can be observed between the measured states of two particles. But experiments have shown time and again that entangled particles are more correlated than Bell’s upper limit, favoring the radical quantum worldview over local realism.

Only there’s a hitch: In addition to locality and realism, Bell made another, subtle assumption to derive his formula — one that went largely ignored for decades. “The three assumptions that go into Bell’s theorem that are relevant are locality, realism and freedom,” said Andrew Friedman of the Massachusetts Institute of Technology, a co-author of the new paper. “Recently it’s been discovered that you can keep locality and realism by giving up just a little bit of freedom.” This is known as the “freedom-of-choice” loophole.

In a Bell test, entangled photons A and B are separated and sent to far-apart optical modulators — devices that either block photons or let them through to detectors, depending on whether the modulators are aligned with or against the photons’ polarization directions. Bell’s inequality puts an upper limit on how often, in a local-realistic universe, photons A and B will both pass through their modulators and be detected. (Researchers find that entangled photons are correlated more often than this, violating the limit.) Crucially, Bell’s formula assumes that the two modulators’ settings are independent of the states of the particles being tested. In experiments, researchers typically use random-number generators to set the devices’ angles of orientation. However, if the modulators are not actually independent — if nature somehow restricts the possible settings that can be chosen, correlating these settings with the states of the particles in the moments before an experiment occurs — this reduced freedom could explain the outcomes that are normally attributed to quantum entanglement.

The universe might be like a restaurant with 10 menu items, Friedman said. “You think you can order any of the 10, but then they tell you, ‘We’re out of chicken,’ and it turns out only five of the things are really on the menu. You still have the freedom to choose from the remaining five, but you were overcounting your degrees of freedom.” Similarly, he said, “there might be unknowns, constraints, boundary conditions, conservation laws that could end up limiting your choices in a very subtle way” when setting up an experiment, leading to seeming violations of local realism.

This possible loophole gained traction in 2010, when Michael Hall, now of Griffith University in Australia, developed a quantitative way of reducing freedom of choice. In Bell tests, measuring devices have two possible settings (corresponding to one bit of information: either 1 or 0), and so it takes two bits of information to specify their settings when they are truly independent. But Hall showed that if the settings are not quite independent — if only one bit specifies them once in every 22 runs — this halves the number of possible measurement settings available in those 22 runs. This reduced freedom of choice correlates measurement outcomes enough to exceed Bell’s limit, creating the illusion of quantum entanglement.

The idea that nature might restrict freedom while maintaining local realism has become more attractive in light of emerging connections between information and the geometry of space-time. Research on black holes, for instance, suggests that the stronger the gravity in a volume of space-time, the fewer bits can be stored in that region. Could gravity be reducing the number of possible measurement settings in Bell tests, secretly striking items from the universe’s menu?

Friedman, Alan Guth and colleagues at MIT were entertaining such speculations a few years ago when Anton Zeilinger, a famous Bell test experimenter at the University of Vienna, came for a visit. Zeilinger also had his sights on the freedom-of-choice loophole. Together, they and their collaborators developed an idea for how to distinguish between a universe that lacks local realism and one that curbs freedom.

In the first of a planned series of “cosmic Bell test” experiments, the team sent pairs of photons from the roof of Zeilinger’s lab in Vienna through the open windows of two other buildings and into optical modulators, tallying coincident detections as usual. But this time, they attempted to lower the chance that the modulator settings might somehow become correlated with the states of the photons in the moments before each measurement. They pointed a telescope out of each window, trained each telescope on a bright and conveniently located (but otherwise random) star, and, before each measurement, used the color of an incoming photon from each star to set the angle of the associated modulator. The colors of these photons were decided hundreds of years ago, when they left their stars, increasing the chance that they (and therefore the measurement settings) were independent of the states of the photons being measured.

And yet, the scientists found that the measurement outcomes still violated Bell’s upper limit, boosting their confidence that the polarized photons in the experiment exhibit spooky action at a distance after all.

Nature could still exploit the freedom-of-choice loophole, but the universe would have had to delete items from the menu of possible measurement settings at least 600 years before the measurements occurred (when the closer of the two stars sent its light toward Earth). “Now one needs the correlations to have been established even before Shakespeare wrote, ‘Until I know this sure uncertainty, I’ll entertain the offered fallacy,’” Hall said.

Next, the team plans to use light from increasingly distant quasars to control their measurement settings, probing further back in time and giving the universe an even smaller window to cook up correlations between future device settings and restrict freedoms. It’s also possible (though extremely unlikely) that the team will find a transition point where measurement settings become uncorrelated and violations of Bell’s limit disappear — which would prove that Einstein was right to doubt spooky action.

“For us it seems like kind of a win-win,” Friedman said. “Either we close the loophole more and more, and we’re more confident in quantum theory, or we see something that could point toward new physics.”

There’s a final possibility that many physicists abhor. It could be that the universe restricted freedom of choice from the very beginning — that every measurement was predetermined by correlations established at the Big Bang. “Superdeterminism,” as this is called, is “unknowable,” said Jan-Åke Larsson, a physicist at Linköping University in Sweden; the cosmic Bell test crew will never be able to rule out correlations that existed before there were stars, quasars or any other light in the sky. That means the freedom-of-choice loophole can never be completely shut.

But given the choice between quantum entanglement and superdeterminism, most scientists favor entanglement — and with it, freedom. “If the correlations are indeed set [at the Big Bang], everything is preordained,” Larsson said. “I find it a boring worldview. I cannot believe this would be true.”

*This article was reprinted on TheAtlantic.com.*

Su opened his talk with the story of Christopher, an inmate serving a long sentence for armed robbery who had begun to teach himself math from textbooks he had ordered. After seven years in prison, during which he studied algebra, trigonometry, geometry and calculus, he wrote to Su asking for advice on how to continue his work. After Su told this story, he asked the packed ballroom at the Marriott Marquis, his voice breaking: “When you think of who does mathematics, do you think of Christopher?”

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Su grew up in Texas, the son of Chinese parents, in a town that was predominantly white and Latino. He spoke of trying hard to “act white” as a kid. He went to college at the University of Texas, Austin, then to graduate school at Harvard University. In 2015 he became the first person of color to lead the MAA. In his talk he framed mathematics as a pursuit uniquely suited to the achievement of human flourishing, a concept the ancient Greeks called *eudaimonia*, or a life composed of all the highest goods. Su talked of five basic human desires that are met through the pursuit of mathematics: play, beauty, truth, justice and love.

If mathematics is a medium for human flourishing, it stands to reason that everyone should have a chance to participate in it. But in his talk Su identified what he views as structural barriers in the mathematical community that dictate who gets the opportunity to succeed in the field — from the requirements attached to graduate school admissions to implicit assumptions about who looks the part of a budding mathematician.

When Su finished his talk, the audience rose to its feet and applauded, and many of his fellow mathematicians came up to him afterward to say he had made them cry. A few hours later *Quanta Magazine* sat down with Su in a quiet room on a lower level of the hotel and asked him why he feels so moved by the experiences of people who find themselves pushed away from math. An edited and condensed version of that conversation and a follow-up conversation follows.

FRANCIS SU: When I think of human flourishing, I’m thinking of something close to Aristotle’s definition, which is activity in accordance with virtue. For instance, each of the basic desires that I mentioned in my talk is a mark of flourishing. If you have a playful mind or a playful spirit, or you’re seeking truth, or pursuing beauty, or fighting for justice, or loving another human being — these are activities that line up with certain virtues. Maybe a more modern way of thinking about it is living up to your potential, in some sense, though I wouldn’t just limit it to that. If I am loving somebody well, that’s living up to a certain potential that I have to be able to love somebody well.

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It builds skills that allow people to do things they might otherwise not have been able to do or experience. If I learn mathematics and I become a better thinker, I develop perseverance, because I know what it’s like to wrestle with a hard problem, and I develop hopefulness that I will actually solve these problems. And some people experience a kind of transcendent wonder that they’re seeing something true about the universe. That’s a source of joy and flourishing.

Math helps us do these things. And when we talk about teaching mathematics, sometimes we forget these larger virtues that we are seeking to cultivate in our students. Teaching mathematics shouldn’t be about sending everybody to a Ph.D. program. That’s a very narrow view of what it means to do mathematics. It shouldn’t mean just teaching people a bunch of facts. That’s also a very narrow view of what mathematics is. What we’re really doing is training habits of mind, and those habits of mind allow people to flourish no matter what profession they go into.

I chose it because it says in a very succinct way what the problem is, what causes injustice — we judge, and we don’t judge correctly. So “read” means “judged,” of course. We read people differently than they actually are.

We do this in lots of different ways. I think part of it is that we have a picture of who actually can succeed in math. Some of that picture has been developed because the only examples we’ve seen so far are people who come from particular backgrounds. We’re not used to, for instance, seeing African-Americans at a math conference, although it’s become more and more common.

We’re not used to seeing kids from lower socioeconomic backgrounds in college or grad school. So what I was trying to say is: If we’re looking for talent, why are we choosing for background? If we really want to have a more diverse set of people in mathematical sciences, we have to take into account the structural barriers that make it hard for people from disadvantaged backgrounds to succeed in math.

That’s right. At every stage we’re losing people. So if you look at some of the studies people are doing now about people who take Calculus 1, and how many of them go on to take Calculus 2, you’ll find basically that we’re losing women and minorities at these critical junctures. This happens for reasons that we can only speculate about. But I’m sure some of it has to do with people in these groups not seeing themselves as belonging in math, possibly because of a negative culture and an unwelcome climate, or because of things that professors or other students are doing to discourage people from continuing.

Math can contribute in a broad way to every person’s life whether that person actually becomes a mathematician or not. The goal of broadly getting people to appreciate math is not at odds with bringing more people into deep mathematics. Connect with people in a deep way and you’re going to draw more people into mathematics. Some of them, more of them, are going to go to graduate school, and that will necessarily happen if you address some of these deep desires — for love, truth, beauty, justice, play. If you address some of these deep themes you’re going to get more people and a more diverse set of people in deep mathematics.

Justice is a desire that people have, and so it leads to a certain virtue which is to become a just person, somebody who cares about fighting for things that defend basic human dignity. I spent the most time discussing justice in my talk mainly because I feel that our mathematics community can do better; we can become more just. I see a lot of ways in which we can do better and become more virtuous as a community.

Being a mathematician in some ways allows us to see things more for what they are. When people learn not to overgeneralize their arguments, they’re going to be very careful not to think that if you’re poor you’re necessarily uneducated or vice versa. Having a mathematical background certainly helps people to be less governed by their biases.

When I was in graduate school at Harvard I realized I loved teaching, and I remember one of my professors from college telling me that the teaching was better at small liberal arts colleges. So when I was on the job market I started looking at those colleges. I was interested in the research track and willing to do that, but I was also very attracted to the liberal arts environment. I chose to go and I love it; I couldn’t see myself being anywhere else.

I think one of the things I didn’t address in the talk, but almost did, is the divide in the community between research universities and liberal arts colleges. There is a cultural divide, and the research universities are in some sense the dominant culture because all of us with Ph.D.s come through research universities. And there’s the whole pattern of the dominant culture being completely unaware of what’s going on at the liberal arts colleges. So people come up to me and say: “So, you’re at Harvey Mudd; are you happy there?” It’s almost like assuming I wouldn’t be. That happens all the time, so I find it a bit frustrating to feel like I have to say: “No, this is actually my dream job.”

Well, the downsides are, for instance, that many of the people at research universities would never consider taking students from an undergraduate college. That’s the downside; they’re missing a lot of talent. So in many ways the issues are analogous to some of the racial issues that are going on.

I think professors at research universities often don’t realize that there are a lot of bright kids coming through the liberal arts colleges. What I’m addressing is the very common practice right now in certain graduate schools of only admitting people who’ve already had a full slate of graduate courses. In other words, they’re expecting undergraduates to have taken graduate courses before they even get considered. If you have that kind of structural situation, you are necessarily going to exclude a bunch of people who otherwise might be successful.

I’m being a little provocative here as well. I think what that communicates is: “This is not an important enough segment of people for me to put my attention to.” I’m certainly not saying everybody who only teaches senior-level courses has this attitude, but I am saying there are a lot of people who think the math major is basically there for the benefit of students who are going to get a Ph.D. That’s a problem.

Definitely, racial inclusiveness has not come as far or as fast as gender inclusiveness. Currently about 27 percent of people with Ph.D.s, faculty members, are women, and about 30 percent of the ones who won awards in teaching and service are women. So we’re actually doing pretty well on that front. With our writing awards, which are awards for research and exposition — the fraction of women winning those awards is lower.

Many of the practices that work to encourage women in math also work for minorities. Part of the issue here is that there just aren’t that many minorities who come into college interested in doing STEM majors. So there’s something that happened at the secondary and primary school level, and it would help a lot if we could figure out what’s going on there.

If you go to an authentic restaurant in a big city in New York or California, if you are not Chinese they will give you a standard menu that has things in English and Chinese. But if you’re Chinese, they’ll give you a different menu. Often it’s a menu that is written completely in Chinese and has some additional options that aren’t on the standard menu. And I think that happens in the math community. If you talk to women and minorities they will often tell you they’ve had experiences where people discouraged them from going on, either because they don’t think a woman should be in math, or for other reasons. So I used the metaphor “secret menu” to mean: Do we have a secret menu? And who gets to look at it?

I think it’s common. Of course we don’t have any data, but I’ve certainly talked to enough people who’ve had those kinds of experiences to know that it’s very frequent and most of those people are women and minorities.

Most of the comments have come from people who are grateful to me for mentioning things that haven’t necessarily been discussed, but also for identifying some of the deep, underlying things that cause us to do what we do. I think a lot of people, especially women and minorities, have expressed to me how important it was for somebody to say that. We’ve been having discussions like this in smaller conversations, and a lot of time it’s preaching to the choir, and so having somebody say that in a big address at the national meeting I think felt important and helpful to them.

*This article was reprinted on Wired.com.*