In the simplest terms, an elliptic curve is a doughnut shape with carefully placed points, explain Emory University mathematicians Ken Ono, left, and John Duncan, right. “The whole game in the math of elliptic curves is determining whether the doughnut has sprinkles and, if so, where exactly the sprinkles are placed,” Duncan says. (Photos by Stephen Nowland, Emory Photo/Video)
By Carol Clark
Mathematicians have opened a new chapter in the theory of moonshine, one which begins to harness the power of the pariahs – sporadic simple groups that previously had no known application.
“We’ve found a new form of moonshine, which in math refers to an idea so farfetched as to sound like lunacy,” says Ken Ono, a number theorist at Emory University. “And we’ve used this moonshine to show the mathematical usefulness of the O’Nan pariah group in a way that moves it from theory to reality. It turns out that the O’Nan group knows deep information about elliptic curves.”
Nature Communications published the representation theory for the O’Nan group developed by Ono, John Duncan (also a number theorist at Emory) and Michael Mertens (a former post-doctoral fellow at Emory who is now at the University of Cologne).
“We’ve shown that the O’Nan group, a very large pariah group, actually organizes elliptic curves in a beautiful and systematic way,” Duncan says. “And not only does it organize them, it allows us to see some of their deepest properties. It sees infinitely many curves, which allows us to then use our moonshine to make predictions about their general behavior. That’s important, because these objects underlie some of the hardest questions at the very horizon of number theory.”
Elliptic curves may sound esoteric, but they are part of our day-to-day lives. They are used in cryptography – the creation of codes that are difficult to break.
An elliptic curve is not an ellipse, rather it is a complex torus, or doughnut shape.
“You can think of it as a doughnut together with specific, delicate configurations of rational points that are very carefully placed,” Duncan says. “So, in the simplest of terms, it’s like a doughnut that you eat, that may have sprinkles on it. The whole game in the math of elliptic curves is determining whether the doughnut has sprinkles and, if so, where exactly the sprinkles are placed.”
Unlike an edible doughnut, however, these mathematical doughnuts are not visible.
“Imagine you are holding a doughnut in the dark,” Ono says. “You wouldn’t even be able to decide whether it has any sprinkles. But the information in our O’Nan moonshine allows us to ‘see’ our mathematical doughnuts clearly by giving us a wealth of information about the points on elliptic curves.”
The findings are especially surprising since none of the pariahs, as six of math’s sporadic simple groups are known, had previously appeared in moonshine theory, or anywhere else in science.
Math’s original moonshine theory dates to a 1979 paper called “Monstrous Moonshine” by John Conway and Simon Norton. The paper described a surprising connection between a massive algebraic object known as the monster group and the j-function, a key object in number theory. In 2015, a group of mathematicians – including Duncan and Ono – presented proof of the Umbral Moonshine Conjecture, which revealed 23 other moonshines, or mysterious connections between the dimensions of symmetry groups and coefficients of special functions.
In theoretical math, symmetry comes in groups. Symmetrical solutions are usually optimal, since they allow you to divide a large problem into equal parts and solve it faster.
The classification of the building blocks of groups is gathered in the ATLAS of Finite Groups, published in 1985. “The ATLAS is like math’s version of the periodic table of the elements, but for symmetry instead of atoms,” Duncan explains.
Both the ATLAS and the periodic table contain quirky characters that may – or may not – exist in nature.
Four super heavy elements with atomic numbers above 100, for example, were discovered in 2016 and added to the periodic table. “People have to work hard to produce these elements in particle accelerators and they vanish immediately after they are constructed,” Ono says. “So you have to wonder if they really are a part of our everyday chemistry.”
The pariah groups pose a similar question in math. Are they natural or simply theoretical constructs?
“Our work proves, for the first time, that a pariah is real,” Ono says. “We found the O’Nan group living in nature. Our theorem shows that it’s connected to elliptic curves, and whenever you find a correspondence between two objects that are seemingly not related, it opens the door to learning more about those objects.”
Emory anthropologist Jessica Thompson next to Malawi rock art paintings, likely made by hunter-gatherers. Thompson's work in Malawi is part of a major new paper in the journal Cell, filling in thousands of years of human prehistory of hunter-gatherers in Africa. (Photo by Suzanne Kunitz)
By Carol Clark
Emory anthropologist Jessica Thompson was at a human origins conference years ago when she heard a presenter lament: “Of course, there is no ancient DNA from Africa because of the poor preservation there.”
That’s when it clicked in Thompson’s mind: She had visited a place in Africa — the highlands of northern Malawi — that had neither extremes of heat or wetness — two main environmental factors that degrade DNA. She also knew that scant archaeological research had been done in the region, although a team had unearthed several ancient skeletons there decades ago.
“It’s a strange and fascinating landscape,” says Thompson, who made that 2005 visit as a tourist and was struck by the surreal beauty of the high mountain grassland.
It’s also remote and off the radar of most of the world. “We saw maybe three other tourists while we were there,” she recalls.
That fateful trip laid the groundwork for discoveries of the oldest-known DNA from Africa. The journal Cell just published an analysis of the new discoveries, filling in thousands of years of human prehistory of hunter-gatherers in Africa, led by Harvard geneticist David Reich.
Thompson is second author of the paper. She contributed and described the cultural context for nearly half of the 15 new DNA finds, including the oldest samples. Her fieldwork in Malawi uncovered human remains that yielded DNA ranging in age from about 2,500 to 6,100 years old. And her work is ongoing at a site where a skeleton recovered in 1950 was just dated to 8,100 years old and also yielded DNA.
The other DNA in the Cell paper ranges in age from 3,000-to-500 years ago and comes from South Africa, Tanzania and Kenya.
“Malawi is positioned in between where living hunter-gatherers survive,” Thompson says. “For the first time, we can see the distribution of ancient hunter-gatherer DNA across Africa, showing how these populations were connected in the past.”
Ancient hunter-gatherers do not have a lot of living representatives in Africa today, and they occur as remnants of people scattered across the continent. The remains of Malawi hunter-gatherers that Thompson is studying may represent a population that was once thriving but subsequently pushed into marginal areas during the expansion of agriculturalists and pastoralists during the past 3,000 years.
Some of this population may have survived until much more recently.
“There are legends in Malawi of the original people who came there, passed down through oral histories,” Thompson says. “They are described as hunters and little people, short in stature. There is also a story of a last, epic battle — that occurred about 200 years ago — when these people got eradicated.”
Mount Hora, where the oldest DNA included in the Cell paper was obtained, from a woman who lived more than 8,000 years ago. (Photo by Jessica Thompson)
Malawi captivated Thompson during that first visit as a tourist, in 2005. She was a graduate student when she spent a summer working on a dig in the Serengeti. She and two companions decided to make a road trip before returning to the United States, including a stop in Malawi.
The landlocked country is located in southeast Africa, bordered by Zambia, Tanzania and Mozambique. It is one of the least-developed and smallest countries in Africa, about the size of the state of Tennessee, and runs north to south along the Rift Valley. An enormous body of water, Lake Malawi, makes up about one-third of the country.
“My traveling companies wanted to relax by the lake in the lowlands,” Thompson recalls. “I had read about the Malawi highlands and really wanted to see this unique ecosystem, so I convinced them to go there instead.”
Her companions complained of the cold — it’s windy and regularly freezes in the highlands of Malawi and summer temperatures peak at around 65 or 70 degrees Fahrenheit. Despite the cold, Thompson admired the rugged, isolated beauty of rocky outcrops and grasslands studded with orchids and fairy ferns where zebra and shaggy antelope grazed.
Thompson, who joined Emory as an assistant professor of anthropology in 2015, dug through the archaeological literature surrounding Malawi and started making exploratory trips there in 2009. She learned of two digs in the Malawi highlands — in 1950 and 1966 — that revealed human skeletons alongside rich cultural evidence of an extinct hunting-and-gathering lifeway.
Dancers at a festival in Malawi. The people living in the country today are the descendants of the Iron Age agriculturalists and pastoralists who swept across the African continent about 3,000 years ago. (Photo by Jessica Thompson)
The 1950 dig turned out to be led by the renowned archaeologist J. Desmond Clark, who Thompson calls her “academic grandfather.” Although Clark died before Thompson could meet him, he served as the mentor to her mentor, Curtis Marean.
On the slopes of Mount Hora — a striking 1,500-meter peak and a major landmark in the highlands — Clark uncovered two skeletons: A woman who had died at around age 22 and a nearby male, who had died in his 40s. The skeletons had been taken out of the country, to the Livingstone Museum in Zambia, and were never dated.
“It was impossible to accurately do radiocarbon dating on bone in 1950,” Thompson explains. “The skeletons became, quite frankly, forgotten over time.”
Guided by the clues from the previous excavations, Thompson began heading digs in the Malawi highlands. A site at a landmark outcrop, known as Fingira Rock, is particularly isolated, requiring the team to hike up a mountainside to more than 2,000 meters on the Nyika Plateau. “Working there you feel the wind, you feel the chill,” Thompson says.
Poachers are a hazard in the area, along with the occasional black mamba — one of the world’s deadliest snakes.
The Fingira site had not been excavated since 1966. “We were appalled to discover that it had been heavily disturbed since then,” Thompson says. Her team uncovered two human leg bones, from two different adult males, which yielded DNA that was about 6,100 years old.
The leg bone of a hunter-gatherer that lived 6,100 years ago, found at the Fingira Rock site. (Photo by Jessica Thompson)
In the back of a cave, they found fragments of a child’s skull in a termite mound. A tiny leg bone next to it indicated that the remains were from a baby younger than age one. DNA analysis revealed that she had been a girl and radiocarbon dating showed that she had died about 2,500 years ago. The analysis also showed that the bones from the infant and the two men were from the same hunter-gatherer population — even though they were separated by thousands of years of time.
The archaeological sediments suggest that Fingira was a place where the dead were buried, although the skeletal material has become scattered over time. Human bones are mixed with the bones of animals that they hunted and ate, as well as with stone tools and shell beads that they used for ornaments.
“When you visit the site,” Thompson says, “you wonder, why were these people living up here when it’s not the most comfortable conditions you can imagine? What was bringing them here? Why were they burying their dead, over and over again, for many thousands of years, in the same place?”
Meanwhile, Thompson tracked down the skeletons that Clark had discovered at Mount Hora in 1950. She learned they had been moved from Zambia to the University of Cape Town in South Africa.
Here’s where Emory graduate student Kendra Ann Sirak enters the story. Sirak had the distinction of being the last graduate student of Emory anthropologist George Armelagos, one of the founders of the field of paleopathology. He spent decades working with graduate students to study the bones of ancient Sudanese Nubians to learn about patterns of health, illness and death in the past. Armelagos sent Sirak to one of the best ancient DNA labs in the world, at University College Dublin (UCD), in Ireland, with samples of the Nubian bones.
After Armelagos died in 2014, at age 77, Thompson stepped in as one of Sirak’s mentors.
Thompson,
left, examines fragments of artifacts from the Malawi excavations in her
lab with Emory graduate student Kendra Ann Sirak. Sirak helped with the
radiocarbon dating and DNA extraction of the "forgotten" 8,100-year-old
skeleton from Mount Hora. (Photo by Ann Borden, Emory Photo/Video)
Thompson contacted the curator of the two skeletons from Mount Hora, to ask about the possibility of getting DNA from them. Alan Morris, now Professor Emeritus at the University of Cape Town, had had the same idea. A sample from the female skeleton was already slated to be sent to the UCD lab where Sirak was working. So Thompson, Morris and Sirak teamed up on the quest.
The petrous bone, which contains components of the inner ear, is the most promising site to drill for ancient DNA. The skeleton's petrous bone had already broken away from the skull, so only this tiny, triangular-shaped piece of the skeleton was sent to Dublin.
"It was extremely fragile," says Sirak, whose job was to drill into the petrous bone and get about 200 millimeters of bone powder without shattering the specimen.
She drank a cup of coffee, donned a hair cover, overalls, a face mask, two pairs of gloves and shoe covers, then entered a small, sterile room where the petrous bone awaited. "I said to myself, 'Here we go, I've got this!'" Sirak recalls.
Sirak was successful. Her colleagues in Dublin processed the sample and then sent it to the genetics team at Harvard Medical School for DNA analysis, which was also successful.
Meanwhile, radiocarbon dating revealed that the skeleton was 8,100 years old.
"It was like Christmas," Sirak says, "knowing that we had DNA data on such an ancient specimen."
The skeleton's genetics connected her to the same population of hunter-gatherers who died thousands of years later and were found 70 kilometers away at Fingira.
Another surprise revealed by the genetic analysis of the Malawi hunter-gatherers: They did not contribute any detectable ancestry to the people living in Malawi today, the descendants of the Iron Age agriculturalists and pastoralists who began sweeping across the African continent about 3,000 years ago.
“In most parts of Africa, you see quite a bit of admixture,” Thompson says. “When you take genetic samples from modern people who are living today, you find that they are a combination of the folks who were expanding into a region and also the folks who were living there before. In Malawi we see that’s not the case. It appears that there was a complete replacement of the original hunter-gatherer people. They are not just gone as a lifeway, they are actually gone as a people as well.”
One of the mysteries Thompson hopes to solve is how that replacement happened. Was it violent? Was it a sudden or a slow process? Did the entrance of strange new technologies, like pottery and iron working, play a role?
“We can’t use genetics to answer these questions,” Thompson says. “We have to use the archaeology.”
Emory anthropology undergraduates assisting with the Malawi excavations this past summer included, from left: Alexa Rome, Alexandra Davis, Suzanne Kunitz and Aditi Majoe. Graduate student Grace Veatch is on the far right.
She continues to excavate in Malawi, aided by local technicians and other collaborators. This summer, five Emory anthropology students accompanied her in the field: Graduate student Grace Veatch, senior Alexandra Davis, juniors Aditi Majoe and Suzanne Kunitz, and sophomore Alexa Rome. They uncovered more human remains at Mount Hora — a charred bone from a human arm and parts of two legs. These bones, recently dated to between 9,500 and 9,300 years old, show that the Hora site still has many secrets to reveal.
While radiocarbon dating of charcoal samples from just above and below the bones establishes their age, it is not clear whether they will yield DNA. “We don’t have high hopes,” Thompson says, “as they were burned and that tends to create even more preservation problems.”
The students assisted in the tedious work of carefully sifting through grey dust and ash, marking coordinates through GPS and other surveying tools, and recording the data into a computer.
Back in her lab at Emory, Thompson uses the data to generate three-dimensional images of the digs and pinpoint where each bone fragment, shell bead or stone tool was found. Her digital model for the this summer’s Mount Hora dig uses different-colored dots to give a glimpse of how hunter-gatherers were depositing both human remains and ordinary objects from their day-to-day lives over time.
“And then at this point,” Thompson says as she moves her cursor on her computer screen, “you see the introduction of pottery and iron technology. And right after that you see this fundamental change in the way that the site was used. People are no longer going there frequently. They’re no longer making these big bonfires. And they’re no longer interring their dead there.”
Thompson and her students are also sorting through hundreds of gallon-sized Ziploc plastic bags containing fragments from the Malawi sites. “As you excavate,” she explains, “you clean away the dirt and you’re left with all these tiny pieces of stone and bone artifacts. The bones are mostly animals. But every once in a while you find something that looks like it might be human. Any one one of them could be a new individual, a new piece to the story.”
She pulls out a small plastic bag labeled “Human distal phalanx.” It contains a piece of bone about the size of a Tic-Tac. “In this case, we think we have a finger bone, most likely from a child,” Thompson says.
Ultimately, Thompson seeks to understand how and when the earliest members of our species — Stone Age Homo sapiens — interacted with one another and with their environments in Africa.
“One thing that’s really easy to forget, when we look at the way people live today, is that for most of our evolution we lived as hunter-gatherers,” she says. “So if we want to understand our own origins as a species, we have to know what those lifeways looked like in the past.”
"I can't imagine not living with dogs. That would be really sad for me," says Emory neuroscientist Gregory Berns, with Callie (left) and Cato. His latest book is called "What It's Like to Be a Dog."
Five years ago, Emory neuroscientist Gregory Berns became the first to capture images of actual canine thought processes. To explore the minds of the oldest domesticated species, the Berns lab trained dogs to remain still and alert while undergoing functional Magnetic Resonance Imaging (fMRI) — the same tool that is unlocking secrets of the human brain. The project opened a new door into canine cognition and social cognition of other species.
Berns went on to conduct a series of experiments on dogs, gathering both behavioral and fMRI data on questions such as: How capable are dogs of self-restraint? Do dogs prefer praise from their owners or food? How do dogs process faces in their brains? What’s going on in a dog’s brain when it smells the scent of its owner?
In 2013, Berns wrote a New York Times bestseller called “How Dogs Love Us.” He described how the death of his beloved pug Newton planted the seeds for his eventual switch from the studying the human brain to focus on non-invasive studies of the cognition of dogs and other animals.
In the following Q&A, Berns talks about his new book, “What It’s Like to Be a Dog: And Other Adventures in Animal Neuroscience,” just published by Basic Books. The book focuses on his hopes that understanding how animals think will revolution how we treat them.
Question: Can you talk about all the dogs you’ve had as pets during your life?
Gregory Berns: When I was a child growing up in Southern California we had two golden retrievers, Pretzel and Popcorn. It’s embarrassing, but my parents always named their dogs after food. I’m not sure why. Most of the children in the area had dogs and horses and we would go traipsing around the hills. Kids and dogs go together.
Berns and Callie
After I was done with medical school and stopped moving around, my wife and I had three pugs, Simon, Newton and Dexter, and then a golden retriever, Lyra.
We now live with our two daughters and have three dogs: Callie, a Feist, which is a Southern squirrel hunting dog; Cato, a Plott hound, which is the state dog of North Carolina; and Argo, a yellow dog of some kind of mix. We also have two bearded dragons and a chameleon.
I can’t imagine not living with dogs. That would be really sad for me.
Q: “What It’s Like to Be a Dog” describes all the experimental work you have done so far with canine cognition. What’s the biggest surprise to come out of your research?
GB: If you take language out of the picture, what we’re finding is that we see a lot of similarities between dogs and humans. In one study, for instance, we used fMRI to measure the relative value of food versus praise to the dogs and found that almost all the dogs’ brains responded to praise as much, and sometimes more, than to food. We ourselves know how it feels when someone praises us, there’s a positive feeling associated with it. That’s perhaps similar to what dogs are feeling.
We also did a study on dogs and delayed gratification. We found that part of the prefrontal cortex is more active in dogs during self-control. And, just like experiments with humans have found, we showed that the dogs who are better at this task use more of their prefrontal lobes.
Now that we are gaining a basic understanding of canine cognition, we are starting to focus more on the individuality of dogs — what it’s like to be this dog, as opposed to that dog.
Q: You’re also using diffusion tensor imaging (DTI) to study the brains of other mammals, mapping the neural pathways in brains from animals that are long deceased and stored in museum collections. How did this project come about?
GB: It started in 2015 when we gained access to the brains of two dolphins that had died, and we showed that we could use DTI to map their sensory and motor systems. Dolphins are incredibly intelligent, social animals but they’ve remained relatively mysterious. We provided the first picture of the entire dolphin brain and all the white matter connections inside of it.
Through a project I call the Brain Ark we’re collecting a digital archive of high-resolution, three-dimensional brain structures of megafauna. It’s publicly available to other researchers to contribute to and draw data from.
Q: What is the ultimate goal of your animal neuroscience research?
GB: The Brain Ark is an attempt to catalog and study brains of large mammal species before they are gone. Or, as in the case of the Tasmanian tiger, after they’re gone. Many megafauna are in danger of extinction because their habitats are being chopped up in ways that don’t allow them to sustain themselves or to migrate.
In the grand scheme of things, I’d also like to explore the commonalities that we have with other animals. That has important ethical implications for how we treat them and for their right to exist in the first place.
Animal welfare laws cover things like abuse — pain and suffering. I think we should go beyond that and acknowledge that animals also have a right to lead a good life — whatever that means for that animal.
