viernes, 15 de septiembre de 2017


Mathematicians Measure Infinities and Find They’re Equal
Two mathematicians have proved that two different infinities are equal in size, settling a long-standing question. Their proof rests on a surprising link between the sizes of infinities and the complexity of mathematical theories.

In a breakthrough that disproves decades of conventional wisdom, two mathematicians have shown that two different variants of infinity are actually the same size. The advance touches on one of the most famous and intractable problems in mathematics: whether there exist infinities between the infinite size of the natural numbers and the larger infinite size of the real numbers.
The problem was first identified over a century ago. At the time, mathematicians knew that “the real numbers are bigger than the natural numbers, but not how much bigger. Is it the next biggest size, or is there a size in between?” said Maryanthe Malliaris of the University of Chicago, co-author of the new work along with Saharon Shelah of the Hebrew University of Jerusalem and Rutgers University.
In their new work, Malliaris and Shelah resolve a related 70-year-old question about whether one infinity (call it p) is smaller than another infinity (call it t). They proved the two are in fact equal, much to the surprise of mathematicians.
“It was certainly my opinion, and the general opinion, that should be less than t,” Shelah said.
Malliaris and Shelah published their proof last year in the Journal of the American Mathematical Society and were honored this past Julywith one of the top prizes in the field of set theory. But their work has ramifications far beyond the specific question of how those two infinities are related. It opens an unexpected link between the sizes of infinite sets and a parallel effort to map the complexity of mathematical theories.
Many Infinities
The notion of infinity is mind-bending. But the idea that there can be different sizes of infinity? That’s perhaps the most counterintuitive mathematical discovery ever made. It emerges, however, from a matching game even kids could understand.
Suppose you have two groups of objects, or two “sets,” as mathematicians would call them: a set of cars and a set of drivers. If there is exactly one driver for each car, with no empty cars and no drivers left behind, then you know that the number of cars equals the number of drivers (even if you don’t know what that number is).
In the late 19th century, the German mathematician Georg Cantor captured the spirit of this matching strategy in the formal language of mathematics. He proved that two sets have the same size, or “cardinality,” when they can be put into one-to-one correspondence with each other — when there is exactly one driver for every car. Perhaps more surprisingly, he showed that this approach works for infinitely large sets as well.
Consider the natural numbers: 1, 2, 3 and so on. The set of natural numbers is infinite. But what about the set of just the even numbers, or just the prime numbers? Each of these sets would at first seem to be a smaller subset of the natural numbers. And indeed, over any finite stretch of the number line, there are about half as many even numbers as natural numbers, and still fewer primes.
Yet infinite sets behave differently. Cantor showed that there’s a one-to-one correspondence between the elements of each of these infinite sets.
(natural numbers)
Because of this, Cantor concluded that all three sets are the same size. Mathematicians call sets of this size “countable,” because you can assign one counting number to each element in each set.
After he established that the sizes of infinite sets can be compared by putting them into one-to-one correspondence with each other, Cantor made an even bigger leap: He proved that some infinite sets are even larger than the set of natural numbers.
Consider the real numbers, which are all the points on the number line. The real numbers are sometimes referred to as the “continuum,” reflecting their continuous nature: There’s no space between one real number and the next. Cantor was able to show that the real numbers can’t be put into a one-to-one correspondence with the natural numbers: Even after you create an infinite list pairing natural numbers with real numbers, it’s always possible to come up with another real number that’s not on your list. Because of this, he concluded that the set of real numbers is larger than the set of natural numbers. Thus, a second kind of infinity was born: the uncountably infinite.
What Cantor couldn’t figure out was whether there exists an intermediate size of infinity — something between the size of the countable natural numbers and the uncountable real numbers. He guessed not, a conjecture now known as the continuum hypothesis.
In 1900, the German mathematician David Hilbert made a list of 23 of the most important problems in mathematics. He put the continuum hypothesis at the top. “It seemed like an obviously urgent question to answer,” Malliaris said.
In the century since, the question has proved itself to be almost uniquely resistant to mathematicians’ best efforts. Do in-between infinities exist? We may never know.
Forced Out
Throughout the first half of the 20th century, mathematicians tried to resolve the continuum hypothesis by studying various infinite sets that appeared in many areas of mathematics. They hoped that by comparing these infinities, they might start to understand the possibly non-empty space between the size of the natural numbers and the size of the real numbers.
Many of the comparisons proved to be hard to draw. In the 1960s, the mathematician Paul Cohen explained why. Cohen developed a method called “forcing” that demonstrated that the continuum hypothesis is independent of the axioms of mathematics — that is, it couldn’t be proved within the framework of set theory. (Cohen’s work complemented work by Kurt Gödel in 1940 that showed that the continuum hypothesis couldn’t be disproved within the usual axioms of mathematics.)
Cohen’s work won him the Fields Medal (one of math’s highest honors) in 1966. Mathematicians subsequently used forcing to resolve many of the comparisons between infinities that had been posed over the previous half-century, showing that these too could not be answered within the framework of set theory. (Specifically, Zermelo-Fraenkel set theory plus the axiom of choice.)
Some problems remained, though, including a question from the 1940s about whether p is equal to t. Both p and t are orders of infinity that quantify the minimum size of collections of subsets of the natural numbers in precise (and seemingly unique) ways.
Briefly, p is the minimum size of a collection of infinite sets of the natural numbers that have a “strong finite intersection property” and no “pseudointersection,” which means the subsets overlap each other in a particular way; t is called the “tower number” and is the minimum size of a collection of subsets of the natural numbers that is ordered in a way called “reverse almost inclusion” and has no pseudointersection.
The details of the two sizes don’t much matter. What’s more important is that mathematicians quickly figured out two things about the sizes of p and t. First, both sets are larger than the natural numbers. Second, p is always less than or equal to t. Therefore, if p is less than t, then p would be an intermediate infinity — something between the size of the natural numbers and the size of the real numbers. The continuum hypothesis would be false.
Briefly, p is the minimum size of a collection of infinite sets of the natural numbers that have a “strong finite intersection property” and no “pseudointersection,” which means the subsets overlap each other in a particular way; t is called the “tower number” and is the minimum size of a collection of subsets of the natural numbers that is ordered in a way called “reverse almost inclusion” and has no pseudointersection.
Mathematicians tended to assume that the relationship between p and t couldn’t be proved within the framework of set theory, but they couldn’t establish the independence of the problem either. The relationship between p and t remained in this undetermined state for decades. When Malliaris and Shelah found a way to solve it, it was only because they were looking for something else.
An Order of Complexity
Around the same time that Paul Cohen was forcing the continuum hypothesis beyond the reach of mathematics, a very different line of work was getting under way in the field of model theory.
H. Jerome Keisler invented “Keisler’s order.”
Courtesy of H. Jerome Keisler
For a model theorist, a “theory” is the set of axioms, or rules, that define an area of mathematics. You can think of model theory as a way to classify mathematical theories — an exploration of the source code of mathematics. “I think the reason people are interested in classifying theories is they want to understand what is really causing certain things to happen in very different areas of mathematics,” said H. Jerome Keisler, emeritus professor of mathematics at the University of Wisconsin, Madison.
In 1967, Keisler introduced what’s now called Keisler’s order, which seeks to classify mathematical theories on the basis of their complexity. He proposed a technique for measuring complexity and managed to prove that mathematical theories can be sorted into at least two classes: those that are minimally complex and those that are maximally complex. “It was a small starting point, but my feeling at that point was there would be infinitely many classes,” Keisler said.
It isn’t always obvious what it means for a theory to be complex. Much work in the field is motivated in part by a desire to understand that question. Keisler describes complexity as the range of things that can happen in a theory — and theories where more things can happen are more complex than theories where fewer things can happen.
A little more than a decade after Keisler introduced his order, Shelah published an influential book, which included an important chapter showing that there are naturally occurring jumps in complexity — dividing lines that distinguish more complex theories from less complex ones. After that, little progress was made on Keisler’s order for 30 years.

Saharon Shelah is a co-author of the new proof.
Yael Shelah
Then, in her 2009 doctoral thesis and other early papers, Malliaris reopened the work on Keisler’s order and provided new evidence for its power as a classification program. In 2011, she and Shelah started working together to better understand the structure of the order. One of their goals was to identify more of the properties that make a theory maximally complex according to Keisler’s criterion.
Malliaris and Shelah eyed two properties in particular. They already knew that the first one causes maximal complexity. They wanted to know whether the second one did as well. As their work progressed, they realized that this question was parallel to the question of whether p and t are equal. In 2016, Malliaris and Shelah published a 60-page paper that solved both problems: They proved that the two properties are equally complex (they both cause maximal complexity), and they proved that p equals t           
“Somehow everything lined up,” Malliaris said. “It’s a constellation of things that got solved.”
This past July, Malliaris and Shelah were awarded the Hausdorff medal, one of the top prizes in set theory. The honor reflects the surprising, and surprisingly powerful, nature of their proof. Most mathematicians had expected that p was less than t, and that a proof of that inequality would be impossible within the framework of set theory. Malliaris and Shelah proved that the two infinities are equal. Their work also revealed that the relationship between p and t has much more depth to it than mathematicians had realized.
“I think people thought that if by chance the two cardinals were provably equal, the proof would maybe be surprising, but it would be some short, clever argument that doesn’t involve building any real machinery,” said Justin Moore, a mathematician at Cornell University who has published a brief overview of Malliaris and Shelah’s proof.

1.     Instead, Malliaris and Shelah proved that p and t are equal by cutting a path between model theory and set theory that is already opening new frontiers of research in both fields. Their work also finally puts to rest a problem that mathematicians had hoped would help settle the continuum hypothesis. Still, the overwhelming feeling among experts is that this apparently unresolvable proposition is false: While infinity is strange in many ways, it would be almost too strange if there weren’t many more sizes of it than the ones we’ve already found.

2.        Is Infinity Real?

Clarification: On September 12, this article was revised to clarify that mathematicians in the first half of the 20th century wondered if the continuum hypothesis was true. As the article states, the question was largely put to rest with the work of Paul Cohen.

sábado, 9 de septiembre de 2017


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lunes, 28 de agosto de 2017

How Neanderthals Gave Us Secret Power

Interbreeding with our fellow hominins appears to have helped humans survive harsh climates. 
·    MAY 31, 2016
Native Tibetans make use of a gene derived from Denisovans to stay healthy at high altitudes.

Final del formulario
Early human history was a promiscuous affair. As modern humans began to spread out of Africa roughly 50,000 years ago, they encountered other species that looked remarkably like them—the Neanderthals and Denisovans, two groups of archaic humans that shared an ancestor with us roughly 600,000 years earlier. This motley mix of humans coexisted in Europe for at least 2,500 years, and we now know that they interbred, leaving a lasting legacy in our DNA. 

The DNA of non-Africans is made up of roughly 1 to 2 percent Neanderthal DNA, and some Asian and Oceanic island populations have as much as 6 percent Denisovan DNA.
Over the last few years, scientists have dug deeper into the Neanderthal and Denisovan sections of our genomes and come to a surprising conclusion. Certain Neanderthal and Denisovan genes seem to have swept through the modern human population—one variant, for example, is present in 70 percent of Europeans—suggesting that these genes brought great advantage to their bearers and spread rapidly.

“In some spots of our genome, we are more Neanderthal than human,” said Joshua Akey, a geneticist at the University of Washington. “It seems pretty clear that at least some of the sequences we inherited from archaic hominins were adaptive, that they helped us survive and reproduce.”

But what, exactly, do these fragments of Neanderthal and Denisovan DNA do? What survival advantage did they confer on our ancestors? Scientists are starting to pick up hints. Some of these genes are tied to our immune system, to our skin and hair, and perhaps to our metabolism and tolerance for cold weather, all of which might have helped emigrating humans survive in new lands.

“What allowed us to survive came from other species,” said Rasmus Nielsen, an evolutionary biologist at the University of California, Berkeley. “It’s not just noise, it’s a very important substantial part of who we are.

Illustration by Lucy Reading-Ikkanda for Quanta Magazine, based on a map by Sriram Sankararaman.
* * *
The Tibetan plateau is a vast stretch of high-altitude real estate isolated by massive mountain ranges. The scant oxygen at 14,000 feet—roughly 40 percent lower than the concentrations at sea level—makes it a harsh environment. People who move there suffer higher rates of miscarriage, blood clots, and stroke on account of the extra red blood cells their bodies produce to feed oxygen-starved tissue. 

Native Tibetans, however, manage just fine. Despite the meager air, they don’t make as many red blood cells as the rest of us would at those altitudes, which helps to protect their health.

In 2010, scientists discovered that Tibetans owe their tolerance of low oxygen levels in part to an unusual variant in a gene known as EPAS1. About 90 percent of the Tibetan population and a smattering of Han Chinese (who share a recent ancestor with Tibetans) carry the high-altitude variant. But it’s completely absent from a database of 1,000 human genomes from other populations.

The unique gene then flourished in those who lived at high altitudes and faded away in descendants who colonized less harsh environments.
In 2014, Nielsen and colleagues found that Tibetans or their ancestors likely acquired the unusual DNA sequence from Denisovans, a group of early humans first described in 2010 that are more closely related to Neanderthals than to us. 

The unique gene then flourished in those who lived at high altitudes and faded away in descendants who colonized less harsh environments. “That’s one of the most clear-cut examples of how [interbreeding] can lead to adaptation,” said Sriram Sankararaman, a geneticist and computer scientist at the University of California, Los Angeles.

The idea that closely related species can benefit from interbreeding, known in evolutionary terms as adaptive introgression, is not a new one. As a species expands into a new territory, it grapples with a whole new set of challenges—different climate, food, predators, and pathogens. 

Species can adapt through traditional natural selection, in which spontaneous mutations that happen to be helpful gradually spread through the population. But such mutations strike rarely, making it a very slow process. A more expedient option is to mate with species that have already adapted to the region and co-opt some of their helpful DNA. (Species are traditionally defined by their inability to mate with one another, but closely related species often interbreed.)

This phenomenon has been well documented in a number of species, including mice that adopted other species’ tolerance to pesticides and butterflies that appropriated other species’ wing patterning. But it was difficult to study adaptive introgression in humans until the first Neanderthal genome was sequenced in 2010, providing scientists with hominin DNA to compare to our own.

Neanderthals and Denisovans would have been a good source of helpful DNA for our ancestors. They had lived in Europe and Asia for hundreds of thousands of years—enough time to adjust to the cold climate, weak sun and local microbes. “What better way to quickly adapt than to pick up a gene variant from a population that had probably already been there for 300,000 years?” Akey said. 

Indeed, the Neanderthal and Denisovan genes with the greatest signs of selection in the modern human genome “largely have to do with how humans interact with the environment,” he said.

Illustration by Lucy Reading-Ikkanda for Quanta Magazine, based on a map by Sriram Sankararaman.

To find these adaptive segments, scientists search the genomes of contemporary humans for regions of archaic DNA that are either more common or longer than expected. Over time, useless pieces of Neanderthal DNA—those that don’t help the carrier—are likely to be lost. And long sections of archaic DNA are likely to be split into smaller segments unless there is selective pressure to keep them intact.

In 2014, two groups, one led by Akey and the other by David Reich, a geneticist at Harvard Medical School, independently published genetic maps that charted where in our genomes Neanderthal DNA is most likely to be found. To Akey’s surprise, both maps found that the most common adaptive Neanderthal-derived genes are those linked to skin and hair growth. One of the most striking examples is a gene called BNC2, which is linked to skin pigmentation and freckling in Europeans. 

Nearly 70 percent of Europeans carry the Neanderthal version.
Scientists surmise that BNC2 and other skin genes helped modern humans adapt to northern climates, but it’s not clear exactly how. Skin can have many functions, any one of which might have been helpful. “Maybe skin pigmentation, or wound healing, or pathogen defense, or how much water loss you have in an environment, making you more or less susceptible to dehydration,” Akey said. “So many potential things could be driving this—we don’t know what differences were most important.”
* * *
One of the deadliest foes that modern humans had to fight as they ventured into new territories was also the smallest—novel infectious diseases for which they had no immunity. “Pathogens are one of the strongest selective forces out there,” said Janet Kelso, a bioinformatician at the Max Planck Institute for Evolutionary Anthropology in Leipzig, Germany.

Earlier this year, Kelso and collaborators identified a large stretch of Neanderthal DNA—143,000 DNA base-pairs long—that may have played a key role in helping modern humans fight off disease. The region spans three different genes that are part of the innate immune system, a molecular surveillance system that forms the first line of defense against pathogens. These genes produce proteins called toll-like receptors, which help immune cells detect foreign invaders and trigger the immune system to attack.

Modern humans can have several different versions of this stretch of DNA. But at least three of the variants appear to have come from archaic humans—two from Neanderthals and one from Denisovans. 

To figure out what those variants do, Kelso’s team scoured public databases housing reams of genomic and health data. They found that people carrying one of the Neanderthal variants are less likely to be infected with H. pylori, a microbe that causes ulcers, but more likely to suffer from common allergies such as hay fever.

Kelso speculates that this variant might have boosted early humans’ resistance to different kinds of bacteria. That would have helped modern humans as they colonized new territories. Yet this added resistance came at a price. “The trade-off for that was a more sensitive immune system that was more sensitive to nonpathogenic allergens,” said Kelso. 

But she was careful to point out that this is just a theory. “At this point, we can hypothesize a lot, but we don’t know exactly how this is working.”

Most of the Neanderthal and Denisovan genes found in the modern genome are more mysterious. Scientists have only a vague idea of what these genes do, let alone how the Neanderthal or Denisovan version might have helped our ancestors. “It’s important to understand the biology of these genes better, to understand what selective pressures were driving the changes we see in present-day populations,” Akey said.

A number of studies like Kelso’s are now under way, trying to link Neanderthal and Denisovan variants frequently found in contemporary humans with specific traits, such as body-fat distribution, metabolism or other factors. 

One study of roughly 28,000 people of European descent, published in Science in February, matched archaic gene variants with data from electronic health records. Overall, Neanderthal variants are linked to higher risk of neurological and psychiatric disorders and lower risk of digestive problems. (That study didn’t focus on adaptive DNA, so it’s unclear how the segments of archaic DNA that show signs of selection affect us today.)

At present, much of the data available for such studies is weighted toward medical problems—most of these databases were designed to find genes linked to diseases such as diabetes or schizophrenia. But a few, such as the U.K. Biobank, are much broader, storing information on participants’ vision, cognitive test scores, mental health assessments, lung capacity and fitness. 

Direct-to-consumer genetics companies also have large, diverse data sets. For example, 23andMe analyzes users’ genetics for clues about ancestry, health risk and other sometimes bizarre traits, such as whether they have a sweet tooth or a unibrow.

Of course, not all the DNA we got from Neanderthals and Denisovans was good. The majority was probably detrimental. Indeed, we tend to have less Neanderthal DNA near genes, suggesting that it was weeded out by natural selection over time. Researchers are very interested in these parts of our genomes where archaic DNA is conspicuously absent. 

“There are some really big places in the genome with no Neanderthal or Denisovan ancestry as far as we can see—some process is purging the archaic material from these regions,” Sankararaman said. “Perhaps they are functionally important for modern humans.”