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Approximately 540 million years ago, life rapidly diversified in an evolutionary burst — a biological “Big Bang” that witnessed the emergence of nearly every modern animal group. Scientists have long sought to determine what caused the Cambrian explosion, and to explain why animal life didn’t take this step at any point about a billion years earlier.

The most popular narrative puts oxygen front and center. The geological record shows a clear link, albeit an often subtle and complicated one, between rises in oxygen levels and early animal evolution. As *Quanta *reported earlier this month, many researchers argue that this suggests low oxygen availability had been holding greater complexity at bay — that greater amounts of oxygen were needed for energy-demanding processes like movement, predation and the development of novel body plans with intricate morphologies.

“It’s a very attractive, intuitive explanation,” said Nicholas Butterfield, a paleobiologist at the University of Cambridge. “And it’s wrong.”

Butterfield — “a lone voice in the wilderness,” he calls himself — has what many others might consider an unusual take on the oxygen story. He’s essentially turned the idea on its head. According to his theory, changes in environmental conditions weren’t the cause, but rather the consequence, of animals migrating and perturbing their surroundings. “We have to appreciate that animals have a powerful, powerful impact on the carbon cycle and on how everything goes around,” he said.

In a paper published in the January issue of *Geobiology*, Butterfield braided fluid dynamics and ecology to present his case for animals driving oxygenation instead of the other way around. First, he argued, if there was enough oxygen to power unicellular eukaryotes 1.6 billion years ago — which was indeed the case — then there would have been enough to run a whole assortment of animals. He believes early multicellular organisms would have consisted of flagellated cells moving in unison, collectively whipping their appendages to create currents that would have made it easier for them to obtain dissolved oxygen. “I make the case that if there’s enough oxygen to run a single-celled eukaryote, there’s enough oxygen to run a fish,” Butterfield said. “So oxygen limitation cannot be invoked to explain the billion-year delay in the evolution of animals.”

Instead, his hypothesis focuses on diurnal vertical migration, a daily process during which sundry organisms, ranging in size and complexity from zooplankton and sponges to fish and squids, migrate between shallow and deeper waters to find food and avoid predators. By feeding up above and metabolizing down below, the animals scrub and help ventilate the ocean, raising oxygen concentrations at the surface while driving anoxic regions to greater depths. This redistribution of oxygen would also have improved the transparency of the water column, allowing light to penetrate farther down and escalating predators’ reliance on vision at deeper and deeper levels when hunting. The subsequent evolution of larger, deeper-diving visual predators would then have pushed the “oxygen minimum zones” to even lower depths, creating a feedback loop.

Eventually, this cascading interplay between animals’ inadvertent re-engineering of ocean structure and their adaptive responses to those changes reached a tipping point. “The system went critical,” in Butterfield’s words, resulting in the sudden eruption of animal diversity and complexity during the Cambrian.

The delayed appearance of animals in the ocean was therefore not caused by a lack of oxygen, according to Butterfield, but rather because blind Darwinian evolution needed time to arrive at that tipping point. “The gene regulatory network to build an animal is the most complex algorithm that evolution has ever produced,” he said. “And it’s only ever happened once, it’s only ever happened once in land plants,” which he points out are the only other lineage of organisms to have derived differentiated tissues, organs and organ systems. “And that took even longer. It followed the evolution of animals by another 100 million years.”

Not everyone is convinced. Timothy Lyons, a geologist at the University of California, Riverside, thinks that multiple independent lines of evidence point to oxygen in the environment as the trigger for the evolutionary cascade Butterfield describes. For example, most major extinction events were tied to low oxygen, he said, and oxygen levels fluctuated throughout the time leading up to the Cambrian (as well as in later eras). Those periods of lower oxygen “planted the seeds” for innovations that allowed certain organisms to take advantage of oxygen more efficiently. When oxygen later rebounded, natural selection would have favored those adaptations and enabled animals with them to blossom and diversify.

Moreover, Lyons and Charles Diamond, a graduate student in Lyons’ lab, find key pieces of evidence to counter Butterfield’s story. They have identified other conditions, not attributable to animals, that would have caused increases in oxygen at exactly the same time as the rapid animal diversification events Butterfield cited. An enormous variety of large fish emerged later, for example, during the Devonian Period (the “Age of Fishes” that started about 419 million years ago), when trees and other vascular plants arose on the continents. Those land plants by themselves greatly increased the amount of oxygen in both the atmosphere and the ocean, Lyons and Diamond said. The timing of the two events casts doubt on Butterfield’s claim that it was newly evolved fish causing the rise in oxygen and not vice versa, Diamond added. “Otherwise, it would have been too much of a coincidence.”

Butterfield disagrees. “Yes, the rise of vascular land plants may have impacted oxygen availability. That’s the standard textbook view,” he wrote in an email. “But it’s based on a bunch of unestablished assumptions” — such as that atmospheric oxygen was previously too low (he thinks the proxy-based measurements of atmospheric oxygen are intrinsically flawed, and that oxygen could have been much higher than estimated) and that it would have been the limiting factor in how large fish evolved. “I am arguing that none of these hold water.”

Notwithstanding those disagreements, Lyons and Diamond do find Butterfield’s ideas — that the evolution of such great complexity was the result of intrinsic biological development, what Butterfield has called an “evolutionary random walk” — to be much more feasible in the case of land plants. Their emergence “couldn’t have been a response to oxygen or carbon dioxide,” Lyons said. But he and Diamond don’t think that explanation can be fairly applied to animals.

For now, Butterfield wants to obtain further support for his theory by looking at how modern extinctions affect vertical migration. As the largest, deepest-diving predators are wiped out, the oxygen minimum zones should rise and precipitate further extinctions, he said. That’s something he can explore in modern oceans, as global climate change wreaks havoc on marine ecosystems.

As for what may have happened millions of years ago during the Cambrian — “Well, at this point the relationship between oxygen and animals is clear,” Lyons said, “but it goes back to the classic chicken-or-the-egg argument.”

Robert Langlands, who developed one of the most original insights of 20th-century mathematics, was named the winner of the 2018 Abel Prize at a ceremony in Norway this morning. The prize, which is modeled on the Nobel, is one of the highest honors in mathematics.

Langlands, 81, an emeritus professor at the Institute for Advanced Study in Princeton, New Jersey, is the progenitor of the “Langlands program,” which explores a deep connection between two pillars of modern mathematics: number theory, which studies arithmetic relationships between numbers, and analysis, which is an advanced form of calculus. The link has far-reaching consequences that mathematicians have used to answer centuries-old questions about the properties of prime numbers.

Langlands first articulated his vision for the program in 1967 — when he was 30 — in a letter to the famed mathematician André Weil. He opened the 17-page missive with a now-legendary stroke of modesty: “If you are willing to read it as pure speculation, I would appreciate that,” he wrote. “If not — I am sure you have a waste basket handy.”

Since then, generations of mathematicians have taken up and expanded upon his vision. The Langlands program now ranges over so many different fields that it is often referred to as the search for a “grand unified theory” of mathematics.

“It’s revolutionary, I think, as far as the history of mathematics is concerned,” said James Arthur, a mathematician at the University of Toronto and former student of Langlands’.

Mathematicians have always been interested in finding patterns in prime numbers — those numbers that are divisible only by one and themselves. Primes are like the atomic elements of number theory, the fundamental pieces from which the study of arithmetic is built. There are an infinite number of them, and they appear to be scattered randomly among all numbers. To find patterns in primes — like how frequently they occur (which is the subject of the famous Riemann hypothesis) — it’s necessary to relate them to something else. Seen correctly, the primes act like a cipher, which turns into a beautiful message when read through the right key.

“They look like random accidents, but especially through the Langlands program, it’s turning out they have an extremely complex structure that relates them to all sorts of other things,” said Arthur.

One question about the structure of primes is which primes can be expressed as a sum of two squares. The first few examples include:

5, a prime number that equals 2^{2} + 1^{2},

13, which equals 3^{2} + 2^{2}, and

29, which equals 5^{2} + 2^{2}.

In the 17th century, number theorists discovered that all primes that can be expressed as a sum of two squares share another property: They leave a remainder of 1 when divided by 4. The work began to reveal a hidden structure to the primes. Then in the late 18th century, Carl Friedrich Gauss generalized this surprising link, formulating a “reciprocity” law that linked certain primes (those that are a sum of two squares) to an identifying characteristic (when divided by 4, they leave a remainder of 1).

In his letter, Langlands proposed a vast extension of the kind of reciprocity law Gauss had discovered. Gauss’s work applied to quadratic equations — those with exponents no higher than the number 2. Langlands suggested that the prime numbers encoded in higher-degree equations (like cubic and quartic equations) should be in a reciprocity relationship with the far-off mathematical land of harmonic analysis, which grows out of calculus and is frequently used to solve problems in physics.

For example, scientists in the 19th century were surprised to discover that when they looked at starlight through a prism, they didn’t find a continuous spectrum of colors. Instead, the spectrum was interrupted here and there by black lines, now called absorption spectra, where the light was missing. Eventually the scientists realized that the missing light had been absorbed by elements in the stars. This discovery provided solid evidence that the stars and our planet are made from the same material.

At the same time, the spectral lines became objects of mathematical interest. The missing wavelengths gave a sequence of numbers — the frequencies of the absent light. Mathematicians could study those numbers through analysis. Or they could work on wholly new kinds of equations — inspired, perhaps, by questions in physics, but arising purely from analysis and geometry. Based on those new equations, they could study a parallel notion of absorption spectra.

The Langlands program relates prime number values of polynomial equations to spectra from the differential equations studied in analysis and geometry. It says that there should be a reciprocity relationship between the two. As a result, you should be able to characterize which prime numbers appear in specific settings by understanding which numbers appear in the corresponding spectra.

The two sets of numbers can’t be compared directly, though. They each have to be translated through different kinds of mathematical objects. In particular, Galois representations, which are based on primes, should pair with objects called automorphic forms, which contain the relevant spectra.

Today mathematicians working in the Langlands program are trying to prove that relationship and many other related conjectures. At the same time, they’re using Langlands-type connections to solve problems that would otherwise seem out of reach. The most celebrated result in this regard is Andrew Wiles’s proof in 1995 of Fermat’s Last Theorem. Wiles’s proof depended in part on exactly the type of relationship between number theory and analysis that Langlands had predicted decades earlier.

The Langlands program has expanded considerably over the years. Yet when you push aside all the complex machinery that’s been created to realize Langlands’ vision, you see that the whole massive enterprise remains motivated by some of the most basic of mathematical concerns.

“Understanding the properties of which primes occur in an equation basically amounts to a fundamental classification of the arithmetic world,” said Arthur.

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