Interesting article about Lie Groups

Math Has No God Particle

Jun. 6, 2017 at 11:01 AM

Ten years ago, Jeffrey Adams, a mathematician at the University of Maryland, made an appearance in The New York Times that prompted a series of angry emails. His correspondents all wanted to know one thing: “Who the hell do you think you are?”

Who Adams is is the leader of a cutting-edge mathematical research project called the Atlas of Lie Groups and Representations. Lie groups are named after Norwegian mathematician Sophus Lie (rhymes with “free,” not “fry”), who studied these crucial mathematical objects. Lie groups are used to map the inner machinery of multidimensional symmetrical objects, and they’re important because symmetry underpins far-flung mathematical concepts, from a third-grade number line to many-dimensional string theory. The Atlas project is a bona fide atlas of these objects, an exhaustive compendium of Lie group information, including tables of data about what they “look” like and what makes them tick. You’d think that cracking the code on these fundamental mathematical ideas would be a big deal. It is, but Adams would rather not dwell on it.

The success of the atlas project poses a tough math problem of a different kind: What should math’s relationship be with the broader, non-expert public? On the one hand, mathematicians in particular and scientists in general relish publicity. It allows them to trumpet good work, lobby for funding and inspire the next generation. On the other, in an ultra-specialized field such as math, publicity can twist finely constructed theorems, proofs and calculations beyond recognition.

In 2007, just before the angry emails started to roll in, the atlas group cleared an early hurdle in its quest, mapping an exotic and supersymmetric Lie group known as E8. They still had years of work before they could declare the atlas complete, yet the milestone was celebrated with a splashy press release from the American Institute of Mathematics explaining that the calculation, “if written out in tiny print, would cover an area the size of Manhattan.” It also provided a pretty picture of the “root system” of E8.

The combination of a superlative calculation and an eye-popping visualization was viral mathematical fuel. The New York Times wrote excitedly that the E8 calculation “may underlie the Theory of Everything that physicists seek to describe the universe.” (Everything! The universe!) E8 news bounced around the internet for months. “All hell broke loose,” Adams said. “We got this incredible tsunami of publicity, and it was only a sort of preliminary, intermediate result. Some people thought it was distasteful.”

Who the hell did he think he was?

“Mathematicians are extremely reluctant to publicize what they do,” Adams said. “The immediate reaction from 90 percent of mathematicians is, ‘It’s too hard, there’s no point in trying to write about this in the popular press.’” (Yet here we are.)

The atlas work, far from complete even amid the tsunami, continued apace. About two months ago — 15 years after it began — the project was finally completed. Adams and his colleagues released Version 1.0 of their atlas software.

This time around, however, there’s been no press release, no pretty picture, no city-size braggadocio, no New York Times story. Adams and his team haven’t trumpeted this latest accomplishment at all. When I reached him at his home, he summarized the milestone plainly, but proudly, in the jargon of his field: “We can now compute the Hermitian form on any irreducible representation.”

Raphaël Rouquier, a mathematician and Lie theorist at UCLA, echoed the ticklish relationship between mathematicians and the press. “There is a general feeling in the pure math community that popularizing mathematics is betraying mathematics,” Rouquier said. But he also argued for the importance of getting the word out. “I think there’s a need for mathematics to be represented in the press,” he said. “And I think we live in a society where people need to be more exposed to science. It’s good for politicians and readers.” The last few decades, up to and including the atlas, have been “an amazing chapter of mathematics,” he said.

Still, for those who do want to bullhorn their research, the difficulty of translation remains, especially compared to the other hard sciences. “We’re not trying to describe the real world,” Rouquier said.

Ah, but then should those of us in the real world care? The hope may be that other scientists, and the rest of us who don’t care about 248-dimensional objects, may profit from this math, but there’s no guarantee. Pure mathematicians do their work with no expectation of concrete application, although applications do have a way of presenting themselves when one least expects it — and often after the mathematician is long dead. In the case of the atlas, symmetry plays an important role in math, but also in physics and biology and astronomy. “There’s always symmetry underlying various systems,” Adams said. “Generally, mathematicians can’t say that what we’re working on is going to be good for society or something,” he said. “Our strong belief is that over time, as we learn these things, we wind up finding applications.”

David Vogan, who’s a mathematician at MIT and was involved with the atlas project, described academic mathematics as a garden. There are showy, flowery fields like number theory. Its beautiful problems and elegant results, such as the prime gap or Fermat’s last theorem, are math’s orchids. There are also the tomatoes — the things you can eat out of the garden, the practical yield. These disciplines, like Fourier analysis with its concrete applications to signal processing of audio, radio and light waves, are businesslike. And then there are the disciplines, often unheralded, that keep the rest of the garden growing — the hoes, the sprinklers. Lie groups, their representations and the atlas project are an example.

“Representation theory,” the field of the atlas group’s research, “is the fertilizer or the rose trellis, depending on the day of the week,” Vogan said.

Even when researchers do want their work shared widely, why don’t we read more about the fuel that makes math grow? “The physicists tell exciting stories,” Vogan said. “In some ways, this is a failure of mathematicians to tell exciting stories.” The physicists also have better names. Black hole and God particle quicken the pulse somewhat more than “irreducible unitary representation.”

The asymmetry in storytelling between math and the other sciences may also be because the research has different start-up costs. You need billions of dollars to build an enormous tunnel to house a particle accelerator to discover evidence of the God particle, also known as the Higgs boson. A good story may secure you coverage, enthusiasm and, if you’re lucky, lots of cash. To map Lie groups, Vogan said, you just need a teaching load light enough to put in extra work on the weekends: “We can do these things with small amounts of money.”

Even after the release of Version 1.0, and even in relative silence, the group has no intention of stopping, and it’ll continue to unravel the symmetric secrets of the mathematical universe. “It’s never complete,” Adams said. “There’s much more to do. I’ll die before I’m happy with everything.”

Source: Math Has No God Particle | FiveThirtyEight


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