# The Atlas of Lie Groups

(written by lawrence krubner, however indented passages are often quotes). You can contact lawrence at: lawrence@krubner.com

SourceAdams 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.

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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.”

August 18, 2018 4:59 am

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