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Magic and symmetry in mathematics

by Angela Herring

We live in a three-​​dimensional world. Despite the many ben­e­fits this presents, it also makes for a com­pli­cated math problem, according to North­eastern asso­ciate pro­fessor of math­e­matics Ivan Loseu. The best a path to a solu­tion, he said, is reducing the number of vari­ables we’re dealing with.

Con­sider the Earth moving around the sun. Three vari­ables are needed to describe the posi­tion of the Earth because the motion occurs in this three-​​dimensional space. Newton’s laws in physics allow you to reduce the number of vari­ables even fur­ther to two, since the Earth never leaves a cer­tain planeBut, hey, one vari­able is even better than two. That’s why physi­cists use the prop­er­ties of grav­i­ta­tional force to track the Earth’s ellip­tical trajectory.

“The for­mula for gravity depends only on the dis­tance between the sun and Earth,” Loseu said. “You rotate the pic­ture, but the phys­ical law remains the same.”

The reason this problem can be solved using only one vari­able, he said, can be described in one word: symmetry.

“A sym­metry is any trans­for­ma­tion that pre­serves your problem,” Loseu explained. The sym­metry people typ­i­cally imagine involves reflecting an image over a single plane to reveal the exact same image—like looking in a mirror. But that’s only one type of sym­metry. There are plenty of others. For instance, rota­tional sym­metry describes the fact that rotating an object—say the Earth’s orbital pattern—around an axis doesn’t change its properties.

Loseu explained that sym­me­tries allow for reducing the dimen­sion of a system because they can be used to pro­duce pre­served quan­ti­ties; in other words, prop­er­ties that do not change no matter how much the system changes.

This idea of using sym­me­tries to reduce the number of vari­ables is the crit­ical ele­ment in Loseu’s research toolbox. He uses it not to solve prob­lems in physics, but rather to solve prob­lems in rep­re­sen­ta­tion theory, a sophis­ti­cated branch of algebra. He was recently named a Sloan 2014 Research Fellow for his con­tri­bu­tions to this field.

And just as num­bers are used in algebra, so too are symmetries.

“Take two sym­met­rical trans­for­ma­tions, apply them con­se­quently, and the com­po­si­tion of the two is again a sym­metry,” Loseu explained.

The more sym­metric a system, he con­tinued, the easier the system is to solve. There­fore, iden­ti­fying sym­me­tries can help sim­plify a problem and trans­form it from an unsolv­able one to a solv­able one.

This idea serves as the foun­da­tion for what he’ll be focusing on in the first year of his two-​​year fel­low­ship, pre­sented by the Alfred P. Sloan Foun­da­tion. “If all of this is a tree, I’ve told you about only its roots,” he said, noting that a mole may think it has a full pic­ture of an oak or a maple, but until it pops its head through the soil, its per­spec­tive is limited.

Loseu’s interest in math­e­matics took shape in ele­men­tary school in Belarus. His par­ents were engi­neers whose work revolved around the applied sci­ences, and he often played with the many math books and cal­cu­la­tors he could find around the house. He quickly learned that math­e­matics was “the thing I loved most of all.”

As an under­grad­uate stu­dent at Belaru­sian State Uni­ver­sity, Loseu ini­tially thought he would pursue work in applied math­e­matics, but the field didn’t retain the beauty that he appre­ci­ated about pure mathematics.

“Any sci­en­tific dis­covery involves some kind of magic,” he said. That is, var­ious pieces that may seem to be com­pletely unre­lated even­tu­ally start to fit together through the fruits of one’s labor. “Since pure math is pure, all this magic is much more clearly seen.

Originally published in news@Northeastern on March 11, 2014.

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