“And we’ve used this moonshine to show the mathematical usefulness of the O’Nan pariah group in a way that moves it from theory to reality. It turns out that the O’Nan group knows deep information about elliptic curves.”
“We’ve shown that the O’Nan group, a very large pariah group, actually organizes elliptic curves in a beautiful and systematic way,” Duncan says. Elliptic curves may sound esoteric, but they are part of our day-to-day lives.
“So, in the simplest of terms, it’s like a doughnut that you eat, that may have sprinkles on it. The whole game in the math of elliptic curves is determining whether the doughnut has sprinkles and, if so, where exactly the sprinkles are placed.”
In 2015, a group of mathematicians – including Duncan and Ono – presented proof of the Umbral Moonshine Conjecture, which revealed 23 other moonshines, or mysterious connections between the dimensions of symmetry groups and coefficients of special functions.
The classification of the building blocks of groups is gathered in the ATLAS of Finite Groups, published in 1985.
“We found the O’Nan group living in nature. Our theorem shows that it’s connected to elliptic curves, and whenever you find a correspondence between two objects that are seemingly not related, it opens the door to learning more about those objects.”