*By Paul Gabrielsen, science writer, University Marketing & Communications*

Christopher Hacon was on vacation in Italy, with a poor internet connection, when he noticed that several prominent mathematicians were trying to get in touch with him. “I was getting the feeling that something important was happening,” he says.

He was right. They were calling to tell Hacon that he had won the 2018 Breakthrough Prize in Mathematics. He had won several prestigious and prominent awards previously, but this one was different. For one thing, none of the previous awards ceremonies had been hosted by Morgan Freeman.

On Dec. 3, Hacon and his wife Aleksandra, a lecturer in math at the U, attended the star-studded, Hollywood-style Breakthrough Prize presentation in Mountain View, California. He joined 19 other scientists, including Hacon’s co-laureate, James McKernan, honored for achievements in life science, physics and math. Hacon and McKernan shared the $3 million award. He donned a tuxedo and he and his wife walked the red carpet as part of the ceremony. “It’s a very unusual situation for a mathematician,” Hacon says.

Hacon was born in England, but moved with his family to Italy at age 3 where his father, mathematician Derek Hacon, began work as a postdoctoral scholar. Hacon says he saw his father’s mathematical research, but was initially interested in physics and engineering. But admission to a top mathematics program at the Scuola Normale Superiore in Pisa, Italy, set him on the path to his current academic career.

He first came to the University of Utah as a postdoctoral scholar in 1998 to work with János Kollár, a lauded researcher now at Princeton University. After two years as a professor at the University of California, Riverside, Hacon came back to the U in 2002 and has been here ever since. He’s now a distinguished professor and holder of the McMinn Presidential Chair in Mathematics. He’s won a slew of awards, including Italy’s highest mathematics honor. The Breakthrough Prize is just the latest entry on Hacon’s impressive resume.

So, what’s all the fuss about? Hacon’s work is based in subjects that nearly everyone encounters in school – algebra and geometry. As you may (or may not recall) from high school, algebra concerns solving polynomial equations – like y = 2x^{2} + 3x + 2, or x^{2} + y^{2} = z^{2}. Algebra teaches how to find the solutions for these equations. “Polynomials are the easiest equations you can write,” Hacon says. “You just need to add and multiply.”

As you may also recall if you owned one of those bulky graphing calculators, those polynomial equations could also be converted into shapes. The first equation describes a parabola (or arch) shape in two dimensions, while the second describes two three dimensional cones. If you continue to add variables to the equation, you continue to add dimensions to the problem and solving the problem becomes much more complex, as do the geometric shapes that the polynomials describe.

That’s where Hacon comes in. Throughout the 20th century, mathematicians in Hacon’s field of algebraic geometry asked what range of shapes were possible in higher and higher dimensions. Some of these shapes can only exist in more than three dimensions. Algebraic geometers supposed that, while there are an infinite number of solutions to high-dimension polynomials, the set of all possible shapes can be enumerated. In 2009, Hacon and McKernan proved that supposition correct.

“The number of solutions is infinite but you can organize them in a way that you can understand them,” Hacon says. Picture a donut shape, he says, with two or three or more holes instead of one. The size of each of the holes can vary, but the general shape is the same. “The theorem we proved is that there are finitely many possibilities of these shapes in any given dimension, and for any given volume,” he says.

His work resonated throughout mathematics, because geometry and algebra touch so many other fields. “Polynomials appear in almost every branch of mathematics and many of the sciences,” he says. “They’re a common problem to have. It’s a subject of wide appeal.”

Hence the Breakthrough Prize. Hacon is back at work in Utah now. Exploration of the math of higher dimensions continues. “It will be hard to have another success like that one, he says. “But one can try.”