Harvard professor Lauren K. Williams has multiple dimensions: teacher, parent and active Cambridge community member. As of this month, she can add to that list: “genius.”
Williams is one of this year’s MacArthur Foundation Fellows, one of 22 “genius grant” recipients. She was recognized for her work on algebraic combinatorics, an area of math that has found surprising applications to topics ranging from shallow water waves to traffic flow to the scattering of elementary particles.
A focus of her research is on the properties of mathematical objects including the positive Grassmannian, which keeps track of all planes within a higher dimensional space. Just as a cube can be decomposed into six two-dimensional faces (squares), 12 one-dimensional faces (edges) and eight zero-dimensional faces (corners), a positive Grassmannian can be decomposed into pieces of different dimension.
As a mathematics graduate student at the Massachusetts Institute of Technology, “the first problem that I looked at was trying to count how many pieces the positive Grassmannian has of each dimension. And now bear in mind there are infinitely many different positive Grassmannians. So I was trying to solve infinitely many counting problems at once, but I came up with this formula that seemed sort of astonishingly simple,” Williams said. She wrote down the formula, then needed nine months to prove it. “The answer looked much more simple and elegant than it seemed to have any right.”
A cross-disciplinary catalyst
After Williams’ work was published in the journal Advances in Mathematics in 2005, academics in unrelated fields found they could use her formula to make breakthroughs in their own research.
“Maybe six months or a year later, I saw a paper of a French mathematician named Sylvie Corteel, and her paper said that one could use my formula to compute probabilities in a model for traffic,” Williams recalled. “That was a big shock.”
Physicists have also used it to tackle complex problems. Mathematical physicist Yuji Kodama found connections to Williams’ work on the Grassmannian to his own on shallow water waves.
One child’s question was: ‘Why is math so fun?’ I just loved that. I didn’t even know how to answer.” Lauren K. Williams
“I found his paper incomprehensible,” she said, but she reached out and invited him to the University of California at Berkeley, where she was on the faculty at the time. During the week Kodama spent there, “we were able to start to understand each other,” she said. Together, they discovered that graphs used to describe how waves interact lined up perfectly with the ‘plabic graphs’ used to describe `pieces’ or `cells’ of the positive Grassmannian diagram. Other forays into physics include Williams’ recent use of the positive Grassmannian to develop proofs for the amplituhedron, a geometric shape that can help predict the outcomes for when elementary particles collide.
“Some kind of ruse”
Williams said she was caught completely off-guard by the MacArthur fellowship. “They actually reached out to me to schedule a phone call under some kind of ruse,” Williams said, telling her they wanted feedback on a different nominee. Instead, they gave her the news. “I was just shocked, just trying to make sure I was really awake,” she said.
The nomination process is anonymous, and Williams had no inkling she might be a candidate. She still does not know who put her up for contention.
The MacArthur grant, worth $800,000 paid out over five years, came at a crucial time for Williams. Her research has been funded by the National Science Foundation, but in May $191 million in foundation grants to Harvard were canceled by the Trump administration. While federal courts have since deemed the cancellation illegal and some Harvard researchers have had funds restored within the past month – including Williams just recently – for how long remains uncertain.
Williams is also keenly aware of the challenges women in mathematics face in particular. She was only the second woman mathematics professor to get tenure at Harvard, and the only woman in the department when she arrived in 2018. There are now two others, including Melanie Matchett Wood, who was awarded the MacArthur fellowship in 2022.
Almost half of her first-year seminar students this year are women, a larger share than in previous years. But Williams said “I’m a little bit worried about what is going to happen going forward, because the current climate is to remove all support for any special interest groups.”
Locally, she’s involved with Girls Angle, a Cambridge-area math club for middle- and high-school aged girls, and Cambridge Math Circle, which serves grades 2-8. In September the Math Circle organized a lecture from her for the 6-and-up crowd, during which she used inflatable pool toys and other objects to illustrate her points and answered questions from the audience. To Williams, making sure students have access to fun, challenging math at a young age can spark a lifelong interest.
“Something that was really formative for me as a very young child is that I went to a public school system with a very strong math program. In fourth grade, they actually tested all the kids, and those of us who already knew the material, they moved us up to sixth grade math,” she said. “From a very early age, the school was supporting my mathematical development. They were testing us, and they were giving us stimulating, interesting math to learn. I worry that most schools do not do this.”
Williams encourages her pupils to consider pursuing math at the undergraduate and graduate level, noting plentiful career doors a math degree can open beyond academia. But aspiring mathematicians must also be driven by something deeper.
“People should do it because they’re drawn to the beauty of it. I had this Q&A with little kids after my public lecture a month ago,” she said. “One child’s question was: ‘Why is math so fun?’ I just loved that. I didn’t even know how to answer. But I think people should do math if they think it’s beautiful, if they think it’s fun.”
This story was updated to correct the name of the journal that published Williams’ work in 2005, that Williams reached out to Kodama, and to clarify that she only skipped a grade of math, not all subjects.
A stronger
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