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Wei Ho Is Drawn to Algebra, Geometry and the Human Side of Math | Quanta Magazine

https://www.quantamagazine.org/wei-ho-is-drawn-to-algebra-geometry-and-the-human-side-of-math-20221122/

Wei Ho, the first director of the Women and Mathematics program at the Institute for Advanced Study, combines algebra and geometry in her work on an ancient class of curves.

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Introduction

Like many people who would go on to become mathematicians, Wei Ho grew up competing in math contests. In eighth grade, she won the Mathcounts state competition in Wisconsin, and her team took third place at nationals.

Unlike many future mathematicians, she wasn’t sure she ever wanted to become one.

“I wanted to do everything, all the time,” Ho said. “I took ballet very seriously until early high school. I edited the literary magazine. I did debate and forensics. I played tennis and soccer and piano and violin.” By contrast, many successful mathematicians appeared to be obsessed with math to the exclusion of everything else. How could she, a person with numerous passions, compete with that level of focus?

Ultimately, Ho was drawn to the rigor of mathematics. She still enjoys ballet, reading novels and doing cryptic crossword puzzles, even as she helps to reinvent the mathematical machinery that underpins fundamental mathematical objects, such as polynomial equations, which have long-standing and perplexing open questions associated with them.

Ho studies familiar geometric objects, but she reformulates the questions to situate them in the realm of the rational numbers — numbers that can be written as fractions. “Then number theory starts to get mixed into all of this,” she said.

She is especially interested in elliptic curves, which are defined by a particular kind of polynomial equation that has applications in different branches of mathematics. Elliptic curves appear in analysis — broadly speaking, the study of continuous things, like the real numbers — and in algebra, which is about finding and defining precise mathematical structures. (Though their focus is different, analysis and algebra are divided more by sensibility than by a strict boundary, as there is plenty of overlap between them.)

Introduction

In a barrier-breaking preprint released in 2018, Ho and her collaborator Levent Alpöge of Harvard University discovered a new upper bound for the number of integer solutions to polynomials that define elliptic curves. Their technique draws upon the decades-old work of Louis Mordell, an American mathematician who emigrated to Britain in 1906. In their paper, Ho and Alpöge were able to glean new information about the distribution of these integer solutions that had evaded other teams studying similar problems.

Ho is spending the year (on leave from her faculty position at the University of Michigan) as a visiting professor at the Institute for Advanced Study, where she was recently named the first director of the IAS’s Women and Mathematics program. She is also a 2023 fellow of the American Mathematical Society and a research scholar at Princeton University.

She’s hopeful that directing the Women and Mathematics program will “at least help the community more, help more people, instead of just me being in my office doing math research by myself or with collaborators,” she said. “I can prove theorems, and maybe someday I can prove a theorem that in 100 years will matter. Maybe, maybe not. But I felt like I wasn’t making enough impact on the world or on people around me.”

Quanta spoke with Ho in a series of videoconferences. The interviews have been condensed and edited for clarity.

How would you describe the way you do mathematics?

Sometimes mathematicians divide ourselves into algebraic and analytic people. The math I do touches both sides, but at heart, I am an algebraist, though I’m geometric in the way I think. I often tend to view algebra and geometry as essentially the same.

That’s not quite accurate, but basically since the work of Descartes and especially in the last century, the two subjects have become really close. There is a rather precise dictionary that can, in some situations, help translate a geometric picture to algebraic consequences.

In my own case, the geometric picture often helps formulate statements and conjectures and give intuition, but then we translate them to algebra when writing. It’s easier to detect mistakes as algebra is typically more rigorous. It can also be easier to use algebra when geometry gets too hard to visualize.

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