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

A lot of people go on to become mathematicians because of their participation in math contests. She won the mathcounts state competition in Wisconsin in eighth grade and her team took third place at nationals.

She wasn't sure if she wanted to be a mathematician.

Ho wanted to do everything. Ballet was very important to me until early high school. The magazine was edited by me. I did a lot of research. I played a lot of instruments. Many successful mathematicians were focused on math to the exclusion of other things. How could a person with so many interests compete with that?

The rigor of mathematics was what drew Ho to it. She still enjoys ballet, reading novels and doing cryptic crosswords, even as she helps to redesign the mathematical machinery that underpins fundamental mathematical objects, such as polynomial equations, which have long-standing and perplexing open questions associated with them.

She reformulates the questions to make them seem like rational numbers, which can be written as fractions. She said that number theory gets mixed into all of this.

She's interested in elliptic curves, which are defined by a particular type of equation that has applications in other branches of mathematics. The study of continuous things, like the real numbers, is one of the things ellipses can be used for. Even though their focus is different, analysis and algebra are not separated by a strict boundary.

Ho and Alpge discovered a new upper bound for the number of solutions to the elliptic curves. Louis Mordell, an American mathematician who migrated to Britain in 1906, was the inspiration for their method. In their paper, Ho and Alpge were able to get new information about the distribution of the solutions that had been avoided by other teams.

Ho is taking a year off from her faculty position at the University of Michigan to teach at the Institute for Advanced Study, where she was recently named the first director of the women and mathematics program. She is a fellow of the American Mathematical Society.

She hopes that directing the Women and Mathematics program will help the community more, help more people, rather than just her being in her office doing math research. Someday I will be able to prove a Theorem that will matter in 100 years. Maybe it isn't. I felt like I wasn't doing enough to make a difference.

In a series of video conferences, Ho was talked to byQuanta. Interviews have been edited for clarity.

How would you describe the way you do mathematics?

Sometimes mathematicians are divided into two groups. I am an algebraist, though I am geometric in the way that I think. I tend to see geometry and algebra in the same way.

The two subjects have become very close since the work of Descartes. In some cases, a dictionary can help translate a geometric picture to a more complex one.

The geometric picture can be used to help formulate statements and give intuition, but it can also be used to translate them to math. It is easier to detect mistakes inalgebra is more difficult. When geometry gets too hard to visualize, it can be easier to use math.

What ideas have you been focusing on in your recent work?

A lot of my work is related to elliptic curves, which are very natural objects in number theory and math.

It shouldn't be easy to have a solution of these equations. Almost all curves should have no solution. It is very difficult to show that.

We looked at the distribution of the number of points. We used a classical construction from a book. We can give an upper bound on the number of points. Others have given higher bounds. There is a bound that is easy to state.

What role did Mordell’s earlier work play in your recent result?

There are points on elliptic curves. It can be related to something else that we can study.

In math, we have to find a proxy to understand an object in order to understand it. Sometimes, that proxy is correct. Sometimes it doesn't have anything. We are able to access it.

When did you decide to focus on mathematics?

There wasn't a tipping point for me. I feel like if things had been different, I would have been happy in many fields. Most mathematicians wouldn't say that, because they like to talk about how passionate they are about math and how they can't think about anything else. I don't believe that's true.

I'm interested in a lot of things. I might have become a mathematician because I was frustrated with the lack of rigor in other areas. I was taught to think like a mathematician when I was a child. I was learning logical reasoning from playing math games with my father. I wanted to know what was true.

I didn't know if I would be a good mathematician.

What's the reason?

I didn't know a lot of math people who were like me. These words are used about role models. I didn't see enough asian american women.

I didn't see a lot of people who were passionate about other things. It made me think a lot. If I don't spend enough time thinking about math, how can I succeed in mathematics? That was what I observed. I thought my peers and people older than me were different in their approach to math. I didn't think it was possible to pursue a career where I wouldn't be like that. I would be interested in other things.

The human aspect isn't something that other people care about as much. I was worried that part of me would be bad at math.

There is a weeklong workshop for women at different career stages. There is a supportive environment in which to learn mathematics.

A lot of undergrads don't know that they want to pursue math, so they meet senior mathematicians and get mentoring all the way up. They can talk to people at different stages of their careers. I don't think there are many programs that are focused on a specific field.

The program is calledPatterns in Integers There will be a lot of people in this area. People from different careers are brought in to meet each other.

Older graduate students who are already working in this area can get a chance to work with junior and senior faculty for a week.

You’re also involved in the Stacks project, which is an extensive online resource. What’s unique about it?

The amount and accessibility of it. If you print it out, it will be more than 7,000 pages. The Columbia University mathematician wrote almost all of it. It is a well-written resource for the study ofgeometry. He has done a great job for the community.

It grows every couple of weeks. It is a good reference for a lot of things. You would need to look at 20 textbooks for a large amount of the geometry covered in it.

Things can be edited and added. They will be caught if they make a mistake.

The tag system is intriguing. Even though this document is growing, you can always reference a specific tag. There are a lot of permanent tags for certain results. The entire back end has been used in other projects. People have adapted it.

The problem is that he won't be able to keep writing this. Someday, we need other people to be more involved in this.

What role do your workshops play in the Stacks project?

Getting younger people involved is the goal. They are writing bits and pieces that will eventually make their way into it. The website needs to be moderated carefully so that it remains correct and high quality as a resource. It takes a lot of work for things to be put into it. It can't be like the internet encyclopedia. If you want this to work, you have to experience that.

There are ways to get more people involved in the Stacks project. Mentoring is being brought in to work on projects with graduate students. They're learning some geometry. Then they put it in writing.

We published a volume of articles that we hope will eventually make their way into the Stacks project.

If enough people get involved, the Stacks project will continue to have a huge impact for hundreds of years.

There is a simple geometry behind brownies bake offs and equal areas.