Harmonizing Algebra and Geometry: The Dance of a Brilliant Mathematician

Harmonizing Algebra and Geometry: Wei Ho, the first director of the Women and Mathematics program at the Institute for Advanced Study, blends geometry and algebra in her research on the oldest class of curves.

As with many who eventually become mathematical scientists, Wei Ho began her career in math competitions. In the eighth grade, she took home the Math counts state championship in Wisconsin and her team won third place at nationals.

In contrast to many mathematicians in the future She wasn’t certain she’d ever want to be one.

“I was a nerd who wanted to be everywhere every day,” Ho said. “I was very serious about ballet until the beginning of high school. Editorial duties included the magazine literary. I participated in debates and forensics. I played soccer and tennis as well as violin and piano.”

In contrast the majority of mathematicians who were successful appeared to be enthralled by math to the exclusion all other activities. What could she do as an individual with many interests be able to compete at that level of dedication?

The truth is that Ho was attracted to mathematics’ rigor. She is still a ballet fan and reading novels, as well as doing mysterious crossword puzzles in her work to improve the mathematical machineries that support the most fundamental mathematical concepts, like polynomial equations which are a recurring and enigmatizing open-ended questions associated with their use.

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Ho is a scientist who studies geometric objects of all kinds however she reframes the issues to place them within the rational numbers – numbers that are able to be written in fractions. “Then the concept of number theory begins to become a part these other subjects,” she said.

She is particularly attracted to elliptic curves that are defined by the polynomial equation, which has applications across different fields of mathematics. 

They are found in the field of analysis — generally that is the study of things that are continuous such as real numbers — as well as within algebra which is concerned with the creation and definition of precise mathematical patterns. (Though their subject matter is different the two are separated more by sensibility rather than the strict boundaries, because there’s a lot of overlap.)

In a preprint that broke the barriers in the year 2018, Ho and her co-worker Levent Alpoge from Harvard University discovered a new upper limit to the total number of polynomials that are the basis of an elliptic curve. 

The method they used is based on the long-standing research conducted by Louis Mordell, an American mathematician, who immigrated from the United States to Britain at the age of 1906. 

In their research, Ho and Alpoge were capable of gaining new knowledge regarding the distribution of these integer solutions that had been elusive to other teams that were studying similar problems.

Ho is presently spending the whole time (on off from her position as a professor in Michigan’s University of Michigan) as an adjunct faculty member in the Institute for Advanced Study, where she was recently appointed to be the inaugural director for the institute’s Women and Mathematics program. She’s also a fellow in 2023 from the American Mathematical Society and a research scholar at Princeton University.

She’s hoping that her Women and Mathematics program will “at minimum, assist the community more, assist more people, rather than only me in my office, doing maths research on my own or with colleagues,” she added. “I have the ability to demonstrate theories and maybe one day I’ll prove a theorem in 100 years it will be important. Maybe, but not sure. However, I felt that I wasn’t having enough impact on the world or the people around me.”

Quanta interviewed Ho in videoconferences. The interviews were cut down and edited to make them more clear.

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What would you say about your method of doing math?

Mathematicians can be divided into analytic and algebraic people. My maths work is on both however, at the core I’m an arithmetic person although I am geometric in my thinking. I tend to think of the two as being the identical.

It’s not entirely accurate However, in general because of the work of Descartes and, more specifically, during the last century, the two fields have become extremely close. There’s a very precise dictionary which can in certain situations assist in translating a geometric image into algebraic results.

In my case the geometric picture usually assists in the formulation of questions and conjectures. It also gives an intuition, but we convert them into math when we are writing. It’s much easier to spot errors because algebra is usually more exact. It’s also easier to apply algebra when geometry becomes difficult to comprehend.

What are the ideas you’ve been thinking about in recent work?

A significant portion of my work has to deal with elliptic curves which are very common objects in arithmetic and number theory geometry.

It shouldn’t be that difficult to find integer solutions for equations such as these. We would expect that, in general that most curves not have integer solutions. However, it’s extremely difficult to prove it.

Levent and I looked into this distribution of that integral points. We employ a classic construction that Mordell wrote in his 1969 volume Diophantine Equations. We can give an upper bound for how many integrals that can be found on an Ellipsic curve. Some people have provided higher limits. We have found a new bound that is easy to establish.

What was the role that Mordell’s previous work contribute to your latest outcome?

The question we’re asking is about integral points on curvatures with elliptic curvatures. Mordell is able of connecting it to other things that we are able to examine.

It’s what we do every day in math. We want to comprehend an object, but we need to locate a proxy for understanding it. Sometimes, the proxy is precise. Sometimes, it isn’t. But, in reality, it’s something that we have access to.

“It could also be simpler to utilize algebra if geometry becomes too difficult to grasp,” Ho said.

Caroline Gutman for Quanta Magazine

When did you make the decision to concentrate on math?

I don’t believe there was a turning point for me. I’m content with my career and life currently, but I think that if the circumstances had been a bit different, I might be content in a variety of jobs or in other areas. 

Perhaps that’s something mathematicians don’t want to say because they love to talk about how fervently they are about math, and how they would never consider any other thing. Personally I don’t believe that’s the case.

I am interested in a lot of things. Maybe I became a mathematician as I was unhappy by the lack of discipline in other fields. As a kid I was taught to see myself as a mathematician certain ways, as that was the way we lived at the home. My dad would play games of math with me, and I learned logic from an early age. I always wanted things to be tested.

However, I wasn’t certain that I’d be a great mathematician.

Why?

When I was a kid I didn’t realize that there were numerous math experts were exactly like me in many ways. We make up a lot of buzzwords about roles models. It’s not that I didn’t have many women. I also didn’t see enough Asian American women.

What I am referring to is I did not see anyone who was interested in anything apart from math. It made me doubt myself quite a bit. How can I succeed in math when I don’t devote 100 percent of my day doing math? 

This is what I observed all around me. I was able to tell that other people were thinking differently about math than I did as well as my colleagues and those who were older. I believed it was hard to find a job I didn’t want to be that way. I’d have other things to do.

Humanity is a topic I did not see other people paying attention to as much. I was worried that a part of me would make me a terrible a mathematician.

You’ve been named director for the Institute of American Studies’ Women and Mathematics program. What is the program’s offer to mathematicians who are women?

This is a week-long course designed for women who are at various levels of their careers, including undergraduate women and postdocs, graduate students and a few senior and junior faculty. The goal is to learn math in a safe and supportive setting.

Students who might not have been aware that they’d like to pursue math are meeting extremely senior mathematicians, and receiving mentoring all the way to the top. 

They will meet with a variety of individuals at different stages of their careers and have conversations with people about their experiences. I don’t believe that there are any other programs with this broad range of experience and are specialized in specific subfields.

The 2023 plan is called “Patterns of integers.” The program will feature many people from the fields of additive combinatorics and analytical number theory. We invite individuals from various career paths to make connections.

For graduate students who are older who are already working in this field They’ll be meeting postdocs as well as senior and junior professors in their field and have the opportunity to work with them for one week.

You’re also part of the Stacks project which is a huge online resource. What’s unique about it?

The sheer quantity and accessibility of the project is staggering and easy to access. It’s this huge with more than 7,500 pages if you print it out, and an on-line collaborative projects. In reality, it’s the Columbia University mathematician] Aise Johan de Jong writes almost all of it. 

It’s a meticulous, well written guide for those who are algebraic geometers. It’s an incredible contribution to the community.

Every week or so the number of users increases. It’s a reliable reference for just about everything. It covers a wide range of algebraic geometry one would have to read through 20 textbooks.

The living is in items can be edited and added. If there are errors that are not corrected, they’ll be discovered.

Another thing that’s interesting about this document is its tag-based system. While the document is growing constantly it is still possible to reference the same tag for a long time. 

There are more than 21,000 permanent tags with specific results that you may want to reference. Pieter Belmans built the whole backend of the system, and it has been utilized in various other projects too. Some other people have modified the technology.

The problem is and Johan is aware of that he’s not likely to be able to continue writing this. In the future, if we wish to see this continue, we’ll need others to get involved.

What role will your workshops contribute to The Stacks initiative?

The goal is to begin having younger children involved. They’re being asked to create pieces and bits of text which could eventually be included in it. 

There are tensions due to the fact that for the site to stay up-to-date and top-quality as an information source, it has to be carefully controlled. 

Therefore, Johan has to continue to do much of the work that goes into creating content for it. It’s not as open as Wikipedia in that anyone is able to access it. This is a bit unfortunate, but must happen if you wish for this to work.

We’re looking for ways to gradually get more participants within this Stacks project. We’re inviting mentors to collaborate on projects with postdocs and graduate students. They study algebraic geometry. They then write something down.

We recently published the volume, which contains several expository pieces which we hope to eventually be included in the Stacks project.

The Stacks project can continue to be extremely influential for many years to come when enough people take part and maintain it.

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