May 24, 2022

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In order to support her students over the course of the pandemic, she recently asked other mathematicians to talk about challenges they’ve had to overcome during their careers. She then transcribed their stories by hand into a small book that she illustrated herself.

Quanta Magazine spoke with Pierce about her research, the relationship between math and music, and her efforts to make math more accessible.

The interview has been condensed and edited for clarity.

You didn’t initially intend to pursue a career in mathematics. What did you want to be?

When I showed up at college, I thought I would be a doctor. It seemed like a safe, exciting and interesting choice. I actually completed all the pre-med requirements. But I also needed to pick a major. I was extremely fortunate that through a series of chance meetings, I met mentors who really welcomed me into mathematics. Over the next three years, I gradually realized that one could be a mathematician, and that that would be a good fit for me.

What ultimately influenced you to become one?

Looking back, I am a little horrified at how close I came to missing meeting these people. As a college freshman, I thought I would be behind everybody else because I’d been home-schooled, so I’d been planning to take Math 101. I think if I had taken that class, it wouldn’t have been a good fit. It’s hard to imagine I would have kept going in math. But my assigned faculty adviser met me over pizza the week before classes started and suggested I take a first-year course in real analysis instead, where you learn how to rigorously prove things about continuity, convergence, the fundamental theorem of calculus — properties of functions that you took for granted in high school. I absolutely loved it.

What did you love about it?

I had attended one year at a normal public high school, where we did proof-based geometry, which I also loved. But I wasn’t sure how I was arriving at the proofs, because they formed so instantaneously in my head. That was uncomfortable. I wanted to be more in control of how my thoughts were arriving at those conclusions.

This real analysis course focused on how to write proofs. In fact, that first fall break, I couldn’t afford to go home. So I was alone in the dorm, and I decided to re-prove every statement in the book so far. That kind of rigorous, repetitive practice — very much like a musician or an athlete — really strengthened my skills.

Speaking of which, you’re also a musician. How did that start?

 Practicing music, performing music, just being surrounded by music — that was the community in which I grew up. Both my parents really loved music. My father had a brass quintet that rehearsed once a week in our living room, and he later led a Renaissance group. By the time I was in high school, that was a really important part of our whole family: rehearsing together and going to one of the historic missions in California once a year to put on a performance involving all kinds of unusual instruments.

I was also taking violin lessons. My parents went to extreme lengths to get me and my three siblings the best music lessons they possibly could, even though we lived in a small town in the middle of a bunch of avocado groves. We had to drive quite a long way to get to a suitable teacher. But that also made me our area’s local violinist. I’d get hired to perform in freelance music groups that would come through my town, which gave me the chance to embed in the professional music scene. I learned a lot by playing in the pit orchestras at musicals and sitting next to full-time professional musicians.

Do you still play?

For the last 10 years, I’ve been too busy to play very much. Over those years, I went from being a first-year postdoc to a full professor, had three kids, and moved many times. It’s been frustrating, because playing chamber music is one of my favorite things to do. But I just don’t get many minutes of the day that are really about me.

I still play when I can, and I think I’m getting close to a time when I can play the way I’d like to again. But meanwhile, I’m thrilled that my two older children are playing beautifully. It makes me so happy to hear them really making music.

Math and music are often compared. Given how deeply you’ve studied both, how do they relate for you?

My intensive study of music early on made me familiar with the idea of having a mental landscape. When you memorize an entire concerto, as you play it you build this mental construction: You have landmarks, you have goals, you have places to watch out for so you don’t accidentally repeat a phrase. That’s actually really similar to how I feel about mathematical terrains that I know very well. After you’ve spent enough years working in certain areas of math, it’s like you get up in the morning and say, oh, I’m going to go to this mountain overlook and see what’s going on there. Can I see something new? The mental practice and internal construction that I inhabit are similar in music and math.

You work in number theory and harmonic analysis, which coincidentally deal with techniques that have some relationship to music too.

I work on problems that can sound like they have nothing to do with each other, but if we step back to methods, then pretty much everything I’m interested in has to do with oscillating functions and trying to quantify how they oscillate, and how over time those oscillations may possibly make things cancel out. When trying to prove something about a well-behaved function, we can use these methods to dissect it into pure waves, and then try to write the proof by looking at those waves instead. This idea is really powerful and shows up in discrete settings, like in number theory, and also in real-variable settings, like in harmonic analysis.

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