In Music and Math, Lillian Pierce Builds Landscapes

magnolia trees forming pink and white arches overhead, as she waited in her family's station wagon for her older brother and sister to get out of school, is what she remembers vividly. Her mother was sitting in the driver's seat, balancing her checkbook while writing sequence of numbers in blue ink. Pierce was in awe.

She showed me a way that the numbers could be used to determine what happened next.

The early interest in numbers led to Pierce becoming a mathematician. She is a professor at Duke University, where she studies the properties of functions and the interplay between them.

A career in math was not something that was preordained.

She was home-schooled in a small town in southern California. She was playing the violin professionally before she was a teenager, and spent a lot of her time focused on music. She said that the point at which she became better at math than she was at music was reached recently or still hasn't been reached.

She was interested in math when she was an undergrad at Princeton University. She decided to see where mathematics could take her after meeting several professors who would later become her mentors.

She spent a lot of time studying proofs, establishing herself as a hard worker. She earned her master's degree at the University of Oxford in 2002 after graduating as the valedictorian of her high school.

She is defined by her work ethic and interest in different disciplines. Over the past two decades, Pierce has come back to an open problem related to how complicated arithmetic can become when different types of numbers are introduced into a numerical system. She said that it was a remarkable voyage, even though she didn't anticipate resolving it in her lifetime.

She has written papers and given talks about making important proofs and techniques more accessible to a broader range of mathematicians as a way of distributing information more widely across the field. She and Melanie Matchett Wood launched a journal with the same goal.

She asked other mathematicians to talk about their challenges in order to support her students during the Pandemic. She illustrated the small book that she transcribed their stories into.

The relationship between math and music and her efforts to make math more accessible were some of the topics discussed by Pierce.

The interview was edited for clarity.

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

I thought I would be a doctor when I showed up at college. It seemed like a good choice. I completed all the pre-med requirements. I had to pick a major. Through a series of chance meetings, I was welcomed into mathematics by my mentors. After three years, I realized that I would be a good fit for a mathematician.

What ultimately influenced you to become one?

I am horrified at how close I came to missing these people. I thought I would be behind other people because I had been home-schooled. It wouldn't have been a good fit for me if I had taken that class. It's hard to imagine I would keep going in math. I was told to take a first-year course in real analysis, where you learn how to rigorously prove the properties of functions that you took for granted, after my faculty adviser met me over pizza the week before classes started. I loved it.

What did you love about it?

I attended a public high school where we did proof-based geometry and I loved it. I wasn't sure how I arrived at the proof because they formed so quickly in my head. That was not comfortable. I wanted to be in control of how I thought.

The course focused on how to write proof. I couldn't afford to go home that first fall break. I was alone in the dorm and decided to re-prove every statement in the book. It really strengthened my skills because of that kind of repetitive practice.

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

I grew up in a community where I was surrounded by music. My parents liked to listen to music. My father led a Renaissance group and had a brass quintet that practiced once a week in our living room. By the time I was in high school, we had to rehearse together and go to one of the historic missions in California once a year to put on a performance involving all kinds of unusual instruments.

I was learning to play the violin. Even though we lived in a small town, my parents went to extreme lengths to get me and my siblings the best music lessons they could. We had to drive a long way to get there. It made me our area's local violinist. I was hired to perform in music groups that would come through my town, which gave me the chance to work in the professional music scene. I learned a lot playing in the pit orchestra at musicals and sitting next to professional musicians.

Do you still play?

I have been too busy to play much for the last 10 years. I was 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 800-273-3217 One of my favorite things to do is play chamber music. I don't get many minutes of the day that are 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 want to. My two older children are playing well. It makes me happy that they are making music.

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

I was familiar with the idea of a mental landscape from my study of music. When you memorize an entire concerto, you build a mental construction: You have landmarks, you have goals, and you have places to watch out for so you don't accidentally repeat a phrase. I feel the same way about mathematical terrains that I do. It's like you get up in the morning and go to the mountain overlook to see what's going on. Can I see something new? Music and math have the same mental practice and internal construction.

There were no tests or grades. We're alive, let's learn, that was the deep, joyful attitude.

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 go back to methods, I would like to know how they work and how long they will last. When trying to prove a function, we can use these methods to look at the waves instead of looking at the function. This idea can be seen in number theory and in real-variable settings.

A few years ago, I was invited to give a seminar and write an exposition about a new breakthrough. I was worried about taking a lot of my time away from original research because I didn't have tenure then. The community needed to digest this new result so it felt important to do it.

Even though I wasn't trying to use it for my own research, it was important for me to spend time learning someone else's work. It was a way of giving back to the community. I realized how valuable it is when each of us takes some time to write down things we understand in our own way, even if it has been understood before by other people.

Where has that approach led?

The paper I wrote was inspired by Jean Bourgain's work. After Bourgain died, I decided to work out the details of how his proof worked so that we could figure out how he thought.

I wrote a paper that tried to start with zero knowledge and show how we could arrive at the conclusion he had arrived at with a lot of cleverness. I wasn't trying to prove something new in the paper. I was trying to explain something that wasn't documented in the literature in a way that everyone could understand. It ended up leading to new work for me.

You and Melanie Matchett Wood recently started a new journal with the same motivation, to publish useful expository work.

The goal is to publish articles that are useful. There is no requirement that they be novel. graduate students all over the world are asking their advisers how it works, but maybe there is a lemma that is described in papers as known to experts. There is a massive new paper that is very important but very technical, and it would help if someone wrote about a special case of that new technique. Neither of those will be novel. It would be difficult to publish them in a journal. We are interested in that.

My mother decided to open a private school when I was 8 years old. My father built a little school on the property after cutting down some trees. My mother taught four other children as well as me and my little brother, but it was just me and my little brother. There were no tests or grades. We're alive, let's learn, that was the deep, joyful attitude. It set me up for a lifetime of being really hungry for knowledge and an atmosphere where I could find food for thought, without limits.

Looking back on your education, would you do it that way again?

I have had the chance to experience this myself because of the Pandemic. My kids have been at home since March 2020. My husband and I have full-time jobs so that's not the optimal home-schooling experience. There is something special about having a learning environment that is tailored to what you need to learn. I make my kids what they call "mathy packets", which are packets of mathematics that they can zoom in on. How amazing is that?

How has your education influenced how you teach and work?

I showed up in an elite math department after coming from a rural area without a high school education. The people there gave me the benefit of doubt. It would have been easy for someone with my background to not make it over the initial thresholds that often prevent people from entering a math major or going on to graduate school. I was very lucky. I want to remove some of the luck from the experience of students now that I'm an educator. I am always thinking about this, everything that I do, every paper I write, every class I teach, every conference I organize, every student application I read.

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