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PhD theses can become popular because they are based on groundbreaking work. They can also go viral because of the way they are written and the very spirit of the whole process, and Piper Harron’s thesis is one such.
Based on her work in algebraic number theory, which she carried out in Princeton University, guided by Manjul Bhargava, Piper Harron wrote a thesis in which each section is divided into three parts. A section of laysplanations for ones who are outsiders to math but willing to put in the effort to read; technical sections for math students and the third part for professional mathematicians. She went into this effort because as she says in this interview, “to be happy with math, you have to really learn to love the process.”
The prologue itself, which may be read at >Piper Harron’s webpage , is remarkable and has been described as a manifesto by some.
To quote from the prologue to her thesis:
“Respected research math is dominated by men of a certain attitude. Even allowing for individual variation, there is still a tendency towards an oppressive atmosphere, which is carefully maintained and even championed by those who find it conducive to success. As any good grad student would do, I tried to fit in, mathematically. I absorbed the atmosphere and took attitudes to heart. I was miserable, and on the verge of failure. The problem was not individuals, but a system of self-preservation that, from the outside, feels like a long string of betrayals, some big, some small, perpetrated by your only support system. When I physically removed myself from the situation, I did not know where I was or what to do. First thought: FREEDOM!!!! Second thought: but what about the others like me, who don’t do math the “right way” but could still greatly contribute to the community? I combined those two thoughts and started from zero on my thesis. What resulted was a thesis written for those who do not feel that they are encouraged to be themselves. People who, for instance, try to read a math paper and think, “Oh my goodness what on earth does any of this mean why can’t they just say what they mean????” rather than, “Ah, what lovely results!” (I can’t even pretend to know how “normal” mathematicians feel when they read math, but I know it’s not how I feel.) My thesis is, in many ways, not very serious, sometimes sarcastic, brutally honest, and very me. It is my art. It is myself. It is also as mathematically complete as I could honestly make it.
I’m unwilling to pretend that all manner of ways of thinking are equally encouraged, or that there aren’t very real issues of lack of diversity. It is not my place to make the system comfortable with itself. This may be challenging for happy mathematicians to read through; my only hope is that the challenge is accepted.”
Piper Harron’s experience in Princeton is unique — for the major part of her time as a PhD student there, she was the only black research scholar studying pure math. And her thesis is unique, in its structure and its description of the process of doing math. In contrast to widespread impressions, it shows how math, too, can be communicated to anyone who has interest.
For example, the introduction begins with a “layscape” of the problem, which begins thus.
“Every thesis is a question and (very long) answer. My question in layspeak is: ‘How many’ ‘shapes’ of certain degree n ‘number fields’ are there? The naive short answer is: Infinitely many! But of course, though true, that is not nearly enough information. What we will show is that the infinitely many shapes we find are actually ‘equidistributed’ with respect to the ‘space of shapes’. In other words, if you think of the collection of possible shapes as being a blob (a ‘space’), then wherever you look in this blob, you will find shapes of number fields in equal quantity.
Equivalently, though somewhat less to my liking, a thesis is a claim and a (very long) proof. My equivalent claim in layspeak is: ‘Shapes’ of certain degree n ‘number fields’ become ‘equidistributed’ when ordered by ‘absolute discriminant.’
In what follows I hope to do enough ‘laysplanations’ to make the whole argument approximately readable by approximately anyone. Approximately. In addition to laysplaining and ‘mathsplaining,’ I will also, where appropriate and not too horrifying, have some ‘weedsplanations’ where I wade into the weeds with examples and explicit calculations, sometimes with extra laysplanations that were not strictly necessary to the main argument….”
The entre thesis may be found >here .
In a world where mathematicians have difficulty in communicating their work even to other mathematicians, here was someone who was aiming to make her thesis so approachable. Here is a transcript of a chat with Piper Harron over email…
How did you choose this topic for research?
I didn’t. I got to grad school knowing I wanted to do “number theory,” and I discovered I hated analysis, so “algebraic number theory” it was.
Beyond these vague notions i did not have any real interests. I spent my pre-exams time figuratively drowning. When I passed, I asked Manjul if he'd be my advisor and he was like: “Are you sure? What is it you want to work on?”
I didn’t have an answer. I had taken eight math classes before I got to grad school — none of them grad courses.
The things I studied for generals didn’t make it obvious what I’d want to work on. Princeton is — I don’t know — it was not made for me, but my “peers” had taken 20-30 classes before starting grad school. They had taken grad courses. They entered with aspirations, it seemed that's what the program was geared for.
I spent a lot of time there trying to justify my existence, and one aspect of that was the dreaded Mathematical Interest thing.
So I met with another Prof. to make sure I didn’t want to work with him. I didn’t. Manjul agreed to be my advisor. He gave me this problem.
How did you find your feet in such a situation?
Heh, what do you mean? I'm not sure I did.
Why do you say that?
I guess I feel like I’ve been operating on survival mode more than anything else. I haven’t felt like I was in charge of my life or making free choices, really, just kind of reacting.
You have written about this in your blog. What was it that was wrong with graduate school?
I mean, my situation was really unique and Princeton is also somewhat unusual.
Lots of people can relate to my words, which means something!
Also do you think this is a problem with the way math is taught.
I think that there are biases.
There's the idea of the genius mathematician. There's the idea that math is fixed, you get it or you don’t; that there’s a right way to think about things. I think this is fairly pervasive with math. I think Princeton accepted me because they thought I might be a genius. They weren’t prepared for me not to be.
They seemed to be willing to make allowances for me, but they wouldn’t tell me upfront. My sanity was not on their radar.
I went to the hospital one day because I almost fainted in the student union. At the end, it was determined to be stress-related, and I was like “Stress? What?!” I thought it was unbelievable, but at the same time tears started falling and I’m like “Oh! Okay, maybe I’m stressed… maybe the fact that I’ve been waking up without an alarm after six hours of sleep isn’t a sign of my overwhelming awesomeness.”
There was nothing in place at Princeton that explained the advisor-advisee relationship. No guidelines, everything was so free, and I assume that was to let the geniuses flourish.
When did you start to make sense of this and how did you handle it then. Did you get help from your advisor?
I didn’t make sense of anything until well after I left Princeton.
When was that?
I was at Princeton 2003 – 2009.
Okay.
I don't know that there was any “racism” in terms of like racial discrimination per se. Being the only black person I saw in the department would have certainly contributed to my feelings of isolation, but I was “color-blind” when I was there. I was still internalising everything.
Sexism was a bigger factor: The culture was sexist.
But you know, there was a woman’s group there, I just didn’t like it.
So I entered grad school still trying to be normal. I didn’t want to be singled out for my race or gender, so the few times someone tried to help me for those reasons I rejected it, which I now regret. But i really didn't understand oppression at the time. Nobody had told me about institutionalised oppression or that intentions didn’t matter.
I think Princeton's "freedom" and lack of guidelines and structure will disproportionately hurt marginalized people... and so, in that sense, I can fairly call it racist or sexist. When you recognise that these are structural issues and not personal issues, it's easy to call out oppression. I still get hung up at times thinking that racism or sexism is about a bad person doing consciously bad things, but it isn't. It's about the exclusion of women and racial minorities by any means. And of course implicit bias against women and people of color is rampant.
You asked about my advisor. One of the bits of cultural knowledge I picked up as a grad student was that you don’t want your advisor to think you're unintelligent.
So try asking for help after that’s been put in your head.
Yeah that’s right, but isn’t it the part of the advisor to sense that his/her student is finding the atmosphere stifling and needs help?
I’d like to think so. I was Manjul’s first student, though, and his experience was far different from mine. When he was taking time off from meeting his advisor, it was so he could be brilliant in some secret way. When I took time off from meeting my advisor it was because I was struggling/lost/giving up.
I don’t think you can rely on individual professors to know how to be good advisors — I mean, especially in Princeton. They hire professors who’ve never had to apply for jobs [ laughs ] They hire professors who, maybe, didn’t need their advisors, you know?
I see what you mean... I think the language bothered you a lot — you say you stopped talking math to your peers.
Yeah, it was emotionally taxing. I am finally at a place where I can talk to my husband about math. And that took a lot of work.
“That's trivial, right?” — that’s the kind of thing a mathematician might say. And it’s like why are you saying this to me? If i thought it was trivial I most definitely would not have said it. What is the "right?" there for. Just to make it sound like you didn't say it?
Or to explain something to me they’ll make an analogy to something even more over my head. Because it’s more important that they force The Right Way of thinking about it on me, than that I actually understand or can make a meaningful connection.
So your way out was that you came out of Princeton... yet you went back to write your thesis — what gave you the energy to do this? And your thesis is so differently structured...
I wrote my thesis on my own, away from Princeton. Every time I left Princeton, I remembered that I did like math. It was never math’s fault that I was miserable. I still wanted to learn it. So, if Princeton wasn’t kicking me out, if Manjul was still going to help me, then I still wanted to try to graduate.
I met with Manjul two years after I left Princeton, and we worked out our paper together.
Initially, my thesis was just supposed to be about quartic number fields (the cubic result was already known), but since Manjul had first given me this problem, he’d published the necessary work for quintic number fields. So, doing the whole argument for n = 3,4,5 was his idea, and so after we wrote that paper, the idea was that I would turn that into a thesis.
He told me I just needed to “add background,” but I didn't even know what that was. What constitutes background? I didn’t know anything. My grasp of any of it was like I could hold a few ideas of it in my head at a time, but if I set it down and went back to it later, I’d be confused all over again. So I had to write something for me to even understand what was going on, and I decided that would be my thesis.
And you know, I had kids. I learned about the world, got angry at systems of inequality, got angry at the status quo. So when I started the last version of my thesis, It was not from a positive, inclusive point of view. It was from a screw-the-system point of view. I stopped caring about doing it the right way. And my friend, at some point, was like, “This is a bad idea,” because it was taking me so long and at any moment Princeton could have said, “Never mind, you're finished”. And, even when I was done, there was the possibility they'd reject it.
But I didn’t feel like I had a choice. I already had one kid, and I was pregnant, and there was no way I could be motivated to write something I didn’t like. Anyway, what would be the point of throwing together a thesis quickly and then embarrassing myself in front of my advisor when I couldn’t defend it?
When I finished it and emailed it to my advisor, I was pretty much like “HERE, ha ha. This is my thesis, I can delete some of it to graduate if you want, but this is what I’m printing for me and my family.”
He was like, “Whoa there — ha ha.”
We had a back and forth about certain things. I didn’t really understand what his problem was at first. It seemed like he just didn’t have a good sense of humor.
But, later, I realised that he was actually thinking that mathematicians and grad students would read it, whereas I had written them off. So I didn’t take out any of the stuff that he mentioned, but I added things. I thought about my audience. I thought about what I wanted grad students to know. And so, in that way, he was instrumental in the final product.
The prologue that people like so much, I just wrote it for myself, to organise my thoughts when I was thinking, “ Okay am I actually going to go through this whole thing and make sure I’m consistent? ”
But then after I wrote it I was like, “ Never mind, this will just be my prologue. ” That was the last thing I wrote.
But it's brilliant... writing different sections for people with different levels of understanding is so cool.
Thanks!
Each section was for me, though. I needed the laysplanations to know what was going on, but I was responsible for the math, too — ha ha.
So I had to write the math sections. And the weeds were extra things I needed to understand from a math perspective, not just big/lay picture.
The weird thing about this situation is people are talking to me now, asking me questions, as if I was ever thinking about other people. I wasn’t! I was thinking about myself, and I think that’s something that happens to marginalised groups — being totally selfish can be revolutionary.
I was thinking about your asking about racism and sexism at Princeton. I think the biggest way I saw their racism and sexism was in their apathy.
The world is racist and sexist. Princeton doesn’t have to do anything to have their math department be oppressive. The problem is the apathy. That they have no problem having so few people of color in their department; they have no problem having so few women. They probably thought they were doing better because there were “more” women than twenty years ago. They don’t want to change anything. They don’t want to look at themselves. Maybe they invite more women, but they do nothing to change the environment to make it hospitable for marginalised people.
As you say, it’s not just Princeton, and it is an apathy that is part of many institutions. What has been the reaction to your thesis? Did people come forth to talk to you
Yeah, I get emails from people. It’s amazing to hear from people who say that my thesis actually helps them. Like they feel less alone or they feel validated. More than one person has said it made them cry.
At my defense, one of the professors on my committee said he really liked how I showed the process; that most students don’t see the process of doing research.
So, I think there are many different levels of appreciation.
It seems a lot of people wish there was more — honestly? — going on. But that doesn’t mean people with power care. It remains to be seen what will happen.
Did you go back to Princeton after writing your thesis — to talk to people and so on?
I was only there for my defense, which was a week before I put it online. Nobody knew what was going to happen, just that I’d done something interesting.
I have found it interesting that there are professors at various schools recommending my thesis to their students. But, I mean, the change that needs to happen is very big and deep — people’s values need to change. A viral thesis is not gonna do that.
How important is it to be able to talk math to your peers and superiors when working as a graduate student?
Well! At Princeton they say it’s really important. It’s part of how they advertise themselves. Certainly, it can be impossible to learn math from texts and research papers, forget about it. Ha ha.
And, yeah, if I hadn’t been married to a number theorist, and if I hadn’t had (female) number theorist friends to talk to, I don't think I would have finished.
I hadn’t considered the importance of social interactions and how inherent racism/sexism makes this a burden for marginalised students. I mean, I don’t know exactly how a department is supposed to deal with that, but to just say that it’s really important to talk to your peers, and then have one woman of color and two other women in a group of fourteen, with very few women on faculty — that’s a problem.
What are you planning to do, now that your thesis is done... more math? Math writing? What lies ahead?
When I finished my thesis, I was saying goodbye to math. I’m home with the kids now, planning to home-school. I figured maybe I’d learn more math when they were a bit older and less needy, and then maybe I’d get a lecturer position or something.
But after my thesis became popular, and I made all these like-minded math friends (after pretty much avoiding math people), I decided to try to get back into academia.
So, we will see if they let me. I plan on applying for grants in the fall and jobs if I don’t already have one.
If academia will take me in as is (not pretending to fit in), that’s my plan. Otherwise, I’ll fall back on my other project — which was, I was gonna write a book about racism in the United States.
I plan to keep writing, though. It takes me so long to learn math, i don’t see myself writing math that isn’t related to my work.
I love universities and campuses and helping students, so I don’t think I’d want to be a math writer outside of academia, and how would I learn the math anyway, heh?
One thing that I’ve learned is how much gets erased. Like how students don't see the process of doing math — that’s erasure — and all the history I didn’t know about.
The history we teach kids in the U.S. is whitewashed, people’s struggles are erased or made more friendly to the status quo.
So many things that would help us — that would help all of us — are just erased before we can benefit from them. It leaves you feeling lost, trying to reinvent the wheel. It’s inefficient. It makes progress difficult.
So true.
And though certain people certainly “benefit” from this, I think overall we’re all less. I watched this >documentary of black American history, and I was shocked at how intellectual black Americans have always been, always talking about their struggles, always analysing their situation. But in my lifetime that was erased and on TV black Americans are just either in trouble with the law or poor.
They never show us black people speaking out on black issues. (Well, now we have Black Lives Matter) and I feel like that cost me personally.
So, one thing i hope to do is to give back what has been erased
That’s why i complain all the time ☺
I think it’s helpful to see what they don’t want you to see, and to know that the obstacles you face are real and are totally unfair and that there's a history behind the obstacles you face.
Would you like to say something to students who love math as kids and are are bright and eager to pursue math in university and research?
Ha ha ha… um… I’m always torn between MATH IS AWESOME and Math Culture Is Terrible. We need more different people in math. We need different ways of thinking in math. It is exciting to do research and to have access to other people doing research.
I would ask them to please get rid of the notion of mathematical genius. They may have great ideas or have a way of thinking that currently makes math harder or easier but that is irrelevant. Math is work, as is everything. They should feel good about the work they put in, regardless of the results.
Results will come over time, but to be happy with math, you have to really learn to love the process. And I think that’s hard because as a kid you might get pushed into math because you’re quick at grasping algorithms; that’s not math.
Kids who think math is easy should seek math that is hard. Kids who think math is hard should keep at it if they enjoy it. I guess I hope for more of a community where we appreciate everyone's contributions.
And that has to include our own.
