Piper Harron discusses her artistic and wonderful math Ph.D. thesis
Piper Harron is a mathematician who is very happy to be here, and yes, is having a great time, despite the fact that she is standing alone awkwardly by the food table hoping nobody will talk to her.
Piper, would you care to write a mathbabe post describing your thesis, and yourself, and anything else you’d care to mention?
When Cathy (Cathy? mathbabe?) asked if I would like to write a mathbabe post describing my thesis, and myself, and anything else I’d care to mention, I said “sure!” because that is objectively the right answer. I then immediately plunged into despair.
Describe my thesis? My thesis is this thing that was initially going to be a grenade launched at my ex-prison, for better or for worse, and instead turned into some kind of positive seed bomb where flowers have sprouted beside the foundations I thought I wanted to crumble. My thesis is that thing I got sick of just when I should have been fine-tuning its organization. It’s where I find typos that have already gone to print. I am a writer; don’t ask me about my writing.
Describe myself? For 7 years I called myself an escaped graduate student. I laughed and made light, but each passing month increased my shame burden. Having kids made it easier to throw my hands up like I was okay with things and not at all failing, but I was never okay. I’m still not okay. After my defense, I had to fill out an exit survey. They asked how many years spent on coursework (3), how many years spent on dissertation (10). TEN YEARS?! WHAT KIND OF PERSON SPENDS TEN YEARS ON ONE DOCUMENT AND STILL HAS TO BUILD UP COURAGE TO DISCUSS IT OPENLY?
When I entered graduate school, I was a sponge for external pressures. Please tell me the rules I must abide by in order to make no waves! Which jokes should I not find offensive? Oh, am I here because of Affirmative Action? Oh, am I here because I’m a woman? Oh, am I here because of a mistake? Okay, haha, I get it. Oh, do my friends think I took the spot of someone who deserved it more? That’s okay. It’s okay if my friends think that. Then there was the actual math. Funny story, I was exceptionally behind my peers when I got to Princeton. I’m tired of talking about it. I should have made myself a Tshirt that said “I only took 7 math classes before coming here, and my peers took 20 – 30, so that’s great.” Funny story, my brain is evidently unusual among mathematicians. Namely, I don’t understand anything they say. I’m strangely literal and I don’t go for hand waving. At all. I can’t just understand the forest, I also need the trees, and the leaves, and the space between, and I need to be able to go forwards and backwards logically between it all. The way people talk to you when you don’t understand what they’re saying. It’s its own terrible language and I had to listen to it for years.
In my second year, my body temporarily lost the ability to properly deal with sugar. I don’t know if this is a thing. My doctor never really figured it out even after she got “really scientific about it.” Whatever it was, I realized stress had sent me to the hospital and I was thoroughly against that on principle, so I gave up stress. This decision seemingly cost me everything. I escaped without graduating in 2009. I had my first child in 2011. I became a conscious feminist. After decades and years of absorbing all the rules, a black teenaged boy was shot dead for no reason and his assailant was found not guilty because black boys are scary. I became consciously anti-racism. Two months later I threw out my previous draft and started on my thesis grenade.
Anything else I want to say? I hope people read my thesis. I hope people who don’t know what research math is like will see that it is not so alien, that it can be accessible, and will make fewer assumptions about who should go into math. I hope graduate students will read it and realize that they are not alone. I hope it could help them be more aware of various and unfair pressures they might be under, so that they could navigate the waters without internalizing things too much (not too much to ask, right?!). I am thrilled at the idea that mathematicians might read and enjoy my thesis, but I didn’t write it for them.
I’d like to say something really important about how to make everything better. Can I get back to you on that?
mathbabe 
Wow! …that’s all, just WOW!
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Love it!!!
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I opened the thesis, and was astonished to find that the first paragraph of section 1.1.1 is all about the Wason card experiment, which, coincidentally, I have been discussing by email for the last three days with a philosophy student in London. So clearly the planets are aligned. I look forward to reading it (the parts I can understand).
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“It’s where I find typos that have already gone to print.” Been there, done that.
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I was expecting (pretty? cool? awesome?) pictures in the thesis and found none. A bit disappointed.
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There’s a comic about giving birth… It’s later on.
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I haven’t read the whole thing yet but I really like the writing so far. I feel we all suffer from this tyranny of boring writing in science and mathematics. I applaud your resistance to that!
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I think this is great on so many levels. Shifting the math community to be more inclusive is a critical part of broadening participation in mathematics. Writing a thesis that can be understood by non-mathematicians is huge! And proving that it can be done while also injecting your personality into it, Wow! You shouldn’t have to answer how to make everything better on your own. Let’s discuss. Will you be at JMM?
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Just wanted to add: With the exception of the Prologue, everything I’ve seen people quote came from my original grenade draft. The difference between my first and final drafts were not huge. All the silly was always there. But initially I assumed no math people would read it, and I may have been a bit bitter about everything. It was my advisor’s support/influence that got me to reconsider my attitude and my potential audience, and in making that transition, I ended up writing the Prologue. That was incredibly important, and from a truth perspective, it should not be omitted from the story.
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Dear Piper,
Some thoughts. I’m not in this area, so this is all probably hopelessly naive (and maybe already covered in your thesis):
(1) What about n=2? Is that well understood (trivial?) or more complicated (too few symmetries)?
(2) Are there similar results for finite fields? Any interesting compare/contrast?
(3 In 1.2.2, you mention S_n being the least “symmetric.” Another way to phrase it is to think about whether the components are generic or special. Take S_4 vs D_4 (use the model of symmetry group the square). If someone asks you to choose a vertex of the square, there’s really no choice, since all the vertices are generic up to symmetry. However, if they ask you to choose another one, suddenly it makes a difference. So, while the vertices are generic, pairs of vertices aren’t. And, maybe surprisingly, triples of vertices (without order) are generic again.
With S_n symmetry, all choices of subsets of your components are generic, so none of the choices matter. And, I guess you’ll agree with this: when none of your choices make a difference, then you don’t really have a choice.
(4) the implication of your result is that, if we wanted to find number fields in a particular subspace of the space of shapes, then there should be some there. (i) Any guesses about how large a discriminant we would need to have a certain confidence of getting some fields (given a particular subspace)? (ii) Are there any interesting subspaces where it is helpful to know that we can find number fields’ shapes living there?
(5) Is the absolute value of the discriminant just a handy way to list out the number fields or is there something particular that is happening as the AVotD grows?
(6) back to the symmetry groups. is there anything interesting we can say about number fields with symmetry groups smaller than S_n? Everyone has their favorites, right: normal subgroups, solvable subgroups, subgroups with index p^m (for prime p), etc? You can tell I don’t know what I’m talking about here . . .
Obviously, no reason to take any of this seriously or even respond.
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first, thank you for reading my post and looking at my thesis!!!!
i have really struggled with whether and how to respond to this. The execution of this message was very nice and respectful, and I genuinely appreciate that. The premise, however, is problematic. Maybe not inherently, but within the context of the sexist society we live in. Men are allowed, and often feel compelled, to think out loud at women, to share unsolicited not necessarily informed thoughts at women. (And usually these men, unlike you, don’t even seem to recognize that their thoughts may not be useful.) Women on the other hand aren’t allowed to be as open. So, if you want to not just be respectful, but actually be anti-oppression, it is better (IMO) not to respond to a woman’s work with the types of thoughts that other men pawn off as insights, if you know what i mean. again, i appreciate your honesty, but i feel obligated to point these things out.
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Do you think that men do not have their theses questioned? Do you think because you are a woman that mathematical discussion should mold itself to your liking? I don’t approve of the culture of mathematics in a lot of ways, but certainly I thought the above comments were at least mildly interesting. Do you genuinely believe that they came from a place of sexism, rather than interest?
Women are certainly allowed to be open in math. The best criticism I have ever received as a mathematician has routinely come from female faculty members and peers.
As someone who is actively involved in mathematics, I think you’re just bitter. Instead of taking the time to answer his questions and contributing to someone’s knowledge, you just lectured about sexism. If every mathematician did this, there’d be no one left in math classes.
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briefly: no, and not true. if every woman pointed out male privilege, women’s careers would be in jeopardy, not men’s.
I could get more into it, but you misread my response so I hardly see the point.
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AWildSheep, these are some poor arguments.
Piper claims that women can’t be as open in math as men. You reply that “The best criticism I have ever received as a mathematician has routinely come from female faculty members and peers.”. That supports Piper’s claim! A natural conclusion from Piper’s claim is that women in math have had to adapt and spend more time considering their comments before making them. They are than more likely to make better comments, when they make them at all.
“If every mathematician did this, there’d be no one left in math classes.”: Srsly? That’s a really big inductive step you’re taking. What if everyone on the internet had to have the word sheep in their screen name? Screen names would be way too long!! So, how many mathematicians do you know who will educate about sexism like this, i.e. when it’s not the most obvious reply? It’s like you’re making a slippery slope argument, but the situation here seems a lot more like an slippery *uphill* slope.
“I think you’re just bitter.”: from the grammar of your comment, my guess is that you are male. This makes you part of the oppressor class (whether or not you want to be, just like a female mathematician is part of the oppressed class, whether or not she wants to be, or whether or not she acts like she is). A member of the oppressed class has expressed some of how it feels to be in the oppressed class and you’ve dismissed it. The oppressor class commends you!
“Do you genuinely believe that they came from a place of sexism, rather than interest?”: You are misunderstanding the claim here. These questions are coming from a place of privilege. Piper claims that men can speak more openly in math. Here is an instance of a man using that privilege. Does this mean men can’t ask women questions in math? No. Here’s a way to think about it. Women have to be more careful about what they say to men. That’s the way the system is. So, how about when you’re going to ask random math questions to a woman, you try to be more thoughtful about what you ask?
I’ll answer your two first questions with questions. Do you think that women do not have their theses questioned disproportionately? Do you think because a mathematician is a woman that mathematical discussion shouldn’t have a place to her liking?
Here’s the thing, if you’re a member of the oppressor class and you’re upset that some of your freedoms are being taken away because of push back from members of the oppressed class, you shouldn’t be upset at the oppressed class. It is the *oppression* that is restricting your freedom. There is so much negative emotion passing from the oppressor class to the oppressed class from “well-meaning people” exactly because they don’t understand this fundamental fact. There’s an old adage that speaks to this: Don’t hate the player, hate the game. Turn your outrage towards the oppressive system, otherwise you *are* contributing to the oppression.
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I’ll pipe in a bit late here on why Joshua’s comment, though somewhat well thought-out and mathematical, was misplaced here. This post is about writing the thesis and who Piper is, rather than the thesis itself. For instance, if this post was an interview with an artist about his life, you probably wouldn’t want to comment with specific questions about the dimensions of a sculpture, because it’s dismissive of the content of the interview/the non-work part of the artist’s life and you want to respect him. [This thought experiment used a man purposefully.] We mathematicians are a friendly and polite lot, and if someone asks us about our math in any context we want to explain, but it’s important to demand respect for ourselves, as Piper is refreshingly doing. If you have serious questions about a mathematician’s work, ever, after reading a paper or seeing a talk or even watching a lecture online, I urge you to email that person if you can’t meet them in person. Mathematical curiosity is good! So is respect! You can have it all!
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Dear Piper,
Last week a couple of my colleagues alerted me to your blog, and
my first thought was of the passage below, by Lewis Thomas, from
his essay `The music of sphere’, published in `Lives of
a cell’ (1974); but digging up a copy took me a few days:
=================
There are, of course, other ways to account for the songs of whales. They
might be simple, down-to-earth statements about navigation, or sources of
krill, or limits of territory. But the proof is not in, and until it is
shown that these long, convoluted, insistent melodies, repeated by
different singers with ornamentations of their own, are the means of
sending through several hundred miles of undersea such ordinary
information as `whale here’, I shall believe otherwise. Now and again,
in the intervals between songs, the whales have been seen to breach,
leaping clear out of the sea and landing on their backs, awash in the
turbulence of their beating flippers. Perhaps they are pleased by the way
the piece went, or perhaps it is celebration at hearing one’s own song
returning after circumnavigation; whatever, it has the look of jubilation.
I suppose that my extraterrestrial Visitor might puzzle over my records in
much the same way, on first listening. The 14th quartet might, for it, be
a communication announcing `Beethoven here’, answered, after passage
through an undersea of time and submerged currents of human thought, by
another long signal a century later, `Bartok here’…
====================
Mathematical conversations too take place across great distances of space
and time, like communicating between correspondents light-years apart,
with decades between call and response. It’s a pleasure to hear your
voice.
sincerely,
❤ Jack (:+{)}
[I was going to email you this, but couldn't figure out how.]
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haha thanks! (i’ve tried to make contact slightly more visible on my site; gmail acct same name as site.)
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typo: it shd have been `The music of *this* sphere…
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Reblogged this on Singapore Maths Tuition and commented:
This is the most “unique” PhD thesis I have ever seen. Very special, and humorous to read, and coming from the most elite institution Princeton, under the guidance of Fields Medalist Manjul Bhargava.
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