Graduate Student Solves Decades-Old Conway Knot Problem
May 20, 2020 7:16 AM   Subscribe

It took Lisa Piccirillo less than a week to answer a long-standing question about a strange knot discovered over half a century ago by the legendary John Conway.

Lisa Piccirillo’s solution to the Conway knot problem helped her land a tenure-track position at the Massachusetts Institute of Technology.
posted by Etrigan (36 comments total) 53 users marked this as a favorite
 
this is so cool. what a fucking great role model. please show this to every young women with the slightest interest in STEM.
posted by zsh2v1 at 7:20 AM on May 20 [13 favorites]


“I don’t think she’d recognized what an old and famous problem this was,” Gordon said.

i love
it when "naive" people aren't intimidated by problems because they don't know the problem's history, then go on to solve them

alas, different types of subject naiveté are harmful in science/math. it's often tough to sort the usefully-on-the-cusp-of-genius naive from the awfully-underprepared naive. not in this case! she rocks
posted by lalochezia at 7:39 AM on May 20 [12 favorites]


When [Piccirillo] first started studying mathematics in college, she didn’t stand out as a “standard golden child math prodigy,” said Elisenda Grigsby, one of Piccirillo’s professors at Boston College. Rather, it was Piccirillo’s creativity that caught Grigsby’s eye.
This is nice. I like stories where victory doesn't rest on the gifted-by-birth hero*ine, but rather on chopping wood and carrying water.
posted by daveliepmann at 7:39 AM on May 20 [25 favorites]


Admit it, "that knot is slice" sounds like slang from a not-particularly-well-written cyberpunk novel from about 1990.
posted by GenjiandProust at 8:01 AM on May 20 [46 favorites]


The engineer in me wants to understand problems like this but I know that my mathematical limitations were pushed hard enough by differential equations and Calculus III such that the understanding of higher (highest?) level proofs and theorems such as this will always have to exist in the abstract rather than technical realm.

I am never-endingly impressed by the individuals whose minds work in such a fashion however. Their grasp of the necessary skills and ingenuity to apply them are one of the key reasons (and justifications to be clearer, perhaps) for humanity's continual advance and improvement. Well done and I wish her luck at MIT.
posted by RolandOfEld at 8:02 AM on May 20 [7 favorites]


It's off topic, but I just want to say how nice it is to see names that I recognize in this story. Cameron Gordon is an amazing guitarist.
posted by of strange foe at 8:05 AM on May 20


Her paper "The Conway knot is not slice", published in the Annals of Mathematics.
posted by blob at 8:07 AM on May 20 [11 favorites]


I'm going to be picky and say she's not a graduate student; she's a "postdoctoral fellow," meaning, she has a Ph.D. and is researching with a professor, probably to get experience for a faculty position.

(Post-docs are an interesting beast -- neither student nor staff nor faculty -- and have none of the protections that come with those classifications. Hopefully this proof will get her into and keep her in a tenure-track position somewhere, if that is her goal.)

@zsh2v1 This is the kind of story that kept me out of graduate school: If I can't solve a 50-year-old problem accidentally, how good at math am I really?
posted by JawnBigboote at 8:10 AM on May 20 [13 favorites]


Wow, really awesome. I wonder what it's like to be her. I hope she has the opportunity to do loads of great things.
posted by Glinn at 8:12 AM on May 20


Hopefully this proof will get her into and keep her in a tenure-track position somewhere,

“The paper, combined with her other work, has secured her a tenure-track job offer from the Massachusetts Institute of Technology that will begin on July 1, only 14 months after she finished her doctorate.”

Also, she was a grad student when this work was done (paper shows a submission date of 15 August 2018) so the headline isn't totally inaccurate.
posted by Maecenas at 8:15 AM on May 20 [10 favorites]


I love the long history of giving students a hard problem that they're either not really supposed to know how to solve yet (high school teachers asking you to add up 1,2,3 ... 98,99,100 for extra credit before you've had algebra) or unknown problems that have been around for a long time that they don't know that it's an unsolved problem so they just solve it anyway. There are a bunch of these in math history, I think it's a bit easier to do this sort of trick question in math than in say biology or philosophy.

And kudos to the women in STEM. I've long known that there's no particular reason why anybody can't do some fancy math once they figure out how to do it their way. Ages and ages ago I tutored a girl failing algebra. We spent a couple of months mostly giggling and having laughs over her little sister's crush on the math tutor while just re-framing algebra into something fun. A couple months in my services were no longer needed because her grades were way up. She went on to get her masters in some branch of math. Probably knows way more than I do.

It's sorta sad that there needs to be a thing that it was a woman who did this thing. It should really be a 50/50 split sort of thing and not really worth mentioning over the fact that someone solved this previously unsolved thing that's been laying around for a long time.
posted by zengargoyle at 8:21 AM on May 20 [4 favorites]


i love it when "naive" people aren't intimidated by problems because they don't know the problem's history, then go on to solve them.

I know that's the framing of the story, and it's reinforced by the photographs (look, she's young! she rides a bike!), but she was very close to completing her PhD (in a field adjacent to knot theory) when she worked this out. So she's only naive in the sense that she didn't recognize the Conway Knot as a problem because she wasn't super-focused on knots. So not an expert-expert, but pretty expert.
posted by GenjiandProust at 8:28 AM on May 20 [13 favorites]


But our world is four-dimensional if we include time as a dimension, so it is natural to ask if there is a corresponding theory of knots in 4D space.

Every time I see these articles about 4D space I get confused by how they are presented. Am I correct to assume that we are really talking about four spatial dimensions here and not three plus time, which is always how four dimensions are introduced?
posted by njohnson23 at 8:44 AM on May 20 [4 favorites]


I like Erica Klarreich's science writing a lot, but this paragraph is a pretty good example of why non-mathematicians like myself have such difficulty with understanding higher-dimension topics:

It’s hard to visualize a knotted sphere in 4D space, but it helps to first think about an ordinary sphere in 3D space. If you slice through it, you’ll see an unknotted loop. But when you slice through a knotted sphere in 4D space, you might see a knotted loop instead (or possibly an unknotted loop or a link of several loops, depending on where you slice).

To understand a thing in nD space, first think about the thing in 3d, but then think about it in nD.

I think I watch Sagan's tesseract demo at least once a year, and I get it, but I still don't really "get it".
posted by Think_Long at 8:46 AM on May 20 [2 favorites]


This video (a 5-minute demo of the 4D Toys iOS app) is the best explanation of four-dimensional objects I've ever seen.
posted by theodolite at 8:50 AM on May 20 [7 favorites]


4d space is a little bit over complicated here, I think.

Consider the usual 2d plane. A neat trick is to put a single point at infinity, to "close up" the plane into a sphere. So any geometry you do on a plane, you can also do on the surface of a sphere, so long as there's some single point on the sphere that you don't need to use. And so long as you don't care about distances, which we don't in knot theory: it's all about stretching things around, and distances never matter.

Now the same trick can be used to turn 3d space into the surface of a 4d sphere: just as a point at infinity. You don't have to try to visualize it; it just works by analogy... So all of the cool knot theory we do can be done on the surface of a 4d sphere with no real changes.

Now, a knot is just a circle that's twisted up in itself somehow. It's also a very 3d thing: given a fourth dimension to work in, you can turn any knot into just a circle.

Ok... So now I can explain what this slice thing is about. Mathematicians care a lot about boundaries. Like a lot, a lot. A circle is the boundary of a disk, and a sphere is the boundary of a ball. (And really, a disk is just a 2 dimensional ball, and a circle is a 1d sphere. So it's enough to say that a sphere is a boundary of a disk.) It turns out that if you take the boundary of a boundary, you get nothing... A circle bounds a disk, but has no edges itself.

So all knots are just circles, right? And circles are the boundary of a disk. So, question: can we find a specific disk that a given knot is a boundary of? Well, easily, if you can have as many dimensions as you want...

But if you restrict things so that the disk had to be in that 4d ball that we talked about up to, the answer is "sometimes." And this is what "sliceness" is about. Can you find the 2d disc in 4d space that this particular knot bounds?

It's basically a high dimensional coloring problem... Can you color in the knot without having any funky crossings, etc, in the disk that you color in?
posted by kaibutsu at 9:26 AM on May 20 [19 favorites]


(As the old joke goes, to imagine 13 dimensional space, picture a 3d object and say "THIRTEEN" very loudly.)
posted by kaibutsu at 9:27 AM on May 20 [16 favorites]


So she's only naive in the sense that she didn't recognize the Conway Knot as a problem because she wasn't super-focused on knots. So not an expert-expert, but pretty expert.

thats what I meant..... "naive to history of attempts at a problem and the gravity of this problem".
posted by lalochezia at 9:30 AM on May 20 [1 favorite]


A neat trick is to put a single point at infinity, to "close up" the plane into a sphere.

You've already lost me.
posted by grumpybear69 at 9:55 AM on May 20 [25 favorites]


thats what I meant..... "naive to history of attempts at a problem and the gravity of this problem".

Sorry if that was snippy, but I see too many "this untrained person did X thing scientists couldn't!" stories where it turns out that they were trained but just didn't have the credential or something. It's bad storytelling, and I think it encourages cranks. This FPP on Harlen Bretz is a case in point. The "high school teacher" who solved the mystery was a high school teacher when he went "huh, I wonder what happened here?" by the time he answer that, he'd earned a PhD in Geology.

Anyway, I didn't mean to be a jerk about it, but I'm editing a book chapter, and it's irritability-inducing.
posted by GenjiandProust at 10:03 AM on May 20 [11 favorites]


>> A neat trick is to put a single point at infinity, to "close up" the plane into a sphere.
> You've already lost me.

Here's three four ways to think about it, some of which might make it clearer.

Mathematicians typically have a lot of stories and analogies about these things, which make them easier to think about. All of these come to mind for me when I think about 'how is a plane like a sphere?' When I try to think about these things, I kinda go through my set of stories about spheres until I find one that's a pretty good match for the discussion at hand. Each story takes a bit of time to play with and draw and internalize, and after that, it's an idea on hand that I can use whenever I need it. Nothing here is about super powers; it's just accumulating a bunch of stories and tools to help understand these things that we have a hard time actually visualizing.

a) Very informal: Start with a round piece of cloth with a 'drawstring' at the edge, and draw the string tight until the boundary is just a point... What shape do you get? A sphere: just fill in that point.

b) Place a sphere on top of the plane. Two points determine a line... For any point on the sphere, draw the line connecting it to the north pole, then extend the line to hit the plane. Now every point on the sphere has a point on the plane. Vice versa, for any point on the plane, connect it to the north pole, and the line intersects the sphere somewhere. (maybe twice; if so, just use the point in the south pole.) So, there's a correspondence between points on the sphere and points on the plane... It's also a 'smooth' correspondence: if you draw a squiggle on the plane, it will 'project' to a kinda deformed squiggle on the sphere. (This is called 'stereographic projection.' Here's a picture.) [This also works in 2d... Draw a circle on top of a line, and connect points on the line to the 'north pole' of the circle. It's much easier to draw!]

c) 'Inversion.' Draw a unit circle (radius 1) on a plane. We're going to 'flip' points inside the circle to points outside the circle. Pick a point anywhere on the plane, and draw the line connecting it to the center of the circle. This line has some length d. Pick the point on the line at distance 1/d from the center. If d > 1, then 1/d < 1, so we send a point outside the circle to the inside. If d < 1, the reverse happens. Inside and outside flip, and points on the circle stay the same. The only point that doesn't have a 'pair' is the center of the circle, which has distance 0... We can then 'fill in' a point at infinity to pair with the center. This shows we can fill in a single point at infinity in a nice way, at least.

d) If you make a sphere really really big, it seems like it's flat. The curvature gets 'diluted' until eventually you can have flat soccer fields and such. There's still a tiny bit of curvature there, until the sphere is infinitely big... And then you're just standing on a plane. (This doesn't explain the 'point at infinity,' though.)
posted by kaibutsu at 10:43 AM on May 20 [5 favorites]


I always liked the Flatland analogy of shadows used to explain higher dimensions. If you take a sphere in our 3d world and shine a light on it, it leaves a circular or oval-like shadow on a 2d surface. If you shine a light on a circle, it will leave the shadow of a line on a 1d line segment. Extending this, if you shined a light on a hypersphere living in a 4d world, it would leave a sphere as a shadow in our world.

More complicated objects leave different shadows as they move. If you shine a light on a cube in 3d, the shadow it leaves on a piece of paper changes as you move and rotate the cube. If you were a Flatlander, a creature on a 2d surface, you might not be able to visualize a cube or other 3d object, but you might get a sense of how it works by the different shadows it leaves in your 2d world.
posted by They sucked his brains out! at 10:44 AM on May 20 [2 favorites]


I would have been interested to know what Conway thought and felt when this problem was solved. This disease is terrible, especially given how avoidable all the deaths could have been.
posted by They sucked his brains out! at 11:01 AM on May 20 [6 favorites]


3blue1brown has an interesting talk about a "slice"-like or slider-based way to think about higher dimensions, and a surprising and weird result about how cramped space can get as dimensions grow from 4 to beyond.
posted by They sucked his brains out! at 11:37 AM on May 20 [2 favorites]


I know every word in this article, yet I understood absolutely none of it. Fortunately I understand tone and context clues enough to feel the thrill!
posted by thejoshu at 3:19 PM on May 20 [3 favorites]


"Admit it, "that knot is slice" sounds like slang from a not-particularly-well-written cyberpunk novel from about 1990."

I think you meant knot-particularly-well-written...
posted by iamkimiam at 3:57 PM on May 20 [3 favorites]


@JawnBigboote i totally get it, but at the same time, these stories are often the thing that get us hooked on this field. i read this and i see a woman so in love with solving problems. I want more of that projected into the world. way more than 'a XYZ stereotype is suddenly generously gifted with this amazing specific ability against all odds of our usual expected ABC type'. it's the love of problems and that innate curiosity that can be so addictive and should be encouraged to anyone that can stand literally untangling a seemingly unsolvable problem. that skillset is so important regardless of your ability to sit through a phd and something that i wish was emphasized to more people at a young (or not-so-young!) age.
posted by zsh2v1 at 6:49 PM on May 20 [1 favorite]


In the "Solved it Because You Didn't Know You Should be Intimidated" department:

In 1994 I took Intro to Differential Equations with Gian-Carlo Rota, a grade-A eccentric character on an international if not intergalactic scale. Talking after one of the quizzes, nobody could figure out WTF was up with #2 on the quiz. Then we get to lecture and Prof. Rota goes through the quiz solutions:

"... and on to number 2. Right. Number 2. I don't know the answer to number 2. Number 2 is an unsolved research problem. I like to put these on quizzes every once in a while because occasionally somebody will solve them. They publish, I get a new grad student, everybody wins."

"This year, unfortunately, nobody won. Maybe next year!"
posted by range at 6:55 PM on May 20 [20 favorites]


I would love to 1) hear her explain how she came up with that particular knot as a trace sibling with the different invariant, and 2) understand it.
posted by away for regrooving at 12:21 AM on May 21 [1 favorite]


There was a great description in one of the books of Liu Cixin's Remembrance of Earth's Past trilogy where a character traveled into four-dimensional space. (It was the third book I think, Death's End.) When looking at 3D objects from 4D space, they could see the entirety of the object, as if the insides and outsides of the object was entirely viewable as if laid out on a single plane. I have no idea if this description was accurate or not but it helped me understand it a bit better.

I really wish I could understand higher maths.
posted by slogger at 10:04 AM on May 21


I would have been interested to know what Conway thought and felt when this problem was solved. This disease is terrible, especially given how avoidable all the deaths could have been.

Piccirillo solved the problem in 2018, and her paper was published in February, so presumably Conway knew about it.
posted by ultraviolet catastrophe at 11:47 AM on May 21 [4 favorites]


>>When [Piccirillo] first started studying mathematics in college, she didn’t stand out as a “standard golden child math prodigy,” said Elisenda Grigsby, one of Piccirillo’s professors at Boston College. Rather, it was Piccirillo’s creativity that caught Grigsby’s eye.>>

This is nice. I like stories where victory doesn't rest on the gifted-by-birth hero*ine, but rather on chopping wood and carrying water.

I know that describing brilliant women as just grinds who had to work real hard for what didn't come naturally is a whole thing, but what you say you like in stories is sort of the opposite of what was said about her in this one: specifically, that she is creative, and did this for fun on her own time (as much as a grad student can be said to have her own time.

"Didn't stand out as" is also, of course, just another way of saying "nobody noticed her for," with agency reassigned to the observed rather than the observers. and what she did stand out for was, again: creativity. Grinding without gifts will get you far, but only so far. it's all right to acknowledge some women as particularly gifted, able to do things that not just anybody would have been able to do, or ever did think of doing previously. and I even find it refreshing.
posted by queenofbithynia at 1:02 PM on May 21 [6 favorites]


it is just extremely depressing to read a very clear statement that she was not a "standard...prodigy" and see so many people seizing on this as if it meant she was not naturally gifted, rather than that her natural gifts were nonstandard. The quote does not say she did not stand out from her peers. It says that not only did she stand out, she stood out in a different way from the usual standouts.

does this matter to the math? not a bit. does it matter to the counterproductive way people like to repurpose popular STEM stories involving women in order to make them "inspirational" to girls who are assumed to be threatened by scary things like needing to be smart, rather than disgusted by constant devaluing of their smartness? I think it might.
posted by queenofbithynia at 1:15 PM on May 21 [3 favorites]


that knot is slice

"Lisa, stop trying trying to make slice happen! It's not going to happen!"
posted by Joe in Australia at 7:51 PM on May 21 [1 favorite]


Piccirillo solved the problem in 2018, and her paper was published in February, so presumably Conway knew about it.

I know! Which is I mentioned it and why I'm sad we don't have that in the article.
posted by They sucked his brains out! at 3:36 PM on May 26


what you say you like in stories is sort of the opposite of what was said about her in this one: specifically, that she is creative
I find diligence to be a key element of creativity, which is specifically what I was pointing out. Some people really are incredible prodigy savants who don't have to try, and I think our culture overvalues that. I feel affinity to people who put in the work, even—no, especially—if it's creative, puzzle-solving knowledge work, because I admire that kind of curiosity paired with the grind.

Or maybe you're right: she's a math savant, I misread it, and the whole story is part of our culture's poisonous obsession with the monomyth.
posted by daveliepmann at 4:49 AM on May 27


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