UT Grad Student Solves 50yo Geometric Problem in a Week

[Doc Reeves 6434]

Neat. Damn Lisa, you dope. 
 

 

IN THE SUMMER of 2018, at a  conference on low-dimensional topology and geometry, Lisa Piccirillo heard about a nice little math problem. It seemed like a good testing ground for some techniques she had been developing as a graduate student at the University of Texas at Austin.

“I didn’t allow myself to work on it during the day,” she said, “because I didn’t consider it to be real math. I thought it was, like, my homework.”

The question asked whether the Conway knot—a snarl discovered more than half a century ago by the legendary mathematician John Horton Conway—is a slice of a higher-dimensional knot. “Sliceness” is one of the first natural questions knot theorists ask about knots in higher-dimensional spaces, and mathematicians had been able to answer it for all of the thousands of knots with 12 or fewer crossings—except one. The Conway knot, which has 11 crossings, had thumbed its nose at mathematicians for decades.

Before the week was out, Piccirillo had an answer: The Conway knot is not “slice.” A few days later, she met with Cameron Gordon, a professor at UT Austin, and casually mentioned her solution.

“I said, ‘What?? That’s going to the Annals right now!’” Gordon said, referring to Annals of Mathematics, one of the discipline’s top journals.

 

“He started yelling, ‘Why aren’t you more excited?’” said Piccirillo, now a postdoctoral fellow at Brandeis University. “He sort of freaked out.”

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

Piccirillo’s proof appeared in Annals of Mathematics in February. 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.

The question of the Conway knot’s sliceness was famous not just because of how long it had gone unsolved. Slice knots give mathematicians a way to probe the strange nature of four-dimensional space, in which two-dimensional spheres can be knotted, sometimes in such crumpled ways that they can’t be smoothed out. Sliceness is “connected to some of the deepest questions in four-dimensional topology right now,” said Charles Livingston, an emeritus professor at Indiana University.

“This question, whether the Conway knot is slice, had been kind of a touchstone for a lot of the modern developments around the general area of knot theory,” said Joshua Greene of Boston College, who supervised Piccirillo’s senior thesis when she was an undergraduate there. “It was really gratifying to see somebody I’d known for so long suddenly pull the sword from the stone.”

Magic Spheres

While most of us think of a knot as existing in a piece of string with two ends, mathematicians think of the two ends as joined, so the knot can’t unravel. Over the past century, these knotted loops have helped illuminate subjects from quantum physics to the structure of DNA, as well as the topology of three-dimensional space.

John Conway in 1990 explaining how in high school he showed why two knots can’t cancel each other out.

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. This isn’t just a matter of taking all the knots we have in 3D space and plunking them down in 4D space: With four dimensions to move around in, any knotted loop can be unraveled if strands are moved over each other in the fourth dimension.

To make a knotted object in four-dimensional space, you need a two-dimensional sphere, not a one-dimensional loop. Just as three dimensions provide enough room to build knotted loops but not enough room for them to unravel, four dimensions provide such an environment for knotted spheres, which mathematicians first constructed in the 1920s.

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). Any knot you can make by slicing a knotted sphere is said to be “slice.” Some knots are not slice—for instance, the three-crossing knot known as the trefoil.

Slice knots “provide a bridge between the three-dimensional and four-dimensional stories of knot theory,” Greene said.

But there’s a wrinkle that lends richness and peculiarity to the four-dimensional story: In 4D topology, there are two different versions of what it means to be slice. In a series of revolutionary developments in the early 1980s (which earned both Michael Freedman and Simon Donaldson Fields Medals), mathematicians discovered that 4D space doesn’t just contain the smooth spheres we intuitively visualize—it also contains spheres so pervasively crumpled that they could never be ironed smooth. The question of which knots are slice depends on whether you choose to include these crumpled spheres.

“These are very, very strange objects, that sort of exist by magic,” said Shelly Harvey of Rice University. (It was at Harvey’s talk in 2018 that Piccirillo first learned about the Conway knot problem.)

These strange spheres are not a bug of four-dimensional topology, but a feature. Knots that are “topologically slice” but not “smoothly slice”—meaning they are a slice of some crumpled sphere, but no smooth one—allow mathematicians to build so-called “exotic” versions of ordinary four-dimensional space. These copies of four-dimensional space look the same as normal space from a topological viewpoint but are irretrievably crumpled. The existence of these exotic spaces sets dimension four apart from all other dimensions.

The question of sliceness is “the lowest-dimensional probe” of these exotic four-dimensional spaces, Greene said.

Over the years, mathematicians discovered an assortment of knots that were topologically but not smoothly slice. Among knots with 12 or fewer crossings, however, there didn’t seem to be any—except possibly the Conway knot. Mathematicians could figure out the slice status of all other knots with 12 or fewer crossings, but the Conway knot eluded them.

Conway, who died of Covid-19 last month, was famous for making influential contributions to one area of mathematics after another. He first became interested in knots as a teenager in the 1950s and came up with a simple way to list essentially all the knots up to 11 crossings. (Previous complete lists had gone up to only 10 crossings.)

 

On the list was one knot that stood out. “Conway, I think, realized that there was something quite special about it,” Greene said.

The Conway knot, as it came to be known, is topologically slice—mathematicians realized this amid the revolutionary discoveries of the 1980s. But they couldn’t figure out whether it was smoothly slice. They suspected that it was not, because it seemed to lack a feature called “ribbonness” that smoothly slice knots typically have. But it also had a feature that made it immune to every attempt to show it was not smoothly slice.

Namely, the Conway knot has a sort of sibling—what’s known as a mutant. If you draw the Conway knot on paper, cut out a certain portion of the paper, flip the fragment over and then rejoin its loose ends, you get another knot known as the Kinoshita-Terasaka knot.

ILLUSTRATION: 5W INFOGRAPHICS/QUANTA MAGAZINE

The trouble is, this new knot happens to be smoothly slice. And because the Conway knot is so closely related to a smoothly slice knot, it manages to hoodwink all the tools (called invariants) that mathematicians use to detect nonslice knots.

“Whenever a new invariant comes along, we try to test it against the Conway knot,” Greene said. “It’s just this one stubborn example that, it seems, no matter what invariant you come up with, it won’t tell you whether or not the thing is slice.”

The Conway knot “sits at the intersection of the blind spots” of these different tools, Piccirillo said.

One mathematician, Mark Hughes of Brigham Young University, created a neural network that uses knot invariants and other information to make predictions about features such as sliceness. For most knots, the network makes clear predictions. But its guess about whether the Conway knot is smoothly slice? Fifty-fifty.

“Over time it stood out as the knot that we couldn’t handle,” Livingston said.

Clever Twists

Piccirillo enjoys the visual intuition that knot theory entails, but she doesn’t think of herself primarily as a knot theorist. “It’s really [three- and four-dimensional shapes] that are exciting for me, but the study of these things is deeply linked with knot theory, so I do a bit of that too,” she wrote in an email.

 

When she 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. “She believed very much in her own point of view, and always has.”

Piccirillo encountered the question about the Conway knot at a time when she was pondering another way two knots can be related besides mutation. Every knot has an associated four-dimensional shape called its trace, which is made by placing the knot on the boundary of a 4D ball and sewing a sort of cap onto the ball along the knot. A knot’s trace “encodes that knot in a very strong way,” Gordon said.

One of Piccirillo’s former professors cited creativity as one of her core strengths as a mathematician.PHOTOGRAPH: IAN MACLELLAN/QUANTA MAGAZINE

Different knots can have the same four-dimensional trace, and mathematicians already knew that these trace siblings, so to speak, always have the same slice status—either they’re both slice, or they’re both not slice. But Piccirillo and Allison Miller, now a postdoctoral fellow at Rice, had shown that these trace siblings don’t necessarily look the same to all the knot invariants used to study sliceness.

That pointed Piccirillo toward a strategy for proving that the Conway knot is not slice: If she could construct a trace sibling for the Conway knot, maybe it would cooperate with one of the slice invariants better than the Conway knot does. Constructing trace siblings is a tricky business, but Piccirillo was an expert. “That’s just, like, a trade I’m in,” she said. “So I just went home and did it.”

Through a combination of clever twists, Piccirillo managed to construct a complicated knot that has the same trace as the Conway knot. For that knot, a tool called Rasmussen’s s-invariant shows it is not smoothly slice—so the Conway knot can’t be either.

“It’s a really beautiful proof,” Gordon said. There was no reason to expect that the knot Piccirillo constructed would yield to Rasmussen’s s-invariant, he said. “But it worked … kind of amazingly.”

Piccirillo’s proof “fits into the mold of short, surprising proofs of elusive results that researchers in the area are able to quickly absorb, admire, and seek to generalize—not to mention wonder how it took so long to come up with,” Greene wrote in an email.

Knot traces are a classical tool that has been around for decades, but one that Piccirillo understood more deeply than anyone else, according to Greene. Her work has shown topologists that knot traces are underappreciated, he said. “She’s picked up some tools that maybe had a bit of dust on them. Others are following suit now.”

[NorthLoop 14214]

How do you like them apples?

[Onboard 2.0 9083]

She needs to stop worrying about knots, and get those mosquito bites on her chest checked out....  (shit, I been here for too long).

[52-80 19162]

so this nerd's inducted in the anals of mathematics?

[MAUFRAIS 6629]

Wood knot

[TwiceHorn 31489]

Man, I am an engineer, so I am far from math-phobic.  My grasp of math was always greatest when it modeled something real world, like the number of apples, or the speed of two trains, or the equations of motion of a mechanical system.  I even enjoyed numerical methods quite a bit.

Start getting into abstract, non-concrete stuff like fields and vectors in fields, fuhgeddaboutit.

I just don't get this shit at all.  Maybe with formal study  . . . . 

Edited May 24, 2020 by TwiceHorn

[Onboard 2.0 9083]

Just now, TwiceHorn said:

Man, I am an engineer, so I am far from math-phobic.  My grasp of math was always greatest when it modeled something real world, like the number of apples, or the speed of two trains, or the equations of motion of a mechanical system.

Start getting into abstract, non-concrete stuff like fields and vectors in fields, fuhgeddaboutit.

I just don't get this shit at all.  Maybe with formal study  . . . . 

So you're saying you're all tied up in knots over this ?

[TornACL 7698]

Yeah when I read about shit like this I think I am either really dumb, or I don't take nearly enough hallucinogens. 

[Onboard 2.0 9083]

Just now, TornACL said:

Yeah when I read about shit like this I think I am either really dumb, or I don't take nearly enough hallucinogens. 

Why not both ? : )

[52-80 19162]

11 minutes ago, TwiceHorn said:

Man, I am an engineer, so I am far from math-phobic.  My grasp of math was always greatest when it modeled something real world, like the number of apples, or the speed of two trains, or the equations of motion of a mechanical system.  I even enjoyed numerical methods quite a bit.

Start getting into abstract, non-concrete stuff like fields and vectors in fields, fuhgeddaboutit.

I just don't get this shit at all.  Maybe with formal study  . . . . 

you are what the math people call a low-dimensional probe.

[Onboard 2.0 9083]

2 minutes ago, 52-80 said:

you are what the math people call a low-dimensional probe.

Sounds more like something an alien would say.

[fattyflattie 4639]

It fascinates me that there are such intelligent people out there. And that people like me, the common clay, you know...morons, can somehow bumble thru life just fine.  I don’t even know what I would do with myself if I was even remotely that intelligent. Bottles the (simple) mind. 

[Mother mopar 3319]

What starts here changes the world.

[ChickenSandwich 9841]

4 minutes ago, Mother mopar said:

What starts here changes the Galactic Empire

 

[South Austin 52515]

I glance at a woman who looks like that in my HEB, and I think "meh."  But something about that level of genius in a woman can turn "meh" into boner material. 

[TonyTexas 10515]

I was told there would be no math. 

[Pato del Muerto 52507]

All the brainpower from all those mathematicians being focused on...knots. 

[Armybrat 40850]

1 hour ago, 52-80 said:

so this nerd's inducted in the anals of mathematics?

Mostly likely not in the NSAA anals.

[Vito Andolini 9126]

1 hour ago, NorthLoop said:

How do you like them apples?

My initial thought was, “Yeah, but can she mop a floor?”

[Prepuce of Doom 21075]

You better knot mention that again! 

[/Button hooked] 

[TexasEd 8067]

Will there be a Big Bang Theory episode on this?

 

 

[RDCanecutter 26446]

Amazing what shutting down the Crown and Anchor will do for grad student productivity.

[TommyGufano 6021]

My brother is a UT mathematics PhD. And I'm fairly certain his thesis had something to do topology. I appreciate you bringing this to my attention so that I can talk shit to him about not being able to solve this.

[Guest Lobo]

13 minutes ago, Prepuce of Doom said:

You better knot mention that again! 

[/Button hooked] 

Here's how good I am at maths, my first thought wasn't "Wow, that's impressive she did that", it was "I'm reminded of an obscure track on a mediocre Adam Sandler comedy album that I last listened to about 20 years ago."  

[immortal13 4059]

I'd like to induct her into some anal

[Blotto 23525]

10 minutes ago, immortal13 said:

I'd like to induct her into some anal balloon knot topology

I gotta believe your chances go up if you speak her language.

Edited May 24, 2020 by Blotto

[Onboard 2.0 9083]

30 minutes ago, Prepuce of Doom said:

You better knot mention that again! 

[/Button hooked] 

Some people just get all tied up dealing with these kinds of things.

[deadshank 24653]

$20 says her checkbook is a wreck. 

[jimmyjazz 45963]

. . . and somewhere, some dumbass is yapping about how worthless her work is.

[Guest Lobo]

The last time I could do math quickly was 7th grade algebra.  My teacher would take points off my tests because I didn't show my work (eventually I stopped showing my work around 10th grade because I had no idea what the hell I was doing anymore).  So how exactly do you show your work on a fucking knot math problem?  I mean, do you bring in your own rope?  

[conVINCEd 21469]

3 minutes ago, Lobo said:

The last time I could do math quickly was 7th grade algebra.  My teacher would take points off my tests because I didn't show my work (eventually I stopped showing my work around 10th grade because I had no idea what the hell I was doing anymore).  So how exactly do you show your work on a fucking knot math problem?  I mean, do you bring in your own rope?  

If you were taking algebra in 7th grade you were probably pretty good at math.

[Guest Lobo]

Yeah, until I hit geometry and the fucking wheels fell off the wagon.  I took one math class (like the natural sciences type, not the type we did at B-school finance shit) at UT and it kicked my ass.  Not sure what it was, but it had to be whatever the easiest one was to fulfill my maths requirement.  

Only thing I remember from it besides my shitty grade was the birthday trick.  Some kind of choice where you win extra credit points according to how big of a chance you'll take on the minimum number of people you'd need in a room to find two people with the same birthday. So you start out at 366 because you're guaranteed two of the same (for this you assume 2/29 celebrates on 2/28).  I think the final student tapped out at like 50 or so people needed.  It's something crazy like just at 30 or 40 people, the odds you'll have two with the same birthday is 97% or something.  It really felt counterintuitive but we all did the extrapolation and it was true.  It basically explains away coincidences with math, which is incredibly important right now to virologists, tracing, hot spots, etc.  Anyway, I'm glad there are smart people at UT and beyond doing math on Covid-19 because that's almost as important as the biology/chemistry aspect of it (and we'll get into how shitty my grades were in those two fields later on).  

I hope Piccirillo stays here for years to come and comes up with her own way to torture grad students.  

[RDCanecutter 26446]

15 minutes ago, conVINCEd said:

If you were taking algebra in 7th grade you were probably pretty good at math.

What if you took Algebra in 7th grade, and flunked it? Asking for my friend, uh, my friend CDRanecutter.

[Lone Star Horn 2018]

19 minutes ago, Lobo said:

Yeah, until I hit geometry and the fucking wheels fell off the wagon.  I took one math class (like the natural sciences type, not the type we did at B-school finance shit) at UT and it kicked my ass.  Not sure what it was, but it had to be whatever the easiest one was to fulfill my maths requirement.  

Only thing I remember from it besides my shitty grade was the birthday trick.  Some kind of choice where you win extra credit points according to how big of a chance you'll take on the minimum number of people you'd need in a room to find two people with the same birthday. So you start out at 366 because you're guaranteed two of the same (for this you assume 2/29 celebrates on 2/28).  I think the final student tapped out at like 50 or so people needed.  It's something crazy like just at 30 or 40 people, the odds you'll have two with the same birthday is 97% or something.  It really felt counterintuitive but we all did the extrapolation and it was true.  It basically explains away coincidences with math, which is incredibly important right now to virologists, tracing, hot spots, etc.  Anyway, I'm glad there are smart people at UT and beyond doing math on Covid-19 because that's almost as important as the biology/chemistry aspect of it (and we'll get into how shitty my grades were in those two fields later on).  

I hope Piccirillo stays here for years to come and comes up with her own way to torture grad students.  

I guess you missed this part.

Quote

Piccirillo’s proof appeared in Annals of Mathematics in February. 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.

 

[Vito Andolini 9126]

Also, already gone, a postdoctoral fellow at Brandeis.

So, Lobo is bad at maths and reading comprehension. 😂

[Guest Lobo]

12 minutes ago, Lone Star Horn said:

I guess you missed this part.

 

You expected me to read an entire 6-page article about a girl solving a math problem?  I barely make it through the pics on the "ASSSSS" thread.  

Well, if you're gonna lose her to somebody, it may as well be MIT.  

[South Austin 52515]

MIT was my safety school.

[Brisketexan 105152]

I have long had a weakness for pretty brunettes who are WAY smarter than I am (ended up marrying one).  Seriously, like, my dating history has a fair number of 'em.  Smart is sexy.  How YOU doin', Lisa?

 

[Guest Lobo]

Plus it’s already published fact that she’s into being tied up in knots.  I mean, if you’re into that kinda thing.  

[atomheartbevo 89686]

4 hours ago, Doc Reeves said:

“I said, ‘What?? That’s going to the Annals right now!’” Gordon said

I've heard that from South Austin's mom so many times.

[Doc Reeves 6434]

45 minutes ago, Lobo said:

Plus it’s already published fact that she’s into being tied up in knots.  I mean, if you’re into that kinda thing.  

What’s the point. No matter what, she could just untie herself

[Grandioso 2592]

 

[Guest Lobo]

16 minutes ago, Doc Reeves said:

What’s the point. No matter what, she could just untie herself

Good point.  I think we can all agree on the new safety word for such roleplaying, "Sliceness!"  

['stache 21817]

Dang, that’s pretty cool, I assume at least because I understood about 3 sentences in that entire article. My brain has a literal roadblock at a certain point in maths. I made 100% in college calc 1 and 2, and physics 1 and 2. I thought I was well on my way. The very next levels I had to drop and moved to business economics like...

 

[conVINCEd 21469]

I wonder if she’ll win the Fields Medal.  It’s a really big deal, it’s like the Nobel Prize for math.

[Brisketexan 105152]

2 hours ago, Lobo said:

Yeah, until I hit geometry and the fucking wheels fell off the wagon. 

Yeah.... I remember taking Geometry in high school.  Headed in thinking "how hard can this be? I know all my shapes and shit."

In my defense, I was just a sophomore.  And really cocky.  And I'd started drinking a lot by then.  That class kicked my ass.

[bschoolprof 6255]

6 minutes ago, conVINCEd said:

I wonder if she’ll win the Fields Medal.  It’s a really big deal, it’s like the Nobel Prize for math.

she'll need a real skinny scarf.

[Guest Lobo]

7 minutes ago, conVINCEd said:

I wonder if she’ll win the Fields Medal.  It’s a really big deal, it’s like the Nobel Prize for math.

It's about my medal conVINCEd?  Oh god, I'll go home and get it for you, you can have it!  

[Paper_jam 831]

Like TwiceHorn, I was decent enough at math, physics, etc. to muddle through an engineering degree at UT. There are many problems and concepts in engineering and science which you can visualize or relate to real-world situations that you can understand intuitively, even if the math is difficult. But when it comes to this level of abstract math, or quantum physics, or relativity--it's all complete greek to me. It might as well be magic or witchcraft.

 

[fattyflattie 4639]

4 hours ago, Brisketexan said:

Yeah.... I remember taking Geometry in high school.  Headed in thinking "how hard can this be? I know all my shapes and shit."

That was trig for me.  Never recovered. I really wish I knew math better.  Now I just ask engineers to do things, and I’m guessing about 33% of the time they are thinking wtf?  I understand many of the concepts in real world work, just not the actual proving.