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Graduate Student Solves Decades-Old Conway Knot Problem
- smitty1e 6y agoOutstanding work. Let us dub Lisa Piccirillo a "Space Age Bo's'n Mate". https://en.wikipedia.org/wiki/Boatswain https://en.wikipedia.org/wiki/Boatswain
- deleted 6y ago[deleted]
- contemporary343 6y agoEven in the most technically demanding and theoretical of disciplines (and in some sense, perhaps especially so) it is creativity and an ability (instinct?) to see possibilities that others don't that distinguish the best researchers. This is a wonderful example of that.
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- melvinroest 6y agoI have immense respect for researchers who venture in these type of disciplines. I don't think I would be able to do it. I do have a bit of a daring question: isn't there a slightly more fine-grained way to quantify that every nook and cranny of such a problem has indeed be researched by researchers? I simply assume that a rigorous research by the best minds of the world has happened, but I never see any data on it, not even anecdata. I mean, I remember a post from Julia Evans, making a Ruby profiler, where she was astonished on how few people were actually working on it [1]. I suspect that in some cases, probably not this one, but in similar theoretical fields, a similar thing might be occuring. And if not, how do we test that? I'm probably not the only one who's curious. [1] I found a talk of her in which she emphasizes on it: > So the three myths that I want to start out by talking about are myth one-- to do something new and innovative you need to be an expert-- myth two-- if it were possible and worthwhile, someone would have done it already so you probably shouldn't try-- and three-- if you want to do a new open source project, you need to code a lot on the weekend and your evenings. https://www.deconstructconf.com/2018/julia-evans-build-impossible-programs https://www.deconstructconf.com/2018/julia-evans-build-impos...
- contemporary343 6y agoAn interesting thought. My understanding is that the library science community has been investigating this for some time in terms of the development of ontologies and libraries to try to categorize research output. Here's a random paper I found on the broader topic: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3129146/ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3129146/ I suspect that this field is vastly under-studied and investigated relative to what it ideally should be. Generally speaking, it is easier in more theory-driven sub-fields to probe new areas. And indeed it is often rewarded. It's harder when $ is needed for experiments since that becomes more grant process-driven (something which is inherently more risk-averse). My observation is that usually a few pioneering people push out into a new topic area. Then, whether a community forms around it and starts getting excited about it depends a lot on timing, luck and also resources. Sometimes nothing happens for decades until the stars align and people realize that there's something there.
- redis_mlc 6y ago> isn't there a slightly more fine-grained way to quantify that every nook and cranny of such a problem has indeed be researched by researchers? Sometimes there is (the map coloring theorem), sometimes there isn't (the rest of math.) > I simply assume that a rigorous research by the best minds of the world has happened, but I never see any data on it, not even anecdata. Most mathematicians work on areas that interest them, ie. alone or with a colleague in another university. Never heard of anything systematic involving "the best minds of the world" outside perhaps military projects, and some cooperative research is being done on forums now. Comparing math and Open Source software development is kind of strange and not helpful. Anybody can expend a lot of time and effort and successfully write a profiler, if they wanted to. Few people make a career in math. If you're not a native English speaker, you might want to get checked for ADHD, since your post wasn't very coherent.
- throwaway_sun 6y ago> Anybody can expend a lot of time and effort and successfully write a profiler, if they wanted to. Few people make a career in math. Anybody can expend a lot of time and effort to write a profiler... but few people make a career of it. Anybody can expend a lot of time and effort on math... but few people make a career of it. > If you're not a native English speaker, you might want to get checked for ADHD, since your post wasn't very coherent. That's a strange suggestion to make after reading a single HN comment, especially when you're basing it off of your own subjective interpretation of said comment. I thought the parent made a coherent point that people may avoid hard problems because of the assumption that 1) they need expertise that they don't have and 2) someone else is already working on it. The question raised was: how, in general, do we verify those assumptions?
- pmiller2 6y agoLink to paper: https://arxiv.org/abs/1808.02923 https://arxiv.org/abs/1808.02923 I have not read the paper, but I have skimmed it briefly, and it looks pretty exciting. This isn't a case of "here's a new invariant, and, oh, BTW, it works to show the Conway knot isn't slice." It's an actual new technique. And, at first glance, it looks like a pretty simple technique. I didn't immediately see anything here that wasn't just a neat combination of low dimensional topology and basic knot theory techniques. I'd be interested to see what this technique could do with knots having more than 12 crossings.
- kmill 6y agoYeah, it's pretty cool! On one hand you have knot traces, which are spaces that completely determine whether a knot is slice, and on the other you have the Rassmussen s-invariant, which can sometimes tell whether a knot isn't slice. It turns out that knot traces contain different information from the s-invariant. One way sliceness is studied is through knot concordance, which is a certain restricted kind of deformation of a knot through 4-d space. All slice knots are concordant to each other. The Rasmussen s-invariant is invariant under concordance. So, the particular pair of knots C and K' cannot be concordant since they have different Rassmussen s-invariants. One consequence is that it's not true that knots with diffeomorphic knot traces are concordant in general. Another interesting thing is that C and K' are related by positive mutation, which makes them topologically concordant (weaker than concordance). Since K' is slice, this implies C is topologically slice (weaker than sliceness), though this was already known by Freedman's work because the Alexander polynomial of C is 1. Altogether, C is topologically slice but not slice (not the first example) while also being a positive mutant of a slice knot (which is what made the problem so difficult for so long). The knot trace X(K) is a 4-dimensional manifold whose boundary is a 3-manifold called the 0-surgery of K. There was a conjecture that knots with diffeomorphic 0-surgeries are concordant (perhaps up to mirror images). While it was already disproved, this pair of knots gives another counterexample. In fact, she gives infinitely many pairs of counterexamples in an earlier paper: https://arxiv.org/abs/1702.03974 https://arxiv.org/abs/1702.03974
- j7ake 6y agoWhat a great story. It’s something we can all learn from.
- grayclhn 6y agoIt's a great story. I'm curious what you think everyone can learn from it, other than "geniuses can occasionally solve longstanding intractable problems very quickly."
- papeda 6y agoMy takeaway was more that fresh eyes and expertise in a related field is a powerful problem-solving combination. She's clearly smart, but I'm reluctant to call anyone a genius if it causes us to view their success as, say, wondrous light from a distant star rather than recognizable human effort.
- grayclhn 6y agoNot to put too fine a point on it, but that problem resisted years of “fresh eyes,” “related expertise,” and “recognizable human effort,” but she still walked in and clobbered it. Some types of success aren’t very relatable.
- xwolfi 6y agoDepends what you define by fresh eyes. You can have fresh people with old ideas, trying constantly the same thing, until an idiot comes along with an exotic expertise and solve your problem. And it happens everyday to many of us, sometimes we are the old ideas people, having a noob show us how it could be done better and faster, and sometimes we are the ones joining an old group and proposing something new we saw elsewhere to apply to one of their stupidly long-standing problem. I find her story very relatable and a good reminder we should never dismiss the noobs in scientific, creative or high-profit endeavors: they can always bring something we didn't think of by exploring an area we overlooked. And as an example, you can look at the high-speed, youth-driven electronic innovation craze in Shenzhen, often overlooked, where everyone share everything, everyone and everything is new somehow, and exploration is encouraged (and financially rewarded) more than production of blueprints.
- JoeAltmaier 6y agoThis story will go down in mathematical history. A graduate student attacks an old unsolved problem, in her spare time, shows up with the solution a week later. Invents a new approach to topology in the process.
- high_derivative 6y agoI wonder how much it helped the creative flow to not know it was an allegedly unsolvable problem. No fear to fail, no stress, not the feeling something complex might be needed because everything simple was already tried.
- exmadscientist 6y agoAnd in addition, it wasn't (quite) her field. (This shows somewhat in the paper linked elsewhere in the comments: you can kind of see the two different jargons in use.) The power of fresh eyes is often underestimated.
- cosmie 6y agoThis. I've built my career off of diversity of experience - changing both the industry and functional role I work in almost every time I change companies. It's amazing how impactful it can be to take an industry-naïve approach to a problem/project. A lot of disciplines have fundamentally similar challenges, but with solutions that evolved in completely different directions. Intractable problems or recurring issues in one industry can frequently be unblocked by plucking mature solutions or approaches from another, but so many people grow linearly within a single industry/discipline that such cross-seeding of concepts rarely actually have an opportunity to occur.
- pwrdbytmr 6y agoI must ask, please do not take this the wrong way, how have you been able to switch consistently with such ease? And what country do you work, where you are being respected for having industry-naïvety?
- melvinroest 6y agoFrom a high level overview what she did feels mostly like a hacker approach: finding a side channel [1]. I wonder to what extent mathematicians think about side channels. Instead of talking to the service/mathematical object (A) directly, you talk to another service/mathematical object (B) that leaks information about (A). Precisely, the information that you want. The way she leaked that information was through a property called traceness that apparently was underappreciated by knot theorists in terms of sliceness problems. Which makes sense, otherwise it wouldn't be an information leak. Finding an info leak in itself, no matter what discipline your in is already amazing. As far as I understood the quantamagazine article, mathematical object (B) still had to be constructed which only a person well-versed in knot theory could do. So not only did she find an info leak, she basically created something entirely new that few people can do (yep, the hacker analogy breaks here, this part is the "incredible builder" analogy). This is so cool. Side channels are everywhere, even in math. Apparently, for knots it's called traceness. [1] Not sure if side channel is the right word, but I view it as: something that leaks information about another thing. For example, air vibrations leaking information on what instructions the CPU is executing (I'm making this up, one would need very fine-grained air vibration data to see if this would be a side channel).
- terramex 6y agoYou could argue that one of the most famous proofs of last 30 years - proof of Fermat's Last Theorem used such approach. Andrew Wiles had proven that if this FLT was false, then certain theorem about elliptical curves (unrelated at first glance and from different area of mathematics) would have to be violated, and mathematicians already knew it was true.
- MaxBarraclough 6y agoI think empath75 is right to point to dualities. I don't think proof-by-contradiction is really analogous to side-channels.
- empath75 6y agoDualities are what you’re thinking about. https://en.m.wikipedia.org/wiki/Duality_(mathematics) https://en.m.wikipedia.org/wiki/Duality_(mathematics)
- m3kw9 6y ago“ To make a knotted object in four-dimensional space, you need a two-dimensional sphere, not a one-dimensional loop. ”. You just need to assume it’s right.
- saagarjha 6y agoYou can't have a one-dimensional knot in the fourth dimension, because it's trivial to pull it apart if you move part of it into the fourth dimension.
- wesammikhail 6y agoToo bad Conway didnt live long enough to see this.
- generationP 6y agoCorrection: Her MIT position (a Moore instructor, at least according to her own website https://sites.google.com/view/lpiccirillo/home https://sites.google.com/view/lpiccirillo/home ) is not a tenure-track job (see https://math.mit.edu/about/employment.php https://math.mit.edu/about/employment.php ). It's still one of the best academic jobs a fresh PhD can get these days (certainly more prestigious than a "mere" postdoc). I don't think you can get a tenure-track job right out of your PhD, even an MIT fake tenure-track job (they say it guarantees you tenure, just not necessarily at MIT).
- mehrdadn 6y ago> I don't think you can get a tenure-track job right out of your PhD, even an MIT fake tenure-track job (they say it guarantees you tenure, just not necessarily at MIT). I know some (very bright) graduate students do get assistant professor positions at top universities right out of their PhDs; I think those are tenure-track?
- generationP 6y agoWho? My guess is that these are "named postdocs" ("[some name] Assistant Professor", e.g. https://www.mathjobs.org/jobs/jobs/14065 https://www.mathjobs.org/jobs/jobs/14065 or https://www.mathjobs.org/jobs/jobs/15707 https://www.mathjobs.org/jobs/jobs/15707 ). Despite their names, they're limited to 3 years and only get renewed in exceptional circumstances.
- mehrdadn 6y agoI'm not going to name names here but I'm aware of at least one person whose title is a "named" Assistant Professor and one whose title is (from what I can tell online) just "Assistant Professor".
- redis_mlc 6y agoGenerally in the US there's no career path to becoming a tenured professor. (A lot of assistant professors, etc. sleep in their cars.) You would have to have the right connections, or publish papers that were undeniably brilliant. (ie. be a celebrity.)
- pinacarlos90 6y agoam I the only one who got super exited with the progress bar on quantamagazine.org? it tracks how far into reading the article you are!! definitely going to steal the idea :)
- andromeduck 6y agoFinancial Times does it too :)
- Sharlin 6y agoIn the olden times we called those ”scroll bars” and you didn’t have to implement them yourself.
- diegoperini 6y agoSolving decades-old math/science problems deserves a name mention in the title.
- pgp001 6y agorip conway
- deleted 6y ago[deleted]
- essem 6y agoI see Piccirillo's paper was published in the Annals of Mathematics in February. Anyone know if Conway got to see it before he passed?