[HN Gopher] Graduate Student Solves Decades-Old Conway Knot Problem
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Graduate Student Solves Decades-Old Conway Knot Problem
Author : theafh
Score : 240 points
Date : 2020-05-19 16:17 UTC (6 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| JoeAltmaier wrote:
| This 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.
| hypersoar wrote:
| It's been known to happen. Would that all of our Ph.D.s were
| that easy.
| high_derivative wrote:
| I 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 wrote:
| And 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 wrote:
| This. 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-
| naive 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.
| kkylin wrote:
| Reminiscent of Dantzig's story:
| https://en.wikipedia.org/wiki/George_Dantzig#Mathematical_st...
| m3kw9 wrote:
| " 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 wrote:
| You 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.
| generationP wrote:
| Correction: Her MIT position (a Moore instructor, at least
| according to her own website
| https://sites.google.com/view/lpiccirillo/home ) is _not_ a
| tenure-track job (see 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 wrote:
| > 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 wrote:
| Who?
|
| My guess is that these are "named postdocs" ("[some name]
| Assistant Professor", e.g.
| https://www.mathjobs.org/jobs/jobs/14065 or
| https://www.mathjobs.org/jobs/jobs/15707 ). Despite their
| names, they're limited to 3 years and only get renewed in
| exceptional circumstances.
| mehrdadn wrote:
| I'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".
| smitty1e wrote:
| Outstanding work.
|
| Let us dub Lisa Piccirillo a "Space Age Bo's'n Mate".
|
| https://en.wikipedia.org/wiki/Boatswain
| wesammikhail wrote:
| Too bad Conway didnt live long enough to see this.
| ttymck wrote:
| If I'm not mistaken, it says her findings were published in
| Annals in February, and Conway passed in April.
| gre wrote:
| The paper is from 2018 so he might have seen it.
| melvinroest wrote:
| From 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).
| sdenton4 wrote:
| My area was algebraic combinatorics, which is all about this
| kind of thing. Combinatorics, representation theory, and
| geometry share a bunch of objects related by some translations
| from one field to another. So you make progress in one field
| until you get stuck, then translate to another to make further
| progress... And repeat.
| empath75 wrote:
| Dualities are what you're thinking about.
|
| https://en.m.wikipedia.org/wiki/Duality_(mathematics)
| terramex wrote:
| You 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.
| [deleted]
| j7ake wrote:
| What a great story. It's something we can all learn from.
| contemporary343 wrote:
| Even 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.
| [deleted]
| melvinroest wrote:
| I 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-impos...
| contemporary343 wrote:
| An 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/
|
| 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.
| pmiller2 wrote:
| Link to paper: 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.
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