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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
by melvinroest 6y ago
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).
- 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)
- sdenton4 6y agoMy 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.
- tnecniv 6y agoThis is pretty common. I had a professor who worked in an applied math domain and said his whole career was based on finding the right change of coordinates.
- redis_mlc 6y agoYour prof might have just been parroting Grothendieck, who was famous for that: https://en.wikipedia.org/wiki/Alexander_Grothendieck https://en.wikipedia.org/wiki/Alexander_Grothendieck
- deleted 6y ago[deleted]