5 ms·

What's some low-hanging fruit in mathematics?

daxfohl 8y ago
For example: https://www.quantamagazine.org/amateur-mathematician-finds-smallest-universal-cover-20181115/ https://www.quantamagazine.org/amateur-mathematician-finds-s... No way you'd get research money for this, but an amateur can sit and poke away at it. These are all unsolved: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_m..., and plenty look like they may be solvable without advanced math, just someone dedicated and a unique approach. Particularly the Collatz Conjecture, afaik basically zero progress has been made on it, no "advanced" math topics are known to apply at all. So perhaps a unique angle is all you need. (Or maybe it's false, in which case all you need is a counterexample. Sometimes it happens, see e.g. https://www.ams.org/journals/mcom/1988-51-184/S0025-5718-1988-0930224-9/S0025-5718-1988-0930224-9.pdf https://www.ams.org/journals/mcom/1988-51-184/S0025-5718-198...)
- intuitionist 8y ago
  For another, more recent counterexample see the Hirsch conjecture: https://en.m.wikipedia.org/wiki/Hirsch_conjecture https://en.m.wikipedia.org/wiki/Hirsch_conjecture
- earthicus 8y ago
  Here's an old post by terry tao explaining why the collatz conjecture is hard, and giving a heuristic argument that it's unlikely to be solved with purely elementary techniques. The essay is surely worth reading if you want to make an attempt on that problem! https://terrytao.wordpress.com/2011/08/25/the-collatz-conjecture-littlewood-offord-theory-and-powers-of-2-and-3/ https://terrytao.wordpress.com/2011/08/25/the-collatz-conjec...
- cjohnson318 8y ago
  I don't think that a list of historically unsolved problems in math, and the Collatz Conjecture in particular, count as low-hanging fruit. The fact that they have remained unsolved for so long suggests that they are rather high-hanging.
  - daxfohl 8y ago
    Heh, fair enough. Was thinking more things that might not need advanced math to solve. Or at least to understand the problem.
xamuel 8y ago
Here's one example. In Darwin's "Origin of Species", a key principle which Darwin talks about at great length is the so-called Knight-Darwin Law, the observation that even in apparently asexual species, sexual reproduction still occasionally occurs, if only after long spans of time. In an article in Acta Biotheoretica, I argued this is in fact an infinite graph-theoretical principle. If G is the directed graph of all organisms, where an edge from x to y means that x is a parent of y, then (Darwin theorizes that) there is no infinite directed path through G all of whose vertices have indegree 1. (A vertex has indegree 1 if it has exactly 1 parent.) The low-hanging fruit (some of which I picked in my paper): Investigate the mathematical implications of the above property (esp. if G also has other biologically motivated properties). The Knight-Darwin law seems to have vanished completely from the literature around the start of the 20th century, until I cited it. Almost as if no-one has actually been reading Darwin in all that time. To generalize: If you want to find low-hanging fruit, read non-fiction classics like Darwin and constantly be asking, "Is this some unrecognized mathematics in disguise?"
- Ericson2314 8y ago
  If the edges go to parent you mean out degree one.
- maxander 8y ago
  I would further generalize; if you want to find low-hanging fruit, know about relevant subjects that aren't presently very popular (such as the Knight-Darwin law) and find relations between disparate fields (such as evolutionary biology and graph theory) where few others are well-versed in both.
  - 0-_-0 8y ago
    I definitely have the impression that many of the "low hanging fruits" are found at the intersection of disparate fields. There are much more people highly proficient in a single field than mediocre in two different fields, so if you want to discover something new and important, look at a new field through the lens of one you're proficient in.
  - 8y ago
svat 8y ago
In his 2017 Christmas lecture (https://news.ycombinator.com/item?id=15898919 https://news.ycombinator.com/item?id=15898919), Donald Knuth mentions at around the 5-minute mark: > You know, people think that mathematicians have been working for hundreds of years, and now there's tens of thousands or hundreds of thousands of mathematicians all spending every day working on problems, so how could you possibly still find a simple down-to-earth problem that hasn't already been studied, you know, way too much? And the answer is: those problems aren't rare at all, they keep coming up several times a year, and this is an example where even this very basic problem of rectangle into rectangles has all kinds of stones not yet turned. You can see in the talk where he starts simple and just keeps working on it, until he gets something that seems new (a conjecture, which he proposed as a problem in the American Mathematical Monthly). Of course, this particular problem is now solved, but if you just keep exploring you'll pretty soon hit the limits of what's known. (Potential exercise that I haven't tried: load random sequences on OEIS http://oeis.org/ http://oeis.org/ until you hit one where there are only conjectures, and not much research.)
Sniffnoy 8y ago
Well, I can't tell you low-hanging fruit that's still out there, because if I knew it, I'd be working on it instead of telling you. :) But I can tell you some low-hanging fruit that I've done! I'll stick to just one example, this[0] paper of mine was, like... I was astonished this was not already in the literature -- everything in this paper is easy to prove once you think to ask the question -- but as best as I and anyone else could tell, somehow nobody had thought to ask the question despite having asked other extremely similar ones. It's especially astonishing that Jacobsthal's exponentiation was independently rediscovered multiple times, but what I call "super-Jacobsthal exponentiation" seems to have only been considered once before and only briefly as a tool for something else (nobody thought to write down its algebraic laws). So, I wrote it up to make sure it was out there. (Honestly I'd say a lot of my work has been LHF, I got through grad school while learning hardly any of the heavy machinery one normally does, unfortunately most of it isn't written up yet and so I can't really easily point to it.) [0]https://arxiv.org/abs/1501.05747 https://arxiv.org/abs/1501.05747 (This, um, will require some knowledge of ordinals to read.)
- semi-extrinsic 8y ago
  There's plenty examples indeed if you just dig around. I have a colleague who published a theoretical paper [0] in one of the top fluid mechanics journals 5 years back. That paper originated from his lecture notes in a course that he gave for the first time, where in one part he followed the classical theory of waves behind a ship done by Lord Kelvin and others back in the olden days. Then he thought "Hmm, how do we extend this so it works if there is a current in the water?" And it turned out to be both possible to solve analytically, and that nobody had done it before. Nobody had thought it was possible to do. It's an example of "low-hanging fruit" that had been sitting since 1887. I think the reason why there's "lots" of LHF is that science has become so incremental, iterating on recent results. If you go back 20-50-100 years and look at the road less traveled, there's plenty of LHF, but going down that path and shaking the trees requires more effort per initial publication than most academics can afford under the current system. But if you can afford (or get lucky enough) to find that first LHF, it usually gives you enough material to work on for quite some time such that it pays off over time. [0] https://arxiv.org/pdf/1401.1682v1 https://arxiv.org/pdf/1401.1682v1
Ar-Curunir 8y ago
I think it's not just a matter of low-hanging fruit, but low-hanging fruit that's interesting to other people and applicable (to other research or to the real world)
Zanni 8y ago
My favorite example of (formerly) low-hanging fruit is the formula for determining how often you can fold a piece of paper in half. Is there anyone who didn't grow up "learning" that you can't fold a piece of paper in half more than six or seven times? Until high school student Britney Gallivan found the formula, with basic algebra, and extended the record to twelve folds. [0] [0] https://en.wikipedia.org/wiki/Britney_Gallivan https://en.wikipedia.org/wiki/Britney_Gallivan
LolWolf 8y ago
Another fun one (from the many examples above!) is a new paper I am working on at the moment as well, but in the intersection of physics and math: it turns out you can phrase a bunch of physics problems as optimization problems, then use some very simple tools in optimization to derive fundamental limits of how well any device could ever perform. The preprint is here: https://arxiv.org/abs/1811.12936 https://arxiv.org/abs/1811.12936 An associated blog post is here: https://guille.site/physics-impossibility-results.html https://guille.site/physics-impossibility-results.html
anonlastname 8y ago
Perhaps people who think mathematics is full of low hanging fruit are the same ones who grew up looking for binomial coefficient identities and calculating the 3n+1 sequence. It might not seem low hanging to everyone, but maybe to mathematicians.

For example: https://www.quantamagazine.org/amateur-mathematician-finds-smallest-universal-cover-20181115/ https://www.quantamagazine.org/amateur-mathematician-finds-s... No way you'd get research money for this, but an amateur can sit and poke away at it. These are all unsolved: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_m..., and plenty look like they may be solvable without advanced math, just someone dedicated and a unique approach. Particularly the Collatz Conjecture, afaik basically zero progress has been made on it, no "advanced" math topics are known to apply at all. So perhaps a unique angle is all you need. (Or maybe it's false, in which case all you need is a counterexample. Sometimes it happens, see e.g. https://www.ams.org/journals/mcom/1988-51-184/S0025-5718-1988-0930224-9/S0025-5718-1988-0930224-9.pdf https://www.ams.org/journals/mcom/1988-51-184/S0025-5718-198...)

intuitionist 8y ago
For another, more recent counterexample see the Hirsch conjecture: https://en.m.wikipedia.org/wiki/Hirsch_conjecture https://en.m.wikipedia.org/wiki/Hirsch_conjecture
earthicus 8y ago
Here's an old post by terry tao explaining why the collatz conjecture is hard, and giving a heuristic argument that it's unlikely to be solved with purely elementary techniques. The essay is surely worth reading if you want to make an attempt on that problem! https://terrytao.wordpress.com/2011/08/25/the-collatz-conjecture-littlewood-offord-theory-and-powers-of-2-and-3/ https://terrytao.wordpress.com/2011/08/25/the-collatz-conjec...
cjohnson318 8y ago
I don't think that a list of historically unsolved problems in math, and the Collatz Conjecture in particular, count as low-hanging fruit. The fact that they have remained unsolved for so long suggests that they are rather high-hanging.
- daxfohl 8y ago
  Heh, fair enough. Was thinking more things that might not need advanced math to solve. Or at least to understand the problem.

xamuel 8y ago

Here's one example. In Darwin's "Origin of Species", a key principle which Darwin talks about at great length is the so-called Knight-Darwin Law, the observation that even in apparently asexual species, sexual reproduction still occasionally occurs, if only after long spans of time. In an article in Acta Biotheoretica, I argued this is in fact an infinite graph-theoretical principle. If G is the directed graph of all organisms, where an edge from x to y means that x is a parent of y, then (Darwin theorizes that) there is no infinite directed path through G all of whose vertices have indegree 1. (A vertex has indegree 1 if it has exactly 1 parent.) The low-hanging fruit (some of which I picked in my paper): Investigate the mathematical implications of the above property (esp. if G also has other biologically motivated properties). The Knight-Darwin law seems to have vanished completely from the literature around the start of the 20th century, until I cited it. Almost as if no-one has actually been reading Darwin in all that time. To generalize: If you want to find low-hanging fruit, read non-fiction classics like Darwin and constantly be asking, "Is this some unrecognized mathematics in disguise?"

Ericson2314 8y ago
If the edges go to parent you mean out degree one.
maxander 8y ago
I would further generalize; if you want to find low-hanging fruit, know about relevant subjects that aren't presently very popular (such as the Knight-Darwin law) and find relations between disparate fields (such as evolutionary biology and graph theory) where few others are well-versed in both.
- 0-_-0 8y ago
  I definitely have the impression that many of the "low hanging fruits" are found at the intersection of disparate fields. There are much more people highly proficient in a single field than mediocre in two different fields, so if you want to discover something new and important, look at a new field through the lens of one you're proficient in.
- 8y ago

svat 8y ago

In his 2017 Christmas lecture (https://news.ycombinator.com/item?id=15898919 https://news.ycombinator.com/item?id=15898919), Donald Knuth mentions at around the 5-minute mark: > You know, people think that mathematicians have been working for hundreds of years, and now there's tens of thousands or hundreds of thousands of mathematicians all spending every day working on problems, so how could you possibly still find a simple down-to-earth problem that hasn't already been studied, you know, way too much? And the answer is: those problems aren't rare at all, they keep coming up several times a year, and this is an example where even this very basic problem of rectangle into rectangles has all kinds of stones not yet turned. You can see in the talk where he starts simple and just keeps working on it, until he gets something that seems new (a conjecture, which he proposed as a problem in the American Mathematical Monthly). Of course, this particular problem is now solved, but if you just keep exploring you'll pretty soon hit the limits of what's known. (Potential exercise that I haven't tried: load random sequences on OEIS http://oeis.org/ http://oeis.org/ until you hit one where there are only conjectures, and not much research.)

Sniffnoy 8y ago

Well, I can't tell you low-hanging fruit that's still out there, because if I knew it, I'd be working on it instead of telling you. :) But I can tell you some low-hanging fruit that I've done! I'll stick to just one example, this[0] paper of mine was, like... I was astonished this was not already in the literature -- everything in this paper is easy to prove once you think to ask the question -- but as best as I and anyone else could tell, somehow nobody had thought to ask the question despite having asked other extremely similar ones. It's especially astonishing that Jacobsthal's exponentiation was independently rediscovered multiple times, but what I call "super-Jacobsthal exponentiation" seems to have only been considered once before and only briefly as a tool for something else (nobody thought to write down its algebraic laws). So, I wrote it up to make sure it was out there. (Honestly I'd say a lot of my work has been LHF, I got through grad school while learning hardly any of the heavy machinery one normally does, unfortunately most of it isn't written up yet and so I can't really easily point to it.) [0]https://arxiv.org/abs/1501.05747 https://arxiv.org/abs/1501.05747 (This, um, will require some knowledge of ordinals to read.)

semi-extrinsic 8y ago
There's plenty examples indeed if you just dig around. I have a colleague who published a theoretical paper [0] in one of the top fluid mechanics journals 5 years back. That paper originated from his lecture notes in a course that he gave for the first time, where in one part he followed the classical theory of waves behind a ship done by Lord Kelvin and others back in the olden days. Then he thought "Hmm, how do we extend this so it works if there is a current in the water?" And it turned out to be both possible to solve analytically, and that nobody had done it before. Nobody had thought it was possible to do. It's an example of "low-hanging fruit" that had been sitting since 1887. I think the reason why there's "lots" of LHF is that science has become so incremental, iterating on recent results. If you go back 20-50-100 years and look at the road less traveled, there's plenty of LHF, but going down that path and shaking the trees requires more effort per initial publication than most academics can afford under the current system. But if you can afford (or get lucky enough) to find that first LHF, it usually gives you enough material to work on for quite some time such that it pays off over time. [0] https://arxiv.org/pdf/1401.1682v1 https://arxiv.org/pdf/1401.1682v1

Ar-Curunir 8y ago

I think it's not just a matter of low-hanging fruit, but low-hanging fruit that's interesting to other people and applicable (to other research or to the real world)

Zanni 8y ago

My favorite example of (formerly) low-hanging fruit is the formula for determining how often you can fold a piece of paper in half. Is there anyone who didn't grow up "learning" that you can't fold a piece of paper in half more than six or seven times? Until high school student Britney Gallivan found the formula, with basic algebra, and extended the record to twelve folds. [0] [0] https://en.wikipedia.org/wiki/Britney_Gallivan https://en.wikipedia.org/wiki/Britney_Gallivan

LolWolf 8y ago

Another fun one (from the many examples above!) is a new paper I am working on at the moment as well, but in the intersection of physics and math: it turns out you can phrase a bunch of physics problems as optimization problems, then use some very simple tools in optimization to derive fundamental limits of how well any device could ever perform. The preprint is here: https://arxiv.org/abs/1811.12936 https://arxiv.org/abs/1811.12936 An associated blog post is here: https://guille.site/physics-impossibility-results.html https://guille.site/physics-impossibility-results.html

anonlastname 8y ago

Perhaps people who think mathematics is full of low hanging fruit are the same ones who grew up looking for binomial coefficient identities and calculating the 3n+1 sequence. It might not seem low hanging to everyone, but maybe to mathematicians.