<p>Maybe this question would make more sense in academia.SE. At least, my answer will be closer to the kind of answers you can read there.</p>
<p>It is hard to say if this is what is happening to you right now, but it often happens to me and other researchers I know that once you have understood the problem you are trying to solve well enough, then everything feels easy in the end. The thing is, mastering something often boils down to make it trivial in some sense: trivial to you because you finally have access to the big picture and ignore tiny details that were blocking the view before, or because you have found the right tool to crack it open. And, if you do it well, trivial to others because the other part of your job as a researcher now starts: how can you communicate this new knowledge so that the problem becomes &quot;trivial&quot; to the readers of your paper or the audience of your talk?</p>
<p>The thing you have to ask yourself is not whether it is trivial or not because everything is trivial at some point:  triviality is a relative concept, relative to some kind of assumed &quot;basic&quot; knowledge among your peers. The right question you have to ask is: is it really trivial to the most people interested in the question? If not, then it is important to communicate on it, and the fact that it now feels trivial is actually a very good thing: it means that you have made your results accessible to many people and that they are now easy to communicate and to build on (at this point of course, Reviewer 2 will complain that it is trivial and the paper will be rejected many times but this is not a problem with the results themtselves and is another story for another day).</p>
<p>As an example: I think that solving linear equations has not always felt trivial to you nor to the humanity in general. Then some dude, apparently called Gauss, came and showed how trivial it was with a method he called Gaussian elimination (this paragraph is completely wrong from a historical point of view and only serves as a punchy example). For a more recent example, have a look at what happened to the sensitivity conjecture 5 years ago: <a href="https://www.quantamagazine.org/mathematician-solves-computer-science-conjecture-in-two-pages-20190725/" rel="noreferrer">https://www.quantamagazine.org/mathematician-solves-computer-science-conjecture-in-two-pages-20190725/</a></p>
<p>Sometimes however, we work on something trivial that has been known (often implicitly) trivial by the community for a long time but we realize it only after some months. Then this is bad luck but this is part of the job. And when it happens, we are better than before, even if we do not write a paper in the end. Although, to be honest, there exists a paper from 1994 rediscovering the rectangle method for integration! But then, it may mean that the problem was <strong>not</strong> trivial for the community targeted by the paper, hence we are back to my original point.</p>
<p>Let me give two personal but related examples as a conclusion. One of my music teacher once said something which has always stayed with me: &quot;<em>if a musician plays something and it sounds hard to play to the audience, then he is not playing it well enough yet</em>&quot;. I guess, this also apply to the musician itself, and to any human activity, writing proofs for example!</p>
<p>My second personal anecdote is about one of my result. A coauthor of mine and me were studying a very natural but open question in enumeration complexity. We were stuck for several years, not even able to decide which one of <span class="math-container">$A$</span> or <span class="math-container">$\neg A$</span> was true. One day, he came to me saying he solved it and showed me a proof on the blackboard: lots of log(FOO/27) and complicated while(FOO) loops. But there was a new core idea. I told him I will try to write a fully formal proof to better understand his idea and make sure it was correct. I started working on it. I tried to simplify some calculations to make the overall presentation clearer. Then I found some recursive way of writing the algorithm that we could then unpack into a <strong>completely trivial-easy-to-write-explain-and-prove-correctness</strong> algorithm. But this was new and interesting. We hence wrote a paper with a now-trivial theorem as main contribution. Reviewer 2 complained a lot but in the end it was accepted!</p>

CC BY-SA 4.0
2024-11-25T09:46:14.397Z
2024-11-25T09:46:14.397Z
holf
<p>I'm working with this professor at a well-known university, and the professor has an h-index of 50+, so to a certain degree I trust his judgement. Do note that the project we are working on is not really his expertise (think an expert on some specific aspect of engineering ML working on a theoretical learning algorithm. This is the main point I'm worried about since I sense a pretty big gap between when engineering domains think a problem is trivial and when theoretical domains think a problem is trivial).</p>
<p>We just figured out a critical part of a research project. I'm not saying it was a trivial process as it did take two months, but once we realized what problem we were asking and the algorithm we really needed to solve that problem (it's entirely theoretical algorithmic learning theory), I feel like it would even be a fair second-semester algo class homework problem (again, if it is stated clearly and if the specific technique is taught, which I think should be for at most a second-semester algo class).</p>
<p>Is this a sign that the problem is too easy? Or is it just how some research problems feel, and it can still be legit. How do I find out?</p>

CC BY-SA 4.0
2024-11-23T23:42:40.720Z
Is it NP-complete?
Daniel Donnelly
CC BY-SA 4.0
2024-11-25T01:06:53.137Z
You mean the problem? It's much more fine-grained. Think about trying to improve some existing O(n^c) algorithm to some O(n^{c-\epsilon}) for some \epsilon > 0.
theoretically nobody
CC BY-SA 4.0
2024-11-25T02:03:53.627Z
That's quite difficult to do and understand.  My hat is off to you.   Remember, there exist no low-hanging fruit any longer.  All the fruit is now impossibly high up there on the tree of problems, and each incremental step at proving something about them is a huge step forward, not to laymen who don't understand it but instead take their technology for granted, but to us, the mathematicians of the world.
Daniel Donnelly
CC BY-SA 4.0
2024-11-25T11:54:31.503Z
You're confusing familiairty with triviality.
Andrej Bauer
<soft-question><research-practice>
How normal is it to feel the core problem of my theoretical research somewhat trivial?
theoretically nobody
# How normal is it to feel the core problem of my theoretical research somewhat trivial?

(Asked by: username_0 on 2024-11-23)

I'm working with this professor at a well-known university, and the professor has an h-index of 50+, so to a certain degree I trust his judgement. Do note that the project we are working on is not really his expertise (think an expert on some specific aspect of engineering ML working on a theoretical learning algorithm. This is the main point I'm worried about since I sense a pretty big gap between when engineering domains think a problem is trivial and when theoretical domains think a problem is trivial).

We just figured out a critical part of a research project. I'm not saying it was a trivial process as it did take two months, but once we realized what problem we were asking and the algorithm we really needed to solve that problem (it's entirely theoretical algorithmic learning theory), I feel like it would even be a fair second-semester algo class homework problem (again, if it is stated clearly and if the specific technique is taught, which I think should be for at most a second-semester algo class).

Is this a sign that the problem is too easy? Or is it just how some research problems feel, and it can still be legit. How do I find out?

**Tags:** soft-question, research-practice
> **Comments:**
> **username_1**: Is it NP-complete?  
> **username_0**: You mean the problem? It's much more fine-grained. Think about trying to improve some existing O(n^c) algorithm to some O(n^{c-\epsilon}) for some \epsilon > 0.  
> **username_1**: That's quite difficult to do and understand.  My hat is off to you.   Remember, there exist no low-hanging fruit any longer.  All the fruit is now impossibly high up there on the tree of problems, and each incremental step at proving something about them is a huge step forward, not to laymen who don't understand it but instead take their technology for granted, but to us, the mathematicians of the world.  
> **username_2**: You're confusing familiairty with triviality.  

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## 1 answers

### Answer by username_3 on 2024-11-25. Score: 5

Maybe this question would make more sense in academia.SE. At least, my answer will be closer to the kind of answers you can read there.

It is hard to say if this is what is happening to you right now, but it often happens to me and other researchers I know that once you have understood the problem you are trying to solve well enough, then everything feels easy in the end. The thing is, mastering something often boils down to make it trivial in some sense: trivial to you because you finally have access to the big picture and ignore tiny details that were blocking the view before, or because you have found the right tool to crack it open. And, if you do it well, trivial to others because the other part of your job as a researcher now starts: how can you communicate this new knowledge so that the problem becomes "trivial" to the readers of your paper or the audience of your talk?

The thing you have to ask yourself is not whether it is trivial or not because everything is trivial at some point: triviality is a relative concept, relative to some kind of assumed "basic" knowledge among your peers. The right question you have to ask is: is it really trivial to the most people interested in the question? If not, then it is important to communicate on it, and the fact that it now feels trivial is actually a very good thing: it means that you have made your results accessible to many people and that they are now easy to communicate and to build on (at this point of course, Reviewer 2 will complain that it is trivial and the paper will be rejected many times but this is not a problem with the results themtselves and is another story for another day).

As an example: I think that solving linear equations has not always felt trivial to you nor to the humanity in general. Then some dude, apparently called Gauss, came and showed how trivial it was with a method he called Gaussian elimination (this paragraph is completely wrong from a historical point of view and only serves as a punchy example). For a more recent example, have a look at what happened to the sensitivity conjecture 5 years ago: <https://www.quantamagazine.org/mathematician-solves-computer-science-conjecture-in-two-pages-20190725/>

Sometimes however, we work on something trivial that has been known (often implicitly) trivial by the community for a long time but we realize it only after some months. Then this is bad luck but this is part of the job. And when it happens, we are better than before, even if we do not write a paper in the end. Although, to be honest, there exists a paper from 1994 rediscovering the rectangle method for integration! But then, it may mean that the problem was **not** trivial for the community targeted by the paper, hence we are back to my original point.

Let me give two personal but related examples as a conclusion. One of my music teacher once said something which has always stayed with me: "*if a musician plays something and it sounds hard to play to the audience, then he is not playing it well enough yet*". I guess, this also apply to the musician itself, and to any human activity, writing proofs for example!

My second personal anecdote is about one of my result. A coauthor of mine and me were studying a very natural but open question in enumeration complexity. We were stuck for several years, not even able to decide which one of $A$ or $\neg A$ was true. One day, he came to me saying he solved it and showed me a proof on the blackboard: lots of log(FOO/27) and complicated while(FOO) loops. But there was a new core idea. I told him I will try to write a fully formal proof to better understand his idea and make sure it was correct. I started working on it. I tried to simplify some calculations to make the overall presentation clearer. Then I found some recursive way of writing the algorithm that we could then unpack into a **completely trivial-easy-to-write-explain-and-prove-correctness** algorithm. But this was new and interesting. We hence wrote a paper with a now-trivial theorem as main contribution. Reviewer 2 complained a lot but in the end it was accepted!


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