Bsrgvty

Admittedly, a soft-question.

I, being a very young researcher (PhD student) have personally faced the following situation many times: You delve into a problem desperately. No progress for a very long while. All of a sudden, you get the light, and boom: the result is proven. Looking in retrospect, though, the result looks extremely obvious (to the extent you sometimes are ashamed of not getting till then, or embarrassed sharing/publishing).

I wonder people's personal opinion on this matter (would also like to hear several real-stories on it, related to published papers).

Note If moderators believe Mathoverflow is not the right venue for my question, I can consider relocating it.

Interpretation #1: P vs. NP

There are many hard-to-solve problems with easily verified solutions. It can be a real talent to present a proof which is especially easy for humans to verify. In like manner, a good counterexample can be very enlightening. Those who have searched for counterexamples can testify to the fact that these can be very hard to find in the first place.

Do not confuse the ease of verifying a solution with the difficulty of finding that solution.

Interpretation #2: Kolmogorov complexity

There are many proofs in the literature that are gigantic case checks, or extremely complicated analytic arguments handling many cases at once. One of the most famous is the four color theorem. Another is Helfgott's proof of Goldbach's weak conjecture. Some proofs require an extraordinary amount of data, which simply cannot be compressed into a smaller space. However, we don't usually celebrate the fact that a computer actually checked all those cases (even though that really does need to be done). Rather, we delight in the fact that we understand how to turn it into a finite task, doable in a small time-frame. The important parts of the proof are the key ideas, which sometimes are quite small in comparison to the rest of the work.

Do not confuse length with importance.

Interpretation #3: Obfuscation

Good mathematical writing makes things clearer. One should "eschew obfuscation, espouse elucidation". My greatest joy in my research is discovering something new and then explaining it to others, in a way that makes it clear and easy to them as well. In my experience, a large amount of time should go into writing the solution well.

Do not confuse lack of clarity with brilliance.

Certainly I myself tend to see anything that I understand well as "being nearly obvious" (although I know better).

Also, I and many other people I know have realized upon completion of a PhD that their advisor "probably could have done this in an afternoon, if they cared...". Right. Much of a PhD (in math) in my opinion is getting-up-to-speed on technique, so that what was impossible before is at least approachable... if only due to acquisition of good technique.

Also, some or many parts of mathematics, perhaps the most useful parts, are very robust, in the sense that once we see the mechanism, we do not have to be particularly careful to have things work out and not fail... So, yes, once one is acquainted with such robust stuff, and has assimilated such things into one's "intuition", lots of things are "easy".

A few things seem to remain permanently delicate... and I myself have a hard time understanding them.

A point that seems often overlooked is that, in my perception of myself and many others, the course of a (successful, substantial) research project involves as much changing oneself as anything else. So one's perception has changed, so that what was unobvious is now obvious. The thing itself probably did not change much...?

Yes, this can be misunderstood as one's own unforgiveable slowness to understand (since after-the-fact it seems so easy), but I think that is a far too naive appraisal of the mechanism.

One issue to keep in mind is that things look obvious after you've been thinking about that thing for weeks, or months, or possibly years. Then when you have a proof, you think about how to present it, and how to generalize the idea in question. After all of that, the basic direction will seem more obvious to you than to someone who hasn't thought about it that much, since you've gone down the same mental paths many times. This also has a consequence in teaching; if there's disconnect for research, it may be even stronger when one is dealing with things one considers basic or obvious because you've used them since you were young.