Personally, I think this is bad advice, because without an undergrad+ background, the projects above will either be impossibly frustrating or you will make up some crackpot bullshit. Plus, the undergraduate curriculum is its own reward.
Not necessarily. I left undergrad after four terms and self-studied to the point of having published research, giving invited talks, and even being a visiting researcher for three months with my expenses paid.
And I think abnry is right that you need some motivating problem / project. In my case it was to prove the Riemann hypothesis. Obviously I have not succeeded but I've learned a lot in the process and it has indirectly led me to some good research questions. I think choosing an outrageously ambitious pie-in-the-sky problem is ok if you are patient and don't try to approach it too directly.
As an aside, your work on numerical cohomology appears to have been useful for a new result pertaining to lattices. Given the authors of the followup work it's likely helpful for the study of lattices in post-quantum cryptography.
Are you referring to the "An Inequality for Gaussians on Lattices" paper? They cite my paper, but it's to give an example of an application of their result (which I use), not because they built on it. But anyway, I think it's very fascinating that the people who discovered a key result that I needed for that paper, which could probably be best classified as arithmetic geometry, are mainly computer scientists!
Ah, thanks for the clarification. The computer scientists who work on quantum computational complexity and post-quantum cryptography tend to be much more mathematical than the norm :)
Sometimes yes, sometimes no. Of course I can go into much more depth in my studies / research while not working. I work the minimum amount necessary to pay my living expenses so I can devote a maximum amount of time to freely pursuing other interests, which includes pure math among other things.
There are plenty of people who devote several years of their life to studying, and must pay not only their living expenses but tuition fees as well. In my opinion, those are the people who you should be asking “how did you manage this”.
Nah, a good way to learn new skills is to pick a destination and then figure out what steps you need to take to get there. This type of “top-down” learning can help one stay motivated through the most frustrating road blocks. This is especially important for self-learning, because unlike an undergrad setting, the person is on their own and can’t rely on peers.
Depending on your background, you probably don't have enough information to pick a long-term goal anyway.
I am afraid that sounds curmudgeonly, but I have also seen students shoot themselves in the foot because they decided they didn't need a class for their not very well informed goals.
>>Depending on your background, you probably don't have enough information to pick a long-term goal anyway.
Nah, it's totally possible for newbies to pick high-level long-term goals.
This can be something like "I want to teach my computer to tell apart dogs and cats", or "I want to create a website where people can buy and sell yarn." From there, Google searches can direct someone towards concepts and various methods of learning them.
I mean, you can disagree all you want, but this is in fact how many people learn things.
The two of you are talking about different things. What forkandwait is talking about is the propensity for people with only an undergraduate education in math (or less) to not actually know what a worthwhile goal is. They usually either lack the mathematical maturity to intuit how difficult a particular problem is (whether it's tractable with available mathematics, whether it's tractable for their ability, etc); or they formulate problems which are "not even wrong."
Of course this is in the context of choosing research problems to strive towards in math. If you tasked yourself with solving an open problem in math, it's more likely than not that, without any collaboration, you'd have no idea how to even work towards the goal due to all the unknown unknowns. If your goal is something concrete that can be augmented with mathematics, then yes I agree that goal setting can be useful. It doesn't take a volume of missing domain knowledge to develop that kind of goal.
Not in math. Unless you are a mathematician, I challenge you to pick a math equivalent of "I want to create a website where people can buy and sell yarn." I’ll wait.
People just assume learning math is the same as learning everything else. That is not even remotely true.
Genuinely curious - why do you think that? I have been self studying math for about 4 years now and find it to be the same as everything else that's worth learning: hard! But I haven't found that it's some entirely different realm divorced from all other intellectual pursuits.
I don’t really have time to give a thoughtful answer (it would be quite long), but the exact post you responded to gave an obvious difference. To roughly summarize that difference, producing anything of value in mathematics requires learning a tremendous amount of prior art, and without a tremendous amount of work you won’t even know what’s of value. It’s no wonder that many crackpots choose to work on high profile number theory problems, like Goldbach’s conjecture and previously Fermat’s Last Theorem, since the formulations are simple enough for laypeople to understand, yet the theories behind developed over hundreds of years are incredibly deep.
> everything else that’s worth learning: hard!
I disagree. I’ve learned many things worth learning that are not hard at all, but to each their own.
This is fair. That's why you need to make the goals your own and make sure they are doable. And sometimes, an unreachable goal will still help you learn and value the fundamental material.
Like, if you are into Rubik's cubes, that's going to make learning group theory a lot more fun and motivating.
I'm with you but I think it depends on the person.
I have actually found people to be very different in this regard.
For instance, I work in data science and a lot of my peers like to learn about new techniques by applying them to real problems or working with datasets and exploring.
I don't like that approach. I always like to learn the theory of something before using it.
Similarly, I had the goal of learning math for statistics and general relativity among other subjects.
But my desire to have a deep understanding, ended up with me essentially learning the equivalent an undergraduate math degree and selected graduate level topics.
In my case, just learning in a bottom up way with maybe a slight direction would have been sufficient.
Every example project suggested by the GP can be easily externally verified, so it has built-in protection against crackpotism. Secondly, it may be very difficult, but not impossible.
I think maybe the most likely outcome, if they were really motivated and somewhat capable, is that they learn lots of mathematics well—but maybe not quite to, "... through graduate level" (that phrasing is a bit ambiguous though; self-teaching 'up to' graduate level is definitely doable).
