mathematics - How can I get feedback for my work in math if I'm not in academia?

I will soon finish my undergraduate degree in math, and I want to study mathematics as a hobby after I graduate.

Online, I see a general trend of people saying

You can't learn real math without collaboration and feedback. Go to graduate school!

But I don't want to go to graduate school, and I have found that I learn better through self-study than by taking notes in class. That being said, it always does help to get my questions answered by professors, and collaboration with other students is sometimes helpful as well.

Question: Is it possible to have this kind of mathematical community without actually being in academia?

What I fear is studying for years by myself obsessively on a subject only to find that I had a horrible misconception the whole time that could have been avoided if only a professor had looked at my work.

Answer

I’ll give an answer based on personal experience. My situation is not exactly like yours, but may be similar enough to provide some insight.

I recently completed my undergrad in physics and applied math. Upon finishing this I decided that I was more interested in pure math, and wanted to pursue this in grad school. I ended up not getting into any of the Ph.D programs I wanted (probably because of a lack of pure math experience), so I was left in limbo, so to speak, still wanting to study math but not yet able to enroll on a graduate program.

My solution was to email some professors at a local university who I had found doing things somewhat related to what I was interested in. I told them my situation and explained that I was just looking to learn more about their research and math in general. This ended up working out and now I attend regular seminars at the university and I meet up with them every once in awhile to just discuss math.

I have found it very helpful to be able to have face to face conversations with other mathematicians, even though I do most studying on my own at home. Though Stack Exchange is a valuable resource, I don’t think it can adequately replace this interaction.

I’d like to point out that I am not sure if my experience is typical. After all, this is a somewhat small university that I am talking about, and it may be harder to find professors at larger universities who are able to take extra time to talk to you (I certainly found this to be the case at UC Berkeley, which is where I did my undergrad). Still, I think that this is an option you may be interested in considering.