[impendia]

I am a math professor. In my observation, there is a huge amount of material available on the web, but it isn't very centralized -- especially at the upper levels.

My advice would be to get a book on any topic which interests you, read through it, and do a significant number of the exercises. You might try Epp's Discrete Mathematics, Hefferon's Linear Algebra, Colley's Vector Calculus, Dudley's Elementary Number Theory, Spivak or Apostol's Calculus (these go far beyond ordinary freshman calculus), Pinter's A Book of Abstract Algebra, among many others. Some of these books are expensive to buy new, but just buy older editions.

Resources like Khan Academy and 3blue1brown are also fantastic, and I have shared some of these with my students. I'd recommend using these as a supplement; if you rely on them solely then you'll develop vague intuition but not much else.

Also, with the pandemic, there are a huge number of traditional university courses that have moved online, and you could probably enroll in one for not too much money. Check the RateMyProfessor reviews -- you want a mix of positive reviews claiming the prof inspired them, and negative reviews complaining that work was expected. I have a RateMyProfessor review which complains bitterly that "homework is graded for accuracy and not completion". :)

[wcarss]

With many books, I'm not sure how to verify that I'm actually doing the exercises properly.

In the (simple!) maths I took, my misunderstandings often led me to do exercises incorrectly, but to think I'd done them right, or to have half a proof and no idea what insight I was missing. Even when solutions were provided (which was rare!), the path connecting the dots could be foggy. "What made them think to introduce this lemma?" kind of questions.

I often needed discussion with others to help figure out my misconceptions and correct them -- so... big ask, but do you have any further suggestions on how to replace talking with a TA, or with a smart group of upper-year friends?

[jimhefferon]

> I'm not sure how to verify that I'm actually doing the exercises properly.

Since the parent mentioned my text, Linear Algebra, I'll note that it is Free, and comes with worked answers to all exercises, including proofs. And since other posters mention videos, I'll also note that there are videos (and the slides are available), along with a lab manual using Sage. https://hefferon.net/linearalgebra/index.html

[akulkarni7]

I like you

[jimhefferon]

:-) I should say that my college, a SLAC, has kindly supported the work for many years, including tolerating the download traffic.

[impendia]

Very good question.

This is difficult. One way to do this is to participate on Math Stack Exchange:

https://math.stackexchange.com/

I'd recommend not only asking, but also answering others' questions. It's not the same as having a group of people that you regularly talk to, but it's something.

You could also hire a graduate student tutor. For example, here is a list of math grad students at the University of Wisconsin willing to do this:

https://www.math.wisc.edu/tutors

An hour every two weeks would go a long way.

[Valk3_]

Another alternative to hanging out in forums is to join a good math discord server. I can highly recommend this one: https://discordnetwork.com/, I have no idea how many people are in the server but I'm guessing at least more than 10k (about 9k online as I'm writing this, and more offline I have to presume). The advantage with such servers is fluid communication, but the downside is because of this in combination with the fact that there might be alot of people, you sometimes have to wait as to not disturb ongoing discussions.

This discord server in particular is good if you want help, although you might not recieve instant help, you will eventually get help if you are patient. There are also other channels in the server just for the purpose of discussion, i.e discussing general topics in math.

This might not give you what you are looking for, but it's a possible alternative I guess.

[rcollyer]

Another method for checking your understanding is to work through the worked out problems in the book and any proofs it may have, too. (Most of the value in a proof is in the techniques used to prove the result.) Work through them yourself, trying to understand each step, and attempt to do the problems in the same way. At first, justify each step you take, don't try to skip the "obvious" steps. Again, this isn't a perfect substitute for an instructor, but it can go a long way towards your goal.