How do I learn advanced mathematics without forgetting?

One way to remember the mathematics that you learn is to create a narrative that explains how the concepts are threaded together. This narrative need not have any historical relevance (and it usually cannot, since historical developments are usually tangled and messy). The goal is to lay down a road from start to finish, passing through all the important checkpoints (definitions, theorems, etc.) in the material.

Repeat this narrative to yourself as you develop it, continuously refining it as you learn more of the material. Of course, you may not necessarily have a linear path through all the topics you need to cover, so create bylanes wherever necessary in the same manner. Ideally, every proof of a theorem should be a mini-narrative on its own. Repeat these narratives to yourself over and over as you create them and as you learn the material.

This should give you a strong base which you can use to recall the mathematics you learned even after a long time. Perhaps the roads you created become dusty and unused, perhaps the small paths become overgrown and slightly hidden from sight, but the broad directions will still lie there waiting for you to traverse them again. With just a bit of effort the next time, you can clear away the paths and speed along confidently from start to finish, or wherever you wish to go.

This is related to what Timothy Chow calls "an exposition problem"¹ (also popularized by Tim Gowers in his blog posts^2,3):

All mathematicians are familiar with the concept of an open research problem. I propose the less familiar concept of an open exposition problem. Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves. The proofs should be "natural" in Donald Newman’s sense⁴:

This term … is introduced to mean not having any ad hoc constructions or brilliancies. A "natural" proof, then, is one which proves itself, one available to the "common mathematician in the streets."

I have found this method to be very useful in not only retaining in my memory the mathematics that I learn, but also in explaining the theory to others on the fly.

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Footnotes

1. T. Y. Chow (2008). "A beginner's guide to forcing." arXiv:0712.1320.
2. W. T. Gowers (2009, May 20). "A solution to an exposition problem." Gowers's Weblog. https://gowers.wordpress.com/2009/05/20/a-solution-to-an-exposition-problem/
3. W. T. Gowers (2011, Nov 20). "Normal subgroups and quotient groups." Gowers's Weblog. https://gowers.wordpress.com/2011/11/20/normal-subgroups-and-quotient-groups/
4. D. J. Newman (1998). "Analytic Number Theory." Springer–Verlag.

So I teach kids and this is what I do. For them two things are most important:

1. To know their bearings in Mathematical Universe
2. Connection with "Real World" entities/concepts etc.

Whether I am teaching kids a new procedure division, multiplication, square root or concepts like sets, matrices etc. I make sure that they know/understand these new things in reference to things already known to them, otherwise they find it difficult to place the knowledge properly and are left hanging with no connection to reference points AND how/where it connects with this real world. This gives a place for kids to arrange that concept in their mind(bearing) and connection with real world helps them understand, formulate ideas and opinions about it and remember. And of course you cannot over emphasize Repetition. Practicing the procedure, reading and thinking about concept multiple times helps kids internalize it.

The same applies to higher Mathematics....

For most people, including myself, there is pretty much no chance that you’ll remember most of the things you’ll see and do in math-land the first time ‘round. It’s just an inevitable relic of the fact that mathematics is... well... hard, and it isn’t something that the human brain was designed to do. The fact that you’ve gotten as far as you have is an accomplishment you should be proud of. You’ve already bridged the statistical gap!

But, as you’ve clearly realized, mathematics is easily the farthest thing from a spectator sport. It gets easier, but there are always going to be challenges, that’s part of the fun. Anyway, back to the point: Math is hard to learn, and even harder to remember. How do you do it right? Take good notes!

I can’t stress how important this really is. I take notes as if I was to forget everything the next day, because chances are I will. The better the notes, the more of the material will fall into place every time you go over it. Every time you go over stuff, add your immediate conclusions and thoughts. In this sense, you will be building up a better and better “imprint” of your own brain’s critical response to the material.

Frame it in your own language, draw pictures, whatever you need to do, just make sure you’ve done everything correctly and that you haven’t sacrificed any of the concept in exchange for intuition. Intuition is great, but not for definitions. The whole point of math is to create a more functional, tangible language to describe things, and watch how the intuition unfolds. Not the other way around.

However, us humans aren’t perfect, optimal and all-knowing holders of ultimate truth and beauty, we have the slightly more disappointing form of a messy set of poorly-calibrated neural networks that are really only capable of expanding conceptually on the relatively basic patterns that only arise after many years of consistent experiences. So the only reason we have wildly brilliant people like Dirac, Einstein, and Ramanujan among us is the fact that they were masters of framing mathematics, more-or-less, in terms of basic experience and intuition. So, as cheesy as I know this is going to sound, “The answer lies within YOU.” Only you know what you understand better than anything else. So, whatever that thing is for you, find it, because it’s going to be the most effective thing you’ll ever be able to use as an analogy.

TL;DR: Mathematics is a beautiful subject, and the best way to learn it is to relate it to what you find to be your own core experiences.

You don't describe engaging in conceptual chunking.

You mention the Sylow theorems as an example and I will also use them as an example. It is easy enough to memorize the theorem statements. If you do not chunk the material in the proofs, then I challenge any claim that you understand the proofs.

(Fix a positive prime integer, $p$.) The Sylow theorems examine the left-multiplication action of a group on its $p$-power-subsets (establishing existence of Sylow $p$-subgroups), the left-multiplication action of a group on its $p$-subgroups (establishing uniqueness of the conjugacy class of Sylow $p$-subgroups), and the conjugation actions of a group and its Sylow $p$-subgroups on its Sylow $p$-subgroups (to obtain counting/combinatorial data).

That pattern -- existence, uniqueness, data/algorithm -- is so pervasive you should notice it quickly. But it is invisible if you do not chunk the information in the theorems/proofs. (This sequence of proof types is an example of one of the frequently repeated "tools" that I discuss further below.)

Chunking is an abstraction process -- If we "hand wave" the lowest level details of this particular paragraph of a proof, express what this paragraph accomplishes in a single sentence. (Note that "paragraph" should not be taken too literally. Sometimes a single mathematical idea projects onto several textual paragraphs. Sometimes a "stream of consciousness" projects several ideas into a single textual paragraph. The point is to capture the ideas one level of abstraction higher than that of the details.) You are, in essence, recreating an outline of the proof process, starting only with the leaves of the tree structure. This compresses a detailed proof into a shorter sequence of abstractions that are easier to remember.

What you will find is that the same chunks appear in multiple proofs. The process of chunking associates an idea with the detailed implementation of that idea. With repetition, you should find your understanding of a step of a proof toggling between "this block of details looks very like 'let group act on its Sylow $p$-subgroups'" and "let group act on its Sylow $p$-subgroup".

There is a next layer of chunking that comes with breadth of reading. Continuing the example, when discussing [special subobjects] of [object], one may see "this block of details looks very like 'let [object] act on its [special subobjects under discussion]'". You eventually recognize the common theme: to understand some special subobjects, let the object act on them. Now you have gone from a particular method to tease structure out of Sylow $p$-subgroups to a general method that you have learned to apply to understand future [special subobjects]. The chunking has taken you from

• an implementation of a tool applied to a specific subject
• to abstraction where you call out and name the tool
• to the tool (detached from its initial context).

Once you have the tool and have seen it used in a few contexts, your "reach" expands through the use of the tool. Many proofs, in some sense, are tutorials on using such a tool. Chunking is a method of recognizing a tool. Like any skill, this takes practice. When starting out you may need a few examples so that a tool's repeated use becomes obvious. With practice, you will chunk at various levels of abstraction almost automatically, so you may recognize the use of a new tool the first time you see it.

I believe you should follow these steps:

• Understand what the theorem says, with some applications of it (which also means to do the exercises, also show your work to TAs)
• Try to prove it yourself and get stuck quickly
• Work through the proof and try to understand it
• Take a fresh sheet of paper and now try again to do the proof
• When you get stuck (which you will again) then read again (and repeat these steps)

When working out a proof like this you usually notice that there are like one or two 'big' ideas at the kernel of the proof. Like in prooving (there are different ones of course) that from f'=g' follows f=g+c you use the mean value theorem. Now make a note next to the theorem which acts as a reminder what was the part at the kernel and not the technical parts which you can easily make up later.

These ideas you need to remember, and this helps you to remember the applicability. Like in the example if you know you need to use mean value theorem it is immediately clear that the domains of the functions have to be an interval.

Of course this is a lot of work. You do not do this in the first reading, only for some theorems. You should plan to go through the whole material at least 3 times (over the course of 6 month)

TL;DR dont learn the theorem, learn the proofs

For context, I have a lot of experience self-learning mathematics. I spent a summer learning additional algebra, point-set topology, linear algebra, and analysis (to extend my undergraduate degree) before entering my current graduate program. This was sufficient to skip a literal year in the program. I am now well into the program and have to self-teach constantly, as very few people who know the material I am learning are available to tell me about it.

From your description, you are not doing any activities that lead to true understanding or retention. That is, learning mathematics requires active thought, and you are passively writing and rewriting the same theorems. This does not lead to the processing that is necessary to retain them, and will ultimately lead to you taking more time to progress.

Do the texts you are working from have exercises? If so, you should do a representative selection of exercises from each chapter before moving to the next. Does the text you are working from give summaries of proofs before or instead of the full detail of the proof? You should see if you can work out the full details of each proof before reading it, given a summary. Are you certain that your solutions are complete, not neglecting any details? If not, you should post your attempted solutions online on, say, math.stackexchange or r/learnmath for feedback.

In general, effective learning cannot occur when the only thing you do is absorb information. You must practice using that information--memorization becomes the side effect. I recommend https://www.learningscientists.org/ as a resource to learn how to effectively learn. They suggest 6 effective strategies for learning. The most important tools in their toolbox, adapted to your situation, are:

• Recall practice. This it appears you are doing, but perhaps too late. You should attempt to recall any new information you learn at multiple stages. For example, say you just learned about the Sylow theorems. As soon as you write down those theorems, close the book and see if you can recall the conditions and conclusions of the theorems, as well as why they "make sense." If you cannot, then you can't expect to remember them much later either. Then, while doing the exercises from the text, aim first to do them without any reference to the text. Only after some struggle should you permit yourself access to the text. This forces you to acknowledge what material you do not currently remember, and helps it stick in your memory. For direct recall of the theorems, you can make yourself simple decks in a program like Anki, which takes little time each day.

• Spaced practice: You should aim to return to each topic a week later, and then a month later. This can be done by setting aside one day each week to be a "review the previous week" study day, where you pick exercises you skipped from each section the first go around. Then one day each month can be a "review the previous month" study day, where you do a similar thing. (Notice: This should not only be repetition of the theorems, but applying those theorems in new problems.)

• Elaborative practice: Any time you learn a new theorem, you should try to explain to yourself or someone else (even a "rubber duck") how it fits in to the old material. Why do you care about it? What questions does it answer? What questions does it not answer?

• Concrete examples: Any time you learn a definition, before learning anything else, you should try to come up with an example of a mathematical object that fits that definition, and one which does not. See if you can justify why.

• Dual Coding: If you are learning an algebraic fact, can you come up with a visual example of that fact? For example, if you are learning about the group D_2n, can you both give algebraic rules for manipulating that group and have a visual example of an object with D_2n as its symmetry group? Where do the elements of the center of D_{2n} show up in terms of the visuals of the symmetry group? If you are asked to come up with the Sylow-2 subgroup(s) of D_{12} and, perhaps, their conjugacy classes in an exercise, can you determine what they are both algebraically and within your visual model?

These strategies do necessarily take more time than simple reading, and your learning will be "slower" as a result -- but you will retain more of the old information. Placing understanding over speed as your primary goal will ultimately end up with you taking longer in the short run, but you will learn and retain faster over time, because you will have a stronger base to go back to.

Memorization per se should not be the primary focus. When I learn something new, I type up notes on the computer in a reverse-indented outline format. Then as time goes on and my understanding improves, I edit the notes to reflect that. When I work an exercise or read a paper, I refer back to my notes. Memorization, to the extent that it happens, is just a side-effect of this process. If a school, teacher, or educational system puts too much emphasis on memorization in math classes, it's a sign of low quality. If you're stuck in such a class, and, e.g., must take closed-notes tests, then just think of that as a side issue and allocate some time the day before an exam to commit some things to short-term memory.

After I have understood the theorems, I practiced them one at a time writing them all.

I'm not sure here if you mean the statements of the theorems or their proofs, and whether writing means copying them down verbatim out of the book or reexpressing them in your own words. But basically this sounds like rote memorization, which is a bad idea.

I tell my students that when they study a proof, they should mainly focus on its inputs and outputs, and try to understand what goes wrong when you try to change those. For example, Newton proved Kepler's laws mathematically. A good exercise for someone trying to understand a presentation of such a proof is to think about what happens if gravity is changed from a $1/r^2$ force to some other exponent like $1/r^7$.

I think it is important to do some drill EVEN (maybe especially) with advanced concepts. This is because the concepts may be more strange and abstract. So you need to do some basic work to get familiarity with it. Quantum mechanics (or E&M) are rather non-intuitive and you just need to work with them. Not everything is as much an "aha" realization as kinematics (or differentiating polynomials).

From the question ("read"), it already sounds like you're concentrating too much on a passive, not active approach. In addition, sometimes advanced topics are made harder (even then they need to be), because less pedagogical effort is put in by the writers (since it is for such a small audience). So you get harder and less problems and less progressive (step by step) learning. This is falsely talked about as "you're in uni now" or "you're in grad school, now". But really is a laziness on the TEACHING side of textbook writers (not the MATH side), by taking less effort. Pedagogy is even harder than math. An extreme example is writers of monographs who don't have any problems. Basically they are teaching a universe of one, themselves, when writing the monographs. [Many math Wiki writers are like this also...not really trying to teach, but to clarify for themselves!]

What you need to do when you encounter these types of materials or courses is to (at a minimum) do all the problems they contain. And also reach out for alternate sources.

Even someone like Andrew Wiles, working with research papers and his own ideas, still felt the need to do little by-hand examples to build familiarity with what he was proposing! Watch the Singh video.

I would also caution you, on the practical side, that some level of forgetting and relearning is to be expected and may even have some positive aspects. Just keep the reference materials next to you and go back to what you did before. It will come back much quicker since you saw it before. And you will know it better and retain it longer.