In many ways, I am atypical in the way that I approach a problem, but it works for me. Specifically, I try to understand an example in as much detail as I possibly can. If the example, is too complicated, then I make a simpler example. As much of the intricate detail that I can bring to bear on the example is brought.
For example, instead of trying to understand Lie groups and Lie algebras in general, start with the circle and the line that is tangent at the point (1,0). What is the exponential map? Oh, OK. Now how about $SU(2)$ and $su(2)$? Can you understand that the Lie group is the $3$-dimensional sphere? Can you understand the coordinates? Can you understand the equators? How do $i,j$ and $k$ really work?
What is the difference between the multiplication rule $i\times i =0$ and $i^2=-1$?
I spend time pondering. And often my notebooks will contain tangential problems or specific computations. I will keep doing the computation until I get it right! If necessary, I will write a program to complete the computation. When I understand the example completely, it is usually easy to abstract.
Then I follow up, usually writing in a notebook or several notebooks before I begin writing on the computer. I have an advantage in that I have long-distance collaborators, so it becomes necessary to explain the idea to the collaborator(s). That is the first writing stage: write for someone who knows your short-hand and your metaphors. the second stage is to write for someone who does not. Then I write with a set of colleagues in mind, but I assume the colleagues do not remember anything from the previous work. I also try to explicate the notation writing for example "the function $f$, the knot $k$, or the tubular neighborhood $N$.
A complex analytical colleague only uses $z$ for a complex number, $x$ for a real variable, and $n$ for an integer. These variable choices are culturally determined, and so one keeps with the culture of the discipline unless there is good reason to deviate. As a final example of this, the variable $A$ in the bracket polynomial is known to everyone in the field. The variables $q$, $t$, $X$ etc. are less known and involve different normalizations. So it is the burden of the author to relate these to the more well known choices.
What you are telling is that you probably forget almost all the advanced undergraduate materials in both algebra and analysis. You have a big gap to fill at this point. You could restart your learning by reading an intro to analysis book along with algebra or linear algebra. I guess that your algebra and linear algebra core knowledge content is long gone. I also propose that you make time to attend a lecture in either of these courses at a university so you can pose questions and get immediate live answer back. Don't try to learn on your own.
There is an old saying that you could liken it to an adage: " a teacher should not give too much wisdom to his students ". I agreed. But in this case, I am willing to go the extra mile.....
Again, base on your inputs, you need to find an advisor who can tell you what to do to get your terminal degree in the shortest possible time frame...
Best Answer
I tend to remember the main point of many theorems, but I often don't remember the details of the statements. When you actually want to use a theorem, you have to look it up.
For example, I know that there's a theorem that the set of homotopy classes of maps $X\to S^1$ is in one-to-one correspondence with the elements of $H^1(X)$, but I have no idea off the top of my head what hypotheses on $X$ are necessary to make this theorem true. For all I know, this may hold for arbitrary topological spaces, there may be some connectedness requirement, or it may even require $X$ to have the homotopy type of a CW complex. What I do know is that this theorem is somewhere in Hatcher's Algebraic Topology, and I would be able to look it up in five minutes or so. (By the way, this is one reason that you should keep around copies of books that you are familiar with. I am much faster at looking things up in Hatcher than I would be with another algebraic topology book.)
For other theorems, it's possible for me to reconstruct the details without looking it up. For example, the Poisson integral formula expresses a holomorphic function inside a disk in terms of the value of the function on the boundary circle. I do not use this theorem often enough to remember it off the top of my head, but I can re-derive it whenever I want using the following procedure:
(1) I know that the value of a holomorphic function in the center of the disk is equal to the average value of the function on the boundary circle. (The same is true of harmonic functions---indeed, the definition of a harmonic function is basically just that this is true for infinitesimal disks.)
(2) I also know that I can use Möbius transformations to map the center to any other point inside the disk in a way that maps the boundary to itself. I'm good at working out the formulas for Möbius transformations, so I can combine this with (1) to find the value of the function at any other point inside the disk.
There are certainly many other ways of deriving the Poisson formula -- you should use whichever one works best for you.
Finally, some theorems are hard to forget. For example, Lagrange's Theorem states that if $G$ is a finite group and $H\leq G$, then the order of $H$ must divide the order of $G$. This theorem is so obvious (picture the cosets), that I've always found it strange that it deserves a name, and I can't imagine ever forgetting that this theorem is true. (That being said, I do sometimes forget the name of this theorem. In particular, it's sometimes hard for me to remember which theorem is Lagrange's Theorem and which theorem is Cauchy's Theorem.)