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I am trying to not forget my old math. I finished my PhD in real algebraic geometry a few years ago and then switched to the industry for financial reasons. Now I get the feeling that I want to do a postdoc and I face the dilemma that I actually have forgotten a lot of my algebraic geometry classics. Sometimes they are minor things and if I browse a book I recall everything, and sometimes they are major (I don't really think I have a mental lapse though!). So I see myself reading a lot of books in order to recall some of my old math.

It gets frustrating that I have to repeat reading 80% of the article that I once used to read and understand. Maybe the new math that I have been feeding myself should be blamed too (I tried learning more differential geometry and fractal theory after doing algebraic geometry and hardly looked back at algebraic geometry after that). I have never tried avoiding to forget old math, especially parts that I do not use in a daily basis (esp. now that I work in the industry). But this can and will be fatal if I do apply for a postdoc. So now I want to read again, yes, but I don't want to forget again.

Is there a magic recipe for this? Usually I do find it helpful to always connect even the most abstract of mathematics with something that is tangible as an example, either in real life or in easier math (e.g. connect invertible sheaves and Picard group to line bundles, vector bundles to tangent bundles and tangent spaces .. etc.). This usually helps me not to forget things, but some of the math that I used to learn is too abstract to make such a connection, or maybe I just didn't learn correctly to apply such a connection. So my approach now, when I start reading something new or old, is to find a practical example ASAP, or ask myself why the originator of the theory first thought of developing this in the first place, before even getting any deeper into the subject. I must be honest though, sometimes this is very difficult to do (esp. if you read references for which such connection is not made).

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    $\begingroup$ A research mathematician remembers results by using them. But judging by the number of upvotes this question has, there must be more to say than that... $\endgroup$ – Mark Grant Sep 27 '13 at 10:04
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    $\begingroup$ What old math ? $\endgroup$ – Andrej Bauer Sep 27 '13 at 10:09
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    $\begingroup$ I think you may be overestimating the cost of relearning. Having spent six years in proprietary trading after my D.Phil. (in geometry) I certainly forgot plenty of things but I also found I could pick them up again very quickly with the bonus that I usually understood them better the second time round. I would even go so far as to say I think this is often better than never forgetting in the first place. $\endgroup$ – Oliver Nash Sep 27 '13 at 10:14
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    $\begingroup$ I find that blogging about material that I would otherwise forget eventually is extremely valuable in this regard. (I end up consulting my own blog posts on a regular basis.) EDIT: and now I remember I already wrote on this topic: terrytao.wordpress.com/career-advice/write-down-what-youve-done $\endgroup$ – Terry Tao Sep 27 '13 at 14:41
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    $\begingroup$ I'd upvote Terry's comment multiple times if I could. In my case, I find that writing up material and uploading it to a place I can always find it (e.g., here, or to the nLab) is a great way to hammer mathematics into the brain. $\endgroup$ – Todd Trimble Sep 27 '13 at 14:55
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You do forget things you are not working on. Nothing can be done about it. I could read German easily by the end of 8th grade and now I can hardly spell "Entshuldigen Sie mir bitte". There are several math. papers I read as a student of which I remember next to nothing. The most frustrating and shameful thing is that I don't remember the details of my own papers written 20 years ago with a few exceptions. After age 40 I also started to lose the ability I always took for granted: to get to the board at any time and start lecturing on some subject in my field with full proofs without any preparation. Now I have to sit for half an hour and to prepare my lectures now and then (thanks God this concerns only advanced graduate courses yet). And I work as a professional mathematician in academia full time!

The only way to cope with this loss of memory I know is to do some reading on systematic basis. Of course, if you read one paper in algebraic geometry (or whatever else) a month (or even two months), you may not remember the exact content of all of them by the end of the year but, since all mathematicians in one field use pretty much the same tricks and draw from pretty much the same general knowledge, you'll keep the core things in your memory no matter what you read (provided it is not patented junk, of course) and this is about as much as you can hope for.

Relating abstract things to "real life stuff" (and vice versa) is automatic when you work as a mathematician. For me, the proof of the Chacon-Ornstein ergodic theorem is just a sandpile moving over a pit with the sand falling down after every shift. I often tell my students that every individual term in the sequence doesn't matter at all for the limit but somehow together they determine it like no individual human is of any real importance while together they keep this civilization running, etc. No special effort is needed here and, moreover, if the analogy is not natural but contrived, it'll not be helpful or memorable. The standard mnemonic techniques are pretty useless in math. IMHO (the famous "foil" rule for the multiplication of sums of two terms is inferior to the natural "pair each term in the first sum with each term in the second sum" and to the picture of a rectangle tiled with smaller rectangles, though, of course, the foil rule sounds way more sexy).

Since it is a "general" question, I suggest making it community wiki (and mark my answer as such).

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    $\begingroup$ What's the famous foil rule, and what makes it sexy? $\endgroup$ – Tom Leinster Sep 27 '13 at 12:27
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    $\begingroup$ The famous foil rule is that to multiple $(a+b)(c+d)$, we take the First elements $ac$, the Outside elements $ad$, the Inside elements $bc$ and the Last elements $bd$, FOIL. It is not a useful mnemonic, as it doesn't generalize well to more terms or more factors. $\endgroup$ – Ben McKay Sep 27 '13 at 12:35
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    $\begingroup$ Dont worry about forgetting how to spell 'entshuldigen'; instead remember 'entschuldigen' :-) $\endgroup$ – user9072 Sep 27 '13 at 13:26
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    $\begingroup$ Good answer - just to fresh up your memory on that, it's "Entschuldigen Sie mich bitte" $\endgroup$ – Tobias Kienzler Sep 27 '13 at 14:12
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    $\begingroup$ Wasn't the German misspelling intentional? $\endgroup$ – Timothy Chow Sep 27 '13 at 15:06
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One thing that I don't think the other respondents have emphasized enough is that you should work on prioritizing what you choose to study and remember.

As others have said, forgetting lots of stuff is inevitable. But there are ways you can mitigate the damage of this information loss. I find that a useful technique is to try to organize your knowledge hierarchically. Start by coming up with a big picture, and make sure you understand and remember that picture thoroughly. Then drill down to the next level of detail, and work on remembering that. For example, if I were trying to remember everything in a particular book, I might start by memorizing the table of contents, and then I'd work on remembering the theorem statements, and then finally the proofs. (Don't take this illustration too literally; it's better to come up with your own conceptual hierarchy than to slavishly follow the formal hierarchy of a published text. But I do think that a hierarchical approach is valuable.)

Organizing your knowledge like this helps you prioritize. You can then consciously decide that certain large swaths of knowledge are not worth your time at the moment, and just keep a "stub" in memory to remind you that that body of knowledge exists, should you ever need to dive into it. In areas of higher priority, you can plunge more deeply. By making sure you thoroughly internalize the top levels of the hierarchy, you reduce the risk of losing sight of entire areas of important knowledge. Generally it's less catastrophic to forget the details than to forget about a whole region of the big picture, because you can often revisit the details as long as you know what details you need to dig up. (This is fortunate since the details are the most memory-intensive.)

Having a hierarchy also helps you accrue new knowledge. Often when you encounter something new, you can relate it to something you already know, and file it in the same branch of your mental tree.

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    $\begingroup$ I agree, and you explained it better than I did. $\endgroup$ – Thomas Kahle Sep 28 '13 at 7:07
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I guess you want personal, and not expert opinions. So, as a non-psychologist who has to use his memory, I'll share my thoughts.

I am lazy and never want to memorize stuff. To go shopping more than one item, I make a list. In school, I avoided matters when I had to use memory. I preferred math because it concentrates potentially infinite knowledge in a small number of axioms. Not something to be proud of, but I am happy that Google and Wikipedia exist now, I can look up things faster than it would take me to recall them, assuming I know them. I did not start programming computers until the advent of editors that pop up lists of suggested keywords (I am a programmer who wants to find a postdoc in mathematical and theoretical physics).

So, with my untrained memory, I was very surprised when I realized that I can remember a lot of things, after I use them long enough.

Here is my advice. Do what little children do when they learn how to speak. They find excuses to use the new words in sentences. You can see this by the fact that sometimes these sentences are a bit forced and useless, obviously being a pretext. And that sometimes they test the boundaries of applicability of the word, by giving it unusual meanings.

I think that to remember things when you will need them during a task, it helps to use them in similar, even though smaller tasks. When you read, look-up terms, definitions, theorems. Write blog entries and essays that force you to remember them. Answer questions on Math Overflow and Math SE, which are in the domains of interest for you. Semi-formal and even informal communication of technical ideas makes us try to be accurate and check what we say, so this is a good motivation to recall things and update them. And, no matter how solid you feel your memory is, when you write papers, double-check anyway the definitions and formulae. Even signs, normalization factors, notations, conventions etc. Keep the formulae handy, in a folder, or printed on sheets of paper on your walls, if this helps more.

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When I was young I could remember everything just by studying it repeatedly. Now I basically cannot remember anything (except what I learned well when young) unless I "discover it myself". The quotes mean that I need not really discover it independently, it is enough to sit down with a pen and try to work it out alone, even if drawing subconsciously on latent memories of previous knowledge of it. It also helps to try to simplify it so much that I can teach it to a bright but unsophisticated person, like a young child.

Nonetheless, I will still forget it again after some time away from it, so it is very helpful, as stated above, to write it out very clearly, in a recoverable medium, once I have mastered it.

I also find that things that are really clearly explained, simply and succinctly, by a great master, are much more easily recalled, since the master's version contains exactly the heart of the matter. E.g. I still cannot forget Lagrange's explanation of the quadratic formula (due to Diophantus?), Euler's explanation of "Cardano's" cubic formula, Auslander's explanation of Nakayama's lemma, Riemann and Roch's exposition of their theorem, Euclid's discussion of the concept of gcd of two integers, Kempf's exposition of Mumford's proof of the Riemann singularities theorem, etc,....

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The short answer is "use it or lose it", but I would add two things:

  • You relearn it faster than you learned it the first time, precisely because you learned it the first time.
  • Since you mention algebraic geometry in particular, I think there's quite a lot of that in this forum. Asking and answering questions here and on math.stackexchange.com might help keep you mentally limber. Some of what gets posted to the latter forum is routine homework stuff, and some is deep and insightful. (And quality varies a lot on mathoverflow too.)
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I believe there is no magic recipe. Evolution made us to remember things that we really need (things that we really use), and forget the less important things. According to an old Greek saying (which has an equivalent in many languages), practice makes the master. Some recent studies in cognitive science indicate that this wisdom needs to be refined as follows: the most efficient way of learning is to try to recall/remember the memories/facts/knowledge regularly. So it is really testing on a regular basis that makes the master.

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If you want to remember concrete things you can try to a flashcard system like Anki.

In my experience, it helps to structure knowledge on several scales, hierarchically. If you remember the cornerstones of some theory, then the details can be attached to this framework, or they can be left out and filled in by researching them when necessary. You could put the cornerstones on Anki cards to never forget them. Since a framework is already present, it will also be easier to remember details. The key to memorizing something is to connect it to other things you already know.

Otherwise I suggest reading material on the brain and the workings of memory. If you want to dig more into how the brain works I also recommend "Thinking fast and slow" by Kahneman.

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  • $\begingroup$ Somehow I missed your post when I wrote mine! I agree wholeheartedly. $\endgroup$ – Timothy Chow Sep 27 '13 at 15:35
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In general, the math I do use -- I do not forget [here I think I repeat everybody who commented above], the only kind of math I do really forget is the math I do not use at all. This is not as sad, because you see I don't use it. But sometimes I have this anxiety feeling that maybe that mathematics that I'm now forgetting could help me at some point of my career. Then I tell it to somebody, I make a course, I share with a friend. The good way to learn something is to make a course on it. I think, it is also a good way not to forget.

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This is not an answer, rather information I think might be useful to you for better clarifying this question and for possibly changing your personal views on what you believe you have forgotten. To preface my response, your brain automatically stores most memories by cues for later retrieval, encountering these cues invokes said memory, for example the smell of chocolate cake could be a retrieval cue for a memory of your sixth birthday.

With that said, if a person describes something and you are unable to recall it, that doesn’t necessarily mean you forgot it, rather that their description was not one of the retrieval cues you used to store it. For example, people often wake up thinking they “forgot” a dream they had that night, but when they encounter some random object during the day it cues back the memory and the contents of their dream come flooding back to them. We all have stored memories that are unconscious to us - that is data our brain has recorded (sort of) but which can only be accessed by certain cues (cues we often don't know exist).

In your case you might believe you have forgotten say Bob's theorem in algebraic geometry simply because the enunciation of "Bob's theorem" does not trigger a recall of its contents. Thus it may be that despite having read a proof of Bob's theorem three years ago, your brain is no longer using the name of that theorem as a memory cue. However this does not rule out the possibility that other cues still exist for content related to Bob's theorem, for example it may be that some obsecure lemma used in the proof is actually a memory cue for the theorem and thus though you can't recall Bob's theorem now, if someone were to show you the lemma used in the proof, you would then be able to recall part of Bob's theorem.

For an overview, see: https://en.wikipedia.org/wiki/Cue-dependent_forgetting

Now lastly I want to comment that as we can see here, even common terms like "forget" are often not rigorous enough to describe what is happening cognitively to people in everyday situations, to make matters worse there are currently many psychological theories about memory that roughly say the same thing but which use different terminology or worse theories that in my opinion say and do nothing at all apart from create complex terminology to state elementary observations. Though like a lot of cognitive science I personally think this is because its almost always impossible to get useful biological measurements from within the brain (with our current knowledge/technology) as so far as they relate to what we see, thus instead we have to reverse engineer what exact processes are occurring within the brain based on the results of how test subjects perform on various tasks, like say someone trying to figure out server side code (this would be some cognitive process) by sending search queries (these would be the psychological tests) to an interface on a website (this would be the test subject) - its an incredibly difficult task as one can create many different scripts (biological processes in the brain) that all respond the same way to search queries (produce the same results in psychological tests) i.e. we have no way of knowing what script(s) (biological processes) are responsible for what. Like a doctor trying to diagnose a patient with only knowledge of their symptoms and not being allowed to administer any tests that involve physically interacting with them. However this is clearly an oversimplification (also I'm getting off-topic) and I should preface this last paragraph by saying that I am by all means a layperson at the will of only my own judgement and online research. The little I know is from reading articles/texts online as well as half a lay persons' book (which I undoubtedly am) by Matthew Walker during an unfortunate stint in a substance abuse program where apart from studying math without the internet that was all I found to do. I would not dare call myself a mathematician as I've not even completed my formal undergraduate education yet, though I can say for myself that worrying about my poor memory/mental-health with respect to my peers has brought me a fair amount of misery so (for whatever my advice is worth) I don't recommend dwelling on it, there are things I and I imagine everyone else has done/experienced be it simply aging or whatever that provide no net benefit no matter how you look at the situation, but I guess that's life.

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