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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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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... –  Mark Grant Sep 27 '13 at 10:04
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What old math ? –  Andrej Bauer Sep 27 '13 at 10:09
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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. –  Oliver Nash Sep 27 '13 at 10:14
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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 –  Terry Tao Sep 27 '13 at 14:41
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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. –  Todd Trimble Sep 27 '13 at 14:55

7 Answers 7

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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What's the famous foil rule, and what makes it sexy? –  Tom Leinster Sep 27 '13 at 12:27
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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. –  Ben McKay Sep 27 '13 at 12:35
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Dont worry about forgetting how to spell 'entshuldigen'; instead remember 'entschuldigen' :-) –  quid Sep 27 '13 at 13:26
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Good answer - just to fresh up your memory on that, it's "Entschuldigen Sie mich bitte" –  Tobias Kienzler Sep 27 '13 at 14:12
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Wasn't the German misspelling intentional? –  Timothy Chow Sep 27 '13 at 15:06

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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I agree, and you explained it better than I did. –  Thomas Kahle Sep 28 '13 at 7:07

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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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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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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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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Somehow I missed your post when I wrote mine! I agree wholeheartedly. –  Timothy Chow Sep 27 '13 at 15:35

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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