nhaliday + fedja   6

soft question - Thinking and Explaining - MathOverflow
- good question from Bill Thurston
- great answers by Terry Tao, fedja, Minhyong Kim, gowers, etc.

Terry Tao:
- symmetry as blurring/vibrating/wobbling, scale invariance
- anthropomorphization, adversarial perspective for estimates/inequalities/quantifiers, spending/economy

fedja walks through his though-process from another answer

Minhyong Kim: anthropology of mathematical philosophizing

Per Vognsen: normality as isotropy
comment: conjugate subgroup gHg^-1 ~ "H but somewhere else in G"

gowers: hidden things in basic mathematics/arithmetic
comment by Ryan Budney: x sin(x) via x -> (x, sin(x)), (x, y) -> xy
I kinda get what he's talking about but needed to use Mathematica to get the initial visualization down.
To remind myself later:
- xy can be easily visualized by juxtaposing the two parabolae x^2 and -x^2 diagonally
- x sin(x) can be visualized along that surface by moving your finger along the line (x, 0) but adding some oscillations in y direction according to sin(x)
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january 2017 by nhaliday
soft question - A Book You Would Like to Write - MathOverflow
- The Differential Topology of Loop Spaces
- Knot Theory: Kawaii examples for topological machines
- An Introduction to Forcing (for people who don't care about foundations.)
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october 2016 by nhaliday
soft question - How do you not forget old math? - MathOverflow
Terry Tao:
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-d‌​one

fedja:
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).

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.

Timothy Chow:
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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june 2016 by nhaliday