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Learning to learn | jiasi
It might sound a bit stupid, but I just realized that a better reading strategy could help me learn faster, almost three times as fast as before.

To enter a research field, we sometimes have to read tens of research papers. We could alternatively read summaries like textbooks and survey papers, which are generally more comprehensive and more friendly for non-experts. But some fields don’t have good summaries out there, for reasons like the fields being too new, too narrow, or too broad.

...

Part 1. Taking good notes and keeping them organized.

Where we store information greatly affects how we access it. If we can always easily find some information — from Google or our own notes — then we can pick it up quickly, even after forgetting it. This observation can make us smarter.

Let’s do the same when reading papers. Now I keep searchable notes as follows:
- For every topic, create a document that contains the notes for all papers on this topic.[1]
- For each paper, take these notes: summaries, quotes, and sufficient bibliographic information for future lookup.[2, pages 95-99]
- When reading a new paper, if it cites a paper that I have already read, review the notes for the cited paper. Update the notes as needed.
This way, we won’t lose what we have read and learned.

Part 2. Skipping technical sections for 93% of the time.

Only 7% of readers of a paper will read its technical sections.[1] Thus, if we want to read like average, it might make sense to skip technical sections for roughly 93% of papers that we read. For example, consider reading each paper like this:
- Read only the big-picture sections — abstract, introduction, and conclusion;
- Scan the technical sections — figures, tables, and the first and the last paragraphs for each section[2, pages 76-77] — to check surprises;
- Take notes;
- Done!
In theory, the only 7% of the papers that we need to read carefully would be those that we really have to know well.
techtariat  scholar  academia  meta:research  notetaking  studying  learning  grad-school  phd  reflection  meta:reading  prioritizing  quality  writing  technical-writing  growth  checklists  metabuch  advice 
september 2019 by nhaliday
Skim / Feature Requests / #138 iphone/ebook support
Skim notes could never work on the iPhone, because SKim notes data depend on AppKit, which is not available in iOS. So any app for iOS would just be some comletely separate PDF app, that has nothing to do with Skim in particular.
tracker  app  pdf  software  tools  ios  mobile  osx  desktop  workflow  scholar  meta:reading  todo 
june 2019 by nhaliday
An Efficiency Comparison of Document Preparation Systems Used in Academic Research and Development
The choice of an efficient document preparation system is an important decision for any academic researcher. To assist the research community, we report a software usability study in which 40 researchers across different disciplines prepared scholarly texts with either Microsoft Word or LaTeX. The probe texts included simple continuous text, text with tables and subheadings, and complex text with several mathematical equations. We show that LaTeX users were slower than Word users, wrote less text in the same amount of time, and produced more typesetting, orthographical, grammatical, and formatting errors. On most measures, expert LaTeX users performed even worse than novice Word users. LaTeX users, however, more often report enjoying using their respective software. We conclude that even experienced LaTeX users may suffer a loss in productivity when LaTeX is used, relative to other document preparation systems. Individuals, institutions, and journals should carefully consider the ramifications of this finding when choosing document preparation strategies, or requiring them of authors.

...

However, our study suggests that LaTeX should be used as a document preparation system only in cases in which a document is heavily loaded with mathematical equations. For all other types of documents, our results suggest that LaTeX reduces the user’s productivity and results in more orthographical, grammatical, and formatting errors, more typos, and less written text than Microsoft Word over the same duration of time. LaTeX users may argue that the overall quality of the text that is created with LaTeX is better than the text that is created with Microsoft Word. Although this argument may be true, the differences between text produced in more recent editions of Microsoft Word and text produced in LaTeX may be less obvious than it was in the past. Moreover, we believe that the appearance of text matters less than the scientific content and impact to the field. In particular, LaTeX is also used frequently for text that does not contain a significant amount of mathematical symbols and formula. We believe that the use of LaTeX under these circumstances is highly problematic and that researchers should reflect on the criteria that drive their preferences to use LaTeX over Microsoft Word for text that does not require significant mathematical representations.

...

A second decision criterion that factors into the choice to use a particular software system is reflection about what drives certain preferences. A striking result of our study is that LaTeX users are highly satisfied with their system despite reduced usability and productivity. From a psychological perspective, this finding may be related to motivational factors, i.e., the driving forces that compel or reinforce individuals to act in a certain way to achieve a desired goal. A vital motivational factor is the tendency to reduce cognitive dissonance. According to the theory of cognitive dissonance, each individual has a motivational drive to seek consonance between their beliefs and their actual actions. If a belief set does not concur with the individual’s actual behavior, then it is usually easier to change the belief rather than the behavior [6]. The results from many psychological studies in which people have been asked to choose between one of two items (e.g., products, objects, gifts, etc.) and then asked to rate the desirability, value, attractiveness, or usefulness of their choice, report that participants often reduce unpleasant feelings of cognitive dissonance by rationalizing the chosen alternative as more desirable than the unchosen alternative [6, 7]. This bias is usually unconscious and becomes stronger as the effort to reject the chosen alternative increases, which is similar in nature to the case of learning and using LaTeX.

...

Given these numbers it remains an open question to determine the amount of taxpayer money that is spent worldwide for researchers to use LaTeX over a more efficient document preparation system, which would free up their time to advance their respective field. Some publishers may save a significant amount of money by requesting or allowing LaTeX submissions because a well-formed LaTeX document complying with a well-designed class file (template) is much easier to bring into their publication workflow. However, this is at the expense of the researchers’ labor time and effort. We therefore suggest that leading scientific journals should consider accepting submissions in LaTeX only if this is justified by the level of mathematics presented in the paper. In all other cases, we think that scholarly journals should request authors to submit their documents in Word or PDF format. We believe that this would be a good policy for two reasons. First, we think that the appearance of the text is secondary to the scientific merit of an article and its impact to the field. And, second, preventing researchers from producing documents in LaTeX would save time and money to maximize the benefit of research and development for both the research team and the public.

[ed.: I sense some salt.

And basically no description of how "# errors" was calculated.]

https://news.ycombinator.com/item?id=8797002
I question the validity of their methodology.
At no point in the paper is exactly what is meant by a "formatting error" or a "typesetting error" defined. From what I gather, the participants in the study were required to reproduce the formatting and layout of the sample text. In theory, a LaTeX file should strictly be a semantic representation of the content of the document; while TeX may have been a raw typesetting language, this is most definitely not the intended use case of LaTeX and is overall a very poor test of its relative advantages and capabilities.
The separation of the semantic definition of the content from the rendering of the document is, in my opinion, the most important feature of LaTeX. Like CSS, this allows the actual formatting to be abstracted away, allowing plain (marked-up) content to be written without worrying about typesetting.
Word has some similar capabilities with styles, and can be used in a similar manner, though few Word users actually use the software properly. This may sound like a relatively insignificant point, but in practice, almost every Word document I have seen has some form of inconsistent formatting. If Word disallowed local formatting changes (including things such as relative spacing of nested bullet points), forcing all formatting changes to be done in document-global styles, it would be a far better typesetting system. Also, the users would be very unhappy.
Yes, LaTeX can undeniably be a pain in the arse, especially when it comes to trying to get figures in the right place; however the combination of a simple, semantic plain-text representation with a flexible and professional typesetting and rendering engine are undeniable and completely unaddressed by this study.
--
It seems that the test was heavily biased in favor of WYSIWYG.
Of course that approach makes it very simple to reproduce something, as has been tested here. Even simpler would be to scan the document and run OCR. The massive problem with both approaches (WYSIWYG and scanning) is that you can't generalize any of it. You're doomed repeating it forever.
(I'll also note the other significant issue with this study: when the ratings provided by participants came out opposite of their test results, they attributed it to irrational bias.)

https://www.nature.com/articles/d41586-019-01796-1
Over the past few years however, the line between the tools has blurred. In 2017, Microsoft made it possible to use LaTeX’s equation-writing syntax directly in Word, and last year it scrapped Word’s own equation editor. Other text editors also support elements of LaTeX, allowing newcomers to use as much or as little of the language as they like.

https://news.ycombinator.com/item?id=20191348
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june 2019 by nhaliday
What's the expected level of paper for top conferences in Computer Science - Academia Stack Exchange
Top. The top level.

My experience on program committees for STOC, FOCS, ITCS, SODA, SOCG, etc., is that there are FAR more submissions of publishable quality than can be accepted into the conference. By "publishable quality" I mean a well-written presentation of a novel, interesting, and non-trivial result within the scope of the conference.

...

There are several questions that come up over and over in the FOCS/STOC review cycle:

- How surprising / novel / elegant / interesting is the result?
- How surprising / novel / elegant / interesting / general are the techniques?
- How technically difficult is the result? Ironically, FOCS and STOC committees have a reputation for ignoring the distinction between trivial (easy to derive from scratch) and nondeterministically trivial (easy to understand after the fact).
- What is the expected impact of this result? Is this paper going to change the way people do theoretical computer science over the next five years?
- Is the result of general interest to the theoretical computer science community? Or is it only of interest to a narrow subcommunity? In particular, if the topic is outside the STOC/FOCS mainstream—say, for example, computational topology—does the paper do a good job of explaining and motivating the results to a typical STOC/FOCS audience?
nibble  q-n-a  overflow  academia  tcs  cs  meta:research  publishing  scholar  lens  properties  cost-benefit  analysis  impetus  increase-decrease  soft-question  motivation  proofs  search  complexity  analogy  problem-solving  elegance  synthesis  hi-order-bits  novelty  discovery 
june 2019 by nhaliday
bibliographies - bibtex vs. biber and biblatex vs. natbib - TeX - LaTeX Stack Exchange
- bibtex and biber are external programs that process bibliography information and act (roughly) as the interface between your .bib file and your LaTeX document.
- natbib and biblatex are LaTeX packages that format citations and bibliographies; natbib works only with bibtex, while biblatex (at the moment) works with both bibtex and biber.)

natbib
The natbib package has been around for quite a long time, and although still maintained, it is fair to say that it isn't being further developed. It is still widely used, and very reliable.

Advantages
...
- The resulting bibliography code can be pasted directly into a document (often required for journal submissions). See Biblatex: submitting to a journal.

...

biblatex
The biblatex package is being actively developed in conjunction with the biber backend.

Advantages
*lots*

Disadvantages
- Journals and publishers may not accept documents that use biblatex if they have a house style with its own natbib compatible .bst file.
q-n-a  stackex  latex  comparison  cost-benefit  writing  scholar  technical-writing  yak-shaving  publishing 
may 2019 by nhaliday
Burrito: Rethinking the Electronic Lab Notebook
Seems very well-suited for ML experiments (if you can get it to work), also the nilfs aspect is cool and basically implements exactly one of the my project ideas (mini-VCS for competitive programming). Unfortunately gnarly installation instructions specify running it on Linux VM: https://github.com/pgbovine/burrito/blob/master/INSTALL. Linux is hard requirement due to nilfs.
techtariat  project  tools  devtools  linux  programming  yak-shaving  integration-extension  nitty-gritty  workflow  exocortex  scholar  software  python  app  desktop  notetaking  state  machine-learning  data-science  nibble  sci-comp  oly  vcs  multi  repo  paste  homepage  research 
may 2019 by nhaliday
Philip Guo - Research Design Patterns
List of ways to generate research directions. Some are pretty specific to applied CS.
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may 2019 by nhaliday
Why books don’t work | Andy Matuschak
https://www.spreaker.com/user/10197011/designing-and-developing-new-tools-for-t
https://twitter.com/andy_matuschak/status/1190675776036687878
https://archive.is/hNIFG
https://archive.is/f9Bwh
hmm: "zettelkasten like note systems have you do a linear search for connections, that gets exponentially more expensive as your note body grows",
https://twitter.com/Meaningness/status/1210309788141117440
https://archive.is/P6PH2
https://archive.is/uD9ls
https://archive.is/Sb9Jq

https://twitter.com/Scholars_Stage/status/1199702832728948737
https://archive.is/cc4zf
I reviewed today my catalogue of 420~ books I have read over the last six years and I am in despair. There are probably 100~ whose contents I can tell you almost nothing about—nothing noteworthy anyway.
techtariat  worrydream  learning  education  teaching  higher-ed  neurons  thinking  rhetoric  essay  michael-nielsen  retention  better-explained  bounded-cognition  info-dynamics  info-foraging  books  communication  lectures  contrarianism  academia  scholar  design  meta:reading  studying  form-design  writing  technical-writing  skunkworks  multi  broad-econ  wonkish  unaffiliated  twitter  social  discussion  backup  reflection  metameta  podcast  audio  interview  impetus  space  open-problems  questions  tech  hard-tech  startups  commentary  postrat  europe  germanic  notetaking  graphs  network-structure  similarity  intersection-connectedness  magnitude  cost-benefit  multiplicative 
may 2019 by nhaliday
Why read old philosophy? | Meteuphoric
(This story would suggest that in physics students are maybe missing out on learning the styles of thought that produce progress in physics. My guess is that instead they learn them in grad school when they are doing research themselves, by emulating their supervisors, and that the helpfulness of this might partially explain why Nobel prizewinner advisors beget Nobel prizewinner students.)

The story I hear about philosophy—and I actually don’t know how much it is true—is that as bits of philosophy come to have any methodological tools other than ‘think about it’, they break off and become their own sciences. So this would explain philosophy’s lone status in studying old thinkers rather than impersonal methods—philosophy is the lone ur-discipline without impersonal methods but thinking.

This suggests a research project: try summarizing what Aristotle is doing rather than Aristotle’s views. Then write a nice short textbook about it.
ratty  learning  reading  studying  prioritizing  history  letters  philosophy  science  comparison  the-classics  canon  speculation  reflection  big-peeps  iron-age  mediterranean  roots  lens  core-rats  thinking  methodology  grad-school  academia  physics  giants  problem-solving  meta:research  scholar  the-trenches  explanans  crux  metameta  duplication  sociality  innovation  quixotic  meta:reading  classic 
june 2018 by nhaliday
What was the hardest part of doing your Ph.D.? - Quora
I think it’s a 5-way tie, each hard in its own way:
- Picking a good topic.
- Figuring out how to bound it.
- Actually getting started.
- Going on when nothing works as you had planned.
- Knowing when to stop.

A good advisor can make some of these things easier, but you’re the one who has to do them all.
q-n-a  qra  grad-school  phd  planning  scholar  success  advice  prioritizing 
january 2017 by nhaliday
Thinking Outside One’s Paradigm | Academically Interesting
I think that as a scientist (or really, even as a citizen) it is important to be able to see outside one’s own paradigm. I currently think that I do a good job of this, but it seems to me that there’s a big danger of becoming more entrenched as I get older. Based on the above experiences, I plan to use the following test: When someone asks me a question about my field, how often have I not thought about it before? How tempted am I to say, “That question isn’t interesting”? If these start to become more common, then I’ll know something has gone wrong.
ratty  clever-rats  academia  science  interdisciplinary  lens  frontier  thinking  rationality  meta:science  curiosity  insight  scholar  innovation  reflection  acmtariat  water  biases  heterodox  🤖  🎓  aging  meta:math  low-hanging  big-picture  hi-order-bits  flexibility  org:bleg  nibble  the-trenches  wild-ideas  metameta  courage  s:**  discovery  context  embedded-cognition  endo-exo  near-far  🔬  info-dynamics  allodium  ideas  questions  within-without  meta:research 
january 2017 by nhaliday
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
ExtraTricky - On Taking Notes in Math Class
Perhaps this fictional story convinced you, and perhaps it didn't. I'm not going to claim I have proof that notes are detrimental to every student, or even on average. I don't know about any research in that area. But if you want to try out not taking notes for yourself, here are my recommendations for how to do it.
- During lecture, try to find the main new ideas being presented. If something is just algebraic manipulation, trust yourself to be able to do that on the homework if you need to.
- If the course doesn't have written materials available, do write down definitions. Keep these very short. Most definitions are only a single sentence. If you're writing more than that you're probably writing something that's not included in the definition.
- Be ready to struggle with the homework. Being stuck on a problem for hours is extremely common for mathematicians, even though it doesn't always seem that way. On one of my problem sets at MIT I was stuck near the end of a solution for around ten hours before realizing that it could be finished in a reasonably simple manner.
- When you get your homework back, make sure you have a complete and correct solution. If it's the one you turned in, great. If the teacher posts homework solutions, read through and keep that. Those solutions are now your notes.
- When exam time comes, go through those homework problems as study materials. If you end up getting stuck on one of those problems again, chances are it'll be in the same place you got stuck the first time, and your mind will connect the dots.
extratricky  oly  math  advice  notetaking  learning  reflection  checklists  metabuch  problem-solving  ground-up  scholar  the-trenches  studying  s:*  org:bleg  nibble  contrarianism  regularizer  hmm  cost-benefit  hi-order-bits 
december 2016 by nhaliday
Fact Posts: How and Why
The most useful thinking skill I've taught myself, which I think should be more widely practiced, is writing what I call "fact posts." I write a bunch of these on my blog. (I write fact posts about pregnancy and childbirth here.)

To write a fact post, you start with an empirical question, or a general topic. Something like "How common are hate crimes?" or "Are epidurals really dangerous?" or "What causes manufacturing job loss?"

It's okay if this is a topic you know very little about. This is an exercise in original seeing and showing your reasoning, not finding the official last word on a topic or doing the best analysis in the world.

Then you open up a Google doc and start taking notes.

You look for quantitative data from conventionally reliable sources. CDC data for incidences of diseases and other health risks in the US; WHO data for global health issues; Bureau of Labor Statistics data for US employment; and so on. Published scientific journal articles, especially from reputable journals and large randomized studies.

You explicitly do not look for opinion, even expert opinion. You avoid news, and you're wary of think-tank white papers. You're looking for raw information. You are taking a sola scriptura approach, for better and for worse.

And then you start letting the data show you things.

You see things that are surprising or odd, and you note that.

You see facts that seem to be inconsistent with each other, and you look into the data sources and methodology until you clear up the mystery.

You orient towards the random, the unfamiliar, the things that are totally unfamiliar to your experience. One of the major exports of Germany is valves? When was the last time I even thought about valves? Why valves, what do you use valves in? OK, show me a list of all the different kinds of machine parts, by percent of total exports.

And so, you dig in a little bit, to this part of the world that you hadn't looked at before. You cultivate the ability to spin up a lightweight sort of fannish obsessive curiosity when something seems like it might be a big deal.

And you take casual notes and impressions (though keeping track of all the numbers and their sources in your notes).

You do a little bit of arithmetic to compare things to familiar reference points. How does this source of risk compare to the risk of smoking or going horseback riding? How does the effect size of this drug compare to the effect size of psychotherapy?

You don't really want to do statistics. You might take percents, means, standard deviations, maybe a Cohen's d here and there, but nothing fancy. You're just trying to figure out what's going on.

It's often a good idea to rank things by raw scale. What is responsible for the bulk of deaths, the bulk of money moved, etc? What is big? Then pay attention more to things, and ask more questions about things, that are big. (Or disproportionately high-impact.)

You may find that this process gives you contrarian beliefs, but often you won't, you'll just have a strongly fact-based assessment of why you believe the usual thing.
ratty  lesswrong  essay  rhetoric  meta:rhetoric  epistemic  thinking  advice  street-fighting  scholar  checklists  🤖  spock  writing  2016  info-foraging  rat-pack  clarity  systematic-ad-hoc  bounded-cognition  info-dynamics  let-me-see  nitty-gritty  core-rats  evidence-based  truth  grokkability-clarity 
december 2016 by nhaliday
Information Processing: Advice to a new graduate student
first 3 points (tough/connected advisor, big picture, benchmarking) are key:

1. There is often a tradeoff between the advisor from whom you will learn the most vs the one who will help your career the most. Letters of recommendation are the most important factor in obtaining a postdoc/faculty job, and some professors are 10x as influential as others. However, the influential prof might be a jerk and not good at training students. The kind mentor with deep knowledge or the approachable junior faculty member might not be a mover and shaker.

2. Most grad students fail to grasp the big picture in their field and get too caught up in their narrowly defined dissertation project.

3. Benchmark yourself against senior scholars at a similar stage in their (earlier) careers. What should you have accomplished / mastered as a grad student or postdoc in order to keep pace with your benchmark?

4. Take the opportunity to interact with visitors and speakers. Don't assume that because you are a student they'll be uninterested in intellectual exchange with you. Even established scholars are pleased to be asked interesting questions by intelligent grad students. If you get to the stage where the local professors think you are really good, i.e., they sort of think of you as a peer intellect or colleague, you might get invited along to dinner with the speaker!

5. Understand the trends and bandwagons in your field. Most people cannot survive on the job market without chasing trends at least a little bit. But always save some brainpower for thinking about the big questions that most interest you.

6. Work your ass off. If you outwork the other guy by 10%, the compound effect over time could accumulate into a qualitative difference in capability or depth of knowledge.

7. Don't be afraid to seek out professors with questions. Occasionally you will get a gem of an explanation. Most things, even the most conceptually challenging, can be explained in a very clear and concise way after enough thought. A real expert in the field will have accumulated many such explanations, which are priceless.
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november 2016 by nhaliday
Thoughts on graduate school | Secret Blogging Seminar
I’ll organize my thoughts around the following ideas.

- Prioritize reading readable sources
- Build narratives
- Study other mathematician’s taste
- Do one early side project
- Find a clump of other graduate students
- Cast a wide net when looking for an advisor
- Don’t just work on one thing
- Don’t graduate until you have to
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september 2016 by nhaliday
Why Information Grows – Paul Romer
thinking like a physicist:

The key element in thinking like a physicist is being willing to push simultaneously to extreme levels of abstraction and specificity. This sounds paradoxical until you see it in action. Then it seems obvious. Abstraction means that you strip away inessential detail. Specificity means that you take very seriously the things that remain.

Abstraction vs. Radical Specificity: https://paulromer.net/abstraction-vs-radical-specificity/
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september 2016 by nhaliday
The capacity to be alone | Quomodocumque
In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective or thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone.
math  reflection  quotes  scholar  mathtariat  lens  optimate  serene  individualism-collectivism  the-monster  humility  the-trenches  virtu  courage  emotion  extra-introversion  allodium  ascetic  heavyweights  psychiatry 
september 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
10 reasons Ph.D. students fail
Once a student has two good publications, if she convinces her committee that she can extrapolate a third, she has a thesis proposal.

Once a student has three publications, she has defended, with reasonable confidence, that she can repeatedly conduct research of sufficient quality to meet the standards of peer review. If she draws a unifying theme, she has a thesis, and if she staples her publications together, she has a dissertation.
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may 2016 by nhaliday
For potential Ph.D. students
Ravi Vakil's advice for PhD students

General advice:
Think actively about the creative process. A subtle leap is required from undergraduate thinking to active research (even if you have done undergraduate research). Think explicitly about the process, and talk about it (with me, and with others). For example, in an undergraduate class any Ph.D. student at Stanford will have tried to learn absolutely all the material flawlessly. But in order to know everything needed to tackle an important problem on the frontier of human knowledge, one would have to spend years reading many books and articles. So you'll have to learn differently. But how?

Don't be narrow and concentrate only on your particular problem. Learn things from all over the field, and beyond. The facts, methods, and insights from elsewhere will be much more useful than you might realize, possibly in your thesis, and most definitely afterwards. Being broad is a good way of learning to develop interesting questions.

When you learn the theory, you should try to calculate some toy cases, and think of some explicit basic examples.

Talk to other graduate students. A lot. Organize reading groups. Also talk to post-docs, faculty, visitors, and people you run into on the street. I learn the most from talking with other people. Maybe that's true for you too.

Specific topics:
- seminars
- giving talks
- writing
- links to other advice
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may 2016 by nhaliday
Answer to What is it like to understand advanced mathematics? - Quora
thinking like a mathematician

some of the points:
- small # of tricks (echoes Rota)
- web of concepts and modularization (zooming out) allow quick reasoning
- comfort w/ ambiguity and lack of understanding, study high-dimensional objects via projections
- above is essential for research (and often what distinguishes research mathematicians from people who were good at math, or majored in math)
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may 2016 by nhaliday
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