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"My two pieces of unsolicited advice for anyone about to start or currently in a PhD program is to (1) write daily and (2) work consistently/intentionally"

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Can say, I definitely didn’t write daily in math. Like, sometimes you just gotta take a reading day, or do some computational examples if you can.

Or just take a break my dudes, good work is done with good rest :)
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Well fuck...
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(1) completely agree with the 2nd point - work consistently / intentionally. It’s really easy to waste a year (or 2…3….more) once you are in the ‘do research’ phase.

(2) I don’t think he was specifically saying ‘write your dissertation’ for an hour. Use this writing to organise your thoughts, plan the next steps, argue with your imaginary committee, plan your Nobel Prize summary of your field, etc. THIS will certainly support Point #1.

(Full disclosure: I didn’t follow this advice, but wish I had….I was one of those ‘hard working’ PhD students … but I got better at crossword puzzles!)
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i heard a story that someone once asked Jacob Lurie how he manages to write so many enormous and influential books, and he said, "well, I only write about five pages of new math a day."

.... i can barely *read* five pages of math in a day
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The advice the author gives on "work consistently/intentionally" goes hand in hand what I would advise on taking real and meaningful breaks. They don't mean work all day, they mean work not too many hours but make them count (= work intentionally). Take real weekends, relax for some meaningful portion of the day, make sure you keep up hobbies you enjoy and take good care of your physical and mental health. Not doing those things can end up being a major negative impact on both your productivity and general well-being.
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Your average PhD dissertation in maths is about 100 pages, worth some 5 years of work (less in Europe, but the point still stands). Surely you don't need to write daily for that.
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Stephen King said the exact same thing about writing. Worked out pretty well for him.
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Learning how to write is one of the most important things. Now, 20 years after my PhD I can write well. I can only imagine I would have been more productive and successful if I mastered writing earlier on.

There are 2 facets though. The natural language portion and the mathematical portion. I mean (1) the normal communication through written language, and (2) the use of natural language writing to communicate math. Toss on top of that the symbolic/notational aspect too.

Best advice for math PhD is to know how to pick a problem just is right for you and to know when to give up on a problem. Even better is to network endlessly and collaborate with others. So above all might be too cultivate social skills. I failed at all of that until much later.
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By writing daily, in math it means you learn something and then write it down explaining yourself or, ideally, others. You only know when you are able to explain.

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