Do topmost professors have something to read daily (in their locally saturated domain)?

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Solution 1

I am going to assume based on your question that you are an undergraduate or beginning graduate student -- I apologize if this assumption is misplaced, but at least in the US this would be the appropriate level for "I am a beginner or in early 20's".

There is a major transition that happens in your education as you go from being an undergraduate and beginning graduate student, to an advanced PhD student and beyond.

As an undergraduate and beginning graduate student, your main job is to receive knowledge from courses, homeworks, lectures, readings, etc. In this context, it makes sense to think that there is essentially a finite amount of knowledge, with some small amount of new material being added on the cutting edge, and a professor is merely someone who has done enough reading that they know the entire field.

This is really not how things work at all! The knowledge you learn in courses has been very carefully organized to present an efficient, compelling, apparently complete story. In reality, "all of human knowledge" on even apparently tiny subjects, like "integration" (just one half of "calculus"!) are really vast and sprawling landscapes of failed ideas, competing formulations, pathways into apparently unrelated areas, disagreements about fine points of key definitions... it is impossible (and probably not useful) to know everything on any subject. Additionally, as pointed out in the comments by @avid, it's not even really possible to carefully define the "boundaries" of a given subject -- often progress in Mathematics (or any field) is made by realizing that a technique from one area can be applied to an apparently completely different area.

On top of that, your main job from the time you are an advanced PhD student onward in the academic world is to produce knowledge. This comes about by doing research, writing papers, disseminating that information into the scientific community, and attracting interest and funding.

As a result, professors don't know everything about their field, partly because no one can, but mainly because that's not what they are trying to do. They are trying to make a lasting contribution to human knowledge in their domain. This requires an enormous amount of work in a narrow field of study to get an understanding of the most important concepts in the field, keep up with the latest developments, and develop a strong vision on where things are headed that fuels research projects.

Applying this to your question, I think you are implicitly assuming a paper is interesting because it is deep, or done by someone famous, or solves a famous problem. It may be the case that results like this are "interesting" in the everyday sense -- enough that it's worth spending an hour or two reading about it. I've watched my share of youtube videos and skimmed papers about results well outside my domain that I found "interesting" but have no hope of actually understanding at a deep level. But just because something is famous doesn't mean it is interesting in the professional sense that it is worth spending a significant amount of your time studying. Most professors would classify something as interesting enough to study, only if there was a chance it could lead to something productive. It would be difficult to justify spending months reading a very difficult paper that will take a long time to fully understand and which is not closely connected with ongoing projects and therefore is unlikely to lead to producing new research on a reasonable timescale. Even if a person was interested in moving into a new area, it would probably be more fruitful to start with something easier, and build up to more complicated papers step by step.

Solution 2

Focus, for now.

I am a 'topmost professor' in my field, by many measures, and I read daily outside of my 'local interests'. This is for fun. I represent it professionally as a personal quest to keep growing, and to become an expert in a larger circle of ideas I find important. Upon finishing my PhD I determined that, alongside my job, I could complete another PhD's worth of work each three years. By my own metric, I have done so. My reputation is for publishing in a number of fields simultaneously, and in their intersections. One is Mathematics.

I achieved this luxury of spending so much of my time learning by first becoming a recognized expert in one thing. It is very difficult to be a polymath from the start in this moment of history. Indeed, some proofs will put you in rarefied company, and require years of study. Your committee, while a bit hyperbolic with the whole madness warning thing, is giving you good advice.

My recommendation: become a master at one thing, and with it do something novel. Use the resulting reputation and resources to go after other things, chosen carefully.

Have fun.

Solution 3

An answer, appropriate for a 20-year-old mathematician in academia. (This relates to points 5 and 6.)

Do not start working on the Riemann Hypothesis now. Wait at least until you are tenured. Reason: it is likely you will get nowhere on the problem, and thus you will be publishing nothing, and will have no job after 10 years.

Many other well-known open problems could be substituted for the Riemann Hypothesis in this advice.

Solution 4

Well, all I can say is that I had an honor of seeing how Jean Bourgain (a Fields medalist in analysis) worked. He certainly was reading a lot. In fact, if you want to become like him, the recipe is very simple. Take some research article (not too long but interesting) that slightly exceeds your current abilities and work it through (meaning try to first quickly get the idea of the general scheme from the paper but try to prove every single lemma yourself and if you fail, go over every detail of the proof and retell it in your own way). Once you finish with one paper, take another, and so on day after day for about 20 years. Increase the slight excess gradually. Spend one half of the time reading and another half doing your own research. In the end (20 years later) you'll be able to take a paper, look at it for 5 minutes, and then to explain how to improve the theorem and simplify the proof (not every single time, of course, there have been nuts even Jean failed to crack, but except for the very top things, which you'll still have to work through slowly like in the beginning, reading research articles in your field will not present any problem to you any more).

This route is open to everyone, but not too many people follow it. Why?

  1. You can under- or over-estimate your abilities and pick up something that is too hard or too easy. In both cases, you can detect it and switch to something else though. The best indicator of a hard thing is that you don't understand the general logic of the paper (or even most of the words on the front page). If it is so, postpone it! Don't try the proof of Fermat's theorem or Poincare conjecture. If you really want to get there, wait a few years reading relevant but simpler things and solving problems in related fields. The indicator of an easy thing is an ability to take a look at the main theorem in a paper and being able to prove it completely in a couple of weeks all by yourself. If it is the case, move up.

  2. Inability to keep the reading schedule. Life is full of distractions, real or imaginary, and this route requires a lot of persistence. It is not the only way to become a decent professional, so if you strongly dislike systematic work or your life circumstances do not allow it, just choose some other way. I'm personally definitely not up to the required amount of self-discipline and I prefer talking to reading, so my way was just to talk to as many people I could and to try to understand and solve their problems. As a result, I'm no match to Jean, of course, but not completely hopeless either.

  3. Not being ambitious enough (as far as mathematics is concerned). We are not all samurai and usually like to balance our priorities. That's completely normal too but it turns the required 20 years into 100, and that may be above one's productive lifespan.

But by itself reading is a good thing if you approach it like running or weight lifting. Just don't try to run a marathon on the first day or lift 250 pounds on the first try. Though neither feat is impossible (and many people can do either one), both require some preliminary training. With it, you'll be there eventually. Without it, you may have a heart attack, break your arm, or, as one of the reviewers said, go insane. The whole secret is in handling the task just the right amount above your current abilities and letting time and patience to do their work. 20 years is not the exact amount of time, of course, but it is not metaphorical either: it has the right order of magnitude. Developing mathematical (or any other brain) abilities is almost universally regarded as a slower process than training your muscles.

Just my 2 cents.

Solution 5

Obviously it depends on the size of their field, doesn't it? Some questions are popular and get a lot of people thinking about them. They all publish papers. If you are in such a field, you will never run out of things to read. Then there are topics that have only a handful of people around the world who are even aware of it, let alone care. Naturally you will not be getting a torrent of new paper alerts in your mailbox from watching such a topic.

As for your misgivings, when you are wanting to bite a big piece, and your professors are saying to bite a small one, I think you are better off listening to them. If you conclude they are wrong but it turns out otherwise, you could easily end up getting stuck for years, wasting a lot of time, not getting anywhere, not graduating, etc. But if you listen to them, take on a smaller task, and it turns out that they were wrong and the small task really is too small for you -- well, that means you will finish it very quickly, and then you can go to the professor and tell them you want to try a bigger challenge now. You're not really losing much.

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Updated on August 01, 2022

Comments

  • hanugm
    hanugm 17 days

    I have this doubt from last year. This doubt arose in me after an academic incident.


    Incident

    During last year, I presented my progress before a panel of four experts and it went almost well. The only issue happened when I kept a slide of "what I will do next?". In that slide, I kept the name of a proof that I want to read. I got opinions and warnings that were shocking to me at that time. The opinions and warnings exhibited by those experts in the panel are as follows:

    1. It is impossible for you to do that in the next academic year.

    2. None of us even think about that since it is beyond our scope currently.

    3. You are overambitious.

    4. Only dozens or at most hundreds of people, on earth, have understood it till now.

    5. The world of exalted mathematicians is entirely different. Don't even try for it. It may lead to mental issues.

    6. Some aspirants I know became mentally ill after attempting such projects.

    Although some are harsh opinions, I got them. The last two are shocking or a revelation for me. Till that time, I was confident enough that I will do. But I am slowly realizing that the opinions may be true.


    Question

    My question is slightly tangential to the incident I faced. I am thinking about the topmost professors who make such proofs or write great textbooks encompassing the overall literature.

    You can consider any domain of interest for answering this question. But the domain has to be locally saturated. I think it happens mostly with math.

    Since I am a beginner or in my early twenties, I may have much to read in my domain. I want to know about the (contemporary) highest people and their world in that domain, especially in the reading aspect. Suppose I spent ten years on it and completed all the available textbooks and the significant papers till that time. Should I need to read anything further in that domain except the occasional seminal research papers by peers?

    Should there be anything in a (say locally saturated) domain for a topmost professor to read?

    Note that I am not considering the option of changing the domain for this question.

    • Bryan Krause
      Bryan Krause 11 months
      Comments are not for extended discussion; this conversation has been moved to chat.
    • Denis Nardin
      Denis Nardin 11 months
      "I think it happens mostly with math". Well, think again: my pile of articles I should read is increasing, my time to read is decreasing and I don't see the situation changing if I ever get a tenure-track position...
    • A rural reader
      A rural reader 11 months
      Very curious to learn what the subject of "Some aspirants I know became mentally ill after attempting such projects" was.
    • Stef
      Stef 11 months
      @Aruralreader If reading a paper is enough cause to become mentally ill, then surely the paper is an extract from the Necronomicon!
  • Bryan Krause
    Bryan Krause 11 months
    I don't believe OP is asking about solving an open problem, but rather understanding an existing proof. It may in fact be that the warnings are more about a logical assumed next step involving an open problem, but that isn't clear.
  • avid
    avid 11 months
    I think it is also worth noting that in many cases progress on Problem A comes from recognising that some result or technique developed for Problem B can be applied. A comprehensive knowledge of A is not sufficient to be able to progress.
  • justhalf
    justhalf 11 months
    Agree with Bryan here. The context seems to imply something like OP was trying to understand the proof for Fermat Last Theorem or Poincare Conjecture.
  • camden_kid
    camden_kid 11 months
    +1 Also, "I am a 'topmost professor' in my field, by many measures" is cool.
  • ProfessorWind-up
    ProfessorWind-up 11 months
    Thank yo. I feel privileged to be such. :) I will add that I am not the most senior professor here, so there is still much ahead.
  • Andrew
    Andrew 11 months
    @avid I incorporated this helpful comment.