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I am currently a PhD student working out research and also doing mathematics for my PhD research. Most of the time these days I spend on studying -- maybe 15 to 16 hours per day. I believe that I joined the PhD as an active student, but now I am not as active as I used to be in the initial years. My productivity is less these days as I try to avoid mistakes in my research. I have started feeling nervous in front of my colleague as I don't have any publications until now.

I always try to prepare my best but these days fail quite frequently and make silly mistakes which are annoying. Sometimes I even forget the basic definitions. Looking at myself during my initial two years I find myself a fool these days. I think this is called the rigorous phase of PhD where your ideas have to pass rigorous mathematical proofs and I am quite independent these days.

Question: How to get through the rigorous phase of PhD?

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    Making mistakes and then correcting them is a big part of the research process. You can't let the fear of making mistakes paralyze you. – user37208 Mar 20 '18 at 18:45
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    You will never stop making silly mistakes which are annoying. You just have to get used to them. – Jair Taylor Mar 21 '18 at 1:21
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    To add on to all the excellent answers, burnout is a thing, and it's absolutely worth knowing the signs of. – Fund Monica's Lawsuit Mar 21 '18 at 4:50
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    Stay away from your laptop/computer. Disconnect from the phone. Go for a walk and come back after a few days. (Do carry your phone with you though). Worked wonders for me back in those days (although I did come back after a few hours :) - the duration depends on how much of "healing" you have to do... – PhD Mar 21 '18 at 20:31
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    15-16hrs/day x (5 days a week? 6? 7?) is burnout territory. Disconnect for a weekend, go hike in the mountains or sit by the sea and watch the waves break, or whatever works for you. Time off is as important as time on. – smci Mar 23 '18 at 1:32
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Take a (short) vacation immediately.

By working 15-16 hours a day you are making yourself too exhausted and tired to be effective. That is also probably why you make mistakes. You should take a break that is long enough to reset, and when you come back aim at reasonable working hours (depends on a person and circumstances but for most people this is 40-60hrs/week).

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    Frankly, I think, as far as actual creative work is concerned, reasonable working hours is even less - at most 20-25 hours a week of creative work. Of course there are plenty of other tasks you have to do that don't take as much mental effort. – Alexander Woo Mar 20 '18 at 17:16
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    @AlexanderWoo: Don’t underestimate how much people’s temperaments vary — I’ve known a few friends/colleagues (in both academia and the arts) who could manage well over 25 hours a week of creative work, for at least several weeks at a stretch, very fruitfully. 15–16 hours a day is probably unsustainable for anyone (and doesn’t sound like it’s working for OP), but some people really are capable of a lot. – PLL Mar 20 '18 at 19:03
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    May also be worth stressing proper nutrition to the OP in this circumstance; if working 16 hrs a day I wonder how much of a break is allowed by he/she for that. – JNS Mar 20 '18 at 19:45
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    @AlexanderWoo I agree about 20-25 hours of creative work. The 40-60 was total work hours. – Boris Bukh Mar 20 '18 at 20:00
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    I worked long hours on my dissertation, getting 4-6 hours of sleep per night, trying to fix the software I was writing. I was stuck on it for a couple of weeks and I'd only made the problem worse. I finally snapped, and walked away from the whole project. After a few nights of 8+ hours of sleep, I looked at it again... and fixed the problem in two hours. Looking back, I can count dozens of occasions where the three extra hours of work at night only caused me to spend two hours in the morning fixing tiredness-induced errors. Those extra few hours are usually better spent resting than working. – anaximander Mar 22 '18 at 11:16
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i spent on study i in a day from 15 to 16 hours I do study [...] My productivity is less these days as I try to avoid mistakes in my research. [...] I always try to prepare my best but fails these day quite frequently and making silly mistakes which are annoying [...]

You have already identified the problem. You're over-worked and under-productive. First, I would take a vacation over a long weekend. You need a break.

After your vacation, you need to better manage your time so you will have a sustainable work schedule. From your post, it sounds like you may need to build better time management skills. I've seen grad students waste time doing the following

1) Spending several hours reading 1 paper. Spend a set amount of time (like 30 minutes) reading each paper and note down what you don't understand. Join a reading group and bring your questions there.

2) Taking too many classes at once to "get them out of the way." Sometimes your advisor will push you to do this. If this is the case, stand firm, and set some other goals with them.

3) Constantly rework a paper, or otherwise pour time into a fruitless research topic. Part of your post mentioned you feel behind on the number of papers published. Bring this up with your advisor and focus on upping that number. This may mean publishing shorter papers or publishing in less prestigious journals. Once you zero in on a good research topic, it'll be much easier to turn out high quality research papers.

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    The only thing I disagree with is, "Take a vacation over a long weekend." A long weekend is not a long enough vacation to recover from this level of burnout! I would suggest at least a week's vacation, including both adjacent weekends, and longer if possible. – Kevin - Reinstate Monica Mar 20 '18 at 23:28
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    @Kevin, I agree that a week would be better, but depending on class schedules, a week may not be doable in the very short term, and I feel like taking Thursday - Sunday off now is better than taking a week off a month from now. – sevensevens Mar 22 '18 at 2:01
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Your experience does not sound incredibly different from my own. (I finished my math PhD less than a year ago.) That said, I can tell you about a few things I did that made a big difference for me during my thesis years.

  1. Deliberately set boundaries for your work. For example, I never did anything academic while eating, and I never worked on Fridays after 6:30 p.m. Those times were deliberately for my brain to rest.

  2. Commit to doing something non-academic to force you to get away from your desk. I volunteered with a local church for a few hours a week. It was only a small commitment, but I found that I benefited greatly from the time away from my work.

  3. It's a cliche, but exercise also made a big difference for me. You don't have to be super intense to get the benefits--I would usually run or lift for about 45 minutes three or four times a week. Take advantage of your university gym if you can!

  4. This is my most important point: Do NOT allow yourself to feel guilty for spending time away from your work. You've been in mathematics long enough now to know that revisiting a problem after a short break can grant you new insights. Just like the people in other answers have said, grinding out 16 hour days can lead to a significant drop in productivity, so you should aspire to have more productive work with less hours spent--that begins with the implementation of breaks.

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Even though I was very obsessed during the year it took to write my dissertation, I had to choose to allow gaps and spaces in my time, i.e., a walk every two hours, or a break for a meal, or a switch to another aspect of the research. What you have described is a form of burnout as a result of being overly focused that leads to numbness, boredom, and dissatisfaction. A day entirely off every once in a while will also work wonders for your mental landscape.

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Being in Ph.D. from the last five years, I will tell you my experience.

Most of the time these days I spend on studying -- maybe 15 to 16 hours per day

This really is a problem. I fell you are missing a very beautiful part of the research. Don't think that you are studying for 15 to 16 hours, but feel you are conducting research for that much of time and enjoy that and take that as fun. Feel that your research is going to be helpful to the community and you will see your name in the world of science where your name will be engraved on scientific journals and will be counted as a good researcher. Have fire and passion inside you and go ahead with that.

Another important thing is that you need to give your brain enough rest so that your mind and body are again ready to go for work.

Some important points:

  • Have a nice sleep. Due to pressure from your peers, you may not be able to have a good sleep. So set timetable of you sleep for e,g sleep before 11 and get up till 6, nearly 7 hours of sleep should be enough for you to reset your brain and body.
  • Play some sports (gym, jogging, etc) daily at-least for 1 hour that is going to increase your productivity a lot.
  • Very important is, don't compare yourself with your peers. Don't bother what they are doing and focus on your goals.
  • Keep yourself calm and don't bother that you don't have any publications till date. Think this is not the end of the road and you will publish a lot more world-class research papers.
  • Have confidence in yourself and say that you are going to do it.

More importantly is to have passion for what you are doing, leave everything aside and take research as fun. Without enjoying your work, it is going to be really difficult with the passage of time, eventually decreasing your productivity.

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This sounds like the classic negative spiral. In broad strokes,here's how it works:

  • You work hard because you're conscientious and because the material is hard
  • You work long hours and start to skimp on physical activity, recreation, etc.
  • Your tiredness makes you make more mistakes than usual and not to learn as efficiently
  • Your mistakes and frustrations increase your anxiety
  • Your anxiety leads you to work harder and longer hours, and further deprive yourself of healthy breaks

Sometimes it's helpful in these situations to check in with one's primary health care provider.

You and your doctor might want to consider medication to support you during this challenging period. Or perhaps not. Regardless, it can still be helpful to check in with him or her, in a number of ways:

  • S/he can have you fill out an anxiety screening questionnaire, and have you come in for a follow-up after a certain number of months, to see how you're doing.

  • S/he might suggest a therapist. Cognitive therapy has a good track record with these anxiety situations.

  • Sometimes an underlying physical health situation is exacerbating the anxiety, and making some improvement in the one can help with the other.

  • The doctor will have your long-term health in mind and may suggest some simple preventative measures intended to improve your general health. It's surprising how much this can help a person's outlook on life.

  • Talking the situation over with the doctor can help one gain perspective.

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