March 6, 2010
1. Work at several problems at a time. If you only work on one problem and get stuck, you might get depressed. It is nice to have an easier back-up problem. The back-up problem will work as an anti-depressant and will allow you to go back to your difficult problem in a better mood. John told me that for him the best approach is to juggle six problems at a time.
2. Pick your problems with specific goals in mind. The problems you work on shouldn’t be picked at random. They should balance each other. Here is the list of projects he suggests you have:
- Big problem. One problem should be both difficult and important. It should be your personal equivalent to the Riemann hypothesis. It is not wise to put all your time into such a problem. It most probably will make you depressed without making you successful. But it is nice to get back to your big problem from time to time. What if you do stumble on a productive idea? That may lead you to become famous without having sacrificed everything.
- Workable problem. You should have one problem where it’s clear what to do. It’s best if this problem requires a lot of tedious work. As soon as you get stuck on other problems, you can go back to this problem and move forward on the next steps. This will revive your sense of accomplishment. It is great to have a problem around that can be advanced when you do not feel creative or when you are tired.
- Book problem. Consider the book you are working on as one of your problems. If you’re always writing a book, you’ll write many of them. If you’re not in the mood to be writing prose, then work on math problems that will be in your book.
- Fun problem. Life is hardly worth living if you are not having fun. You should always have at least one problem that you do for fun.
3. Enjoy your life. Important problems should never interfere with having fun.
This advice from J.H. Conway is excerpted from the blog post of Tanya Khovanova
January 8, 2010
If an ape can make a discovery, so can you.
Richard P. Feynman
as quoted in this book
What do you think about this quote?
January 6, 2010
1. Raise your quality standards as high as you can live with, avoid wasting your time on routine problems, and always try to work as closely as possible at the boundary of your abilities. Do this, because it is the only way of discovering how that boundary should be moved forward.
2. We all like our work to be socially relevant and scientifically sound. If we can find a topic satisfying both desires, we are lucky; if the two targets are in conflict with each other, let the requirement of scientific soundness prevail.
3. Never tackle a problem of which you can be pretty sure that (now or in the near future) it will be tackled by others who are, in relation to that problem, at least as competent and well-equipped as you.
The original text of the rules together with the author’s comments can be found here (HTML) or here (PDF).
November 12, 2009
A list by Dmitry Podolsky
Update: another such list (this time of 24 problems) by Sean Carroll, the three most important open problems in physics by the Nobel Prize winner Vitaly Ginzburg, and a more extensive list (see also the updated book version of this list) by the same author.
November 7, 2009
I have found (hat tip: Yuri Kryakin) a great interview in Russian with Ivan Panin, where he reminesces, inter alia, about his teacher, a prominent mathematician Andrei Suslin and the way he works. The whole text is pretty long and very interesting but it is quite difficult to find reasonably self-contained excerpts to translate into English for those who don’t speak Russian, so let me give you just one bit as a teaser:
Suslin tackled the problems roughly as follows: first we see [the problem or the result to prove], then we believe [that we can solve it or that we can prove the result], and then we prove it. Because if you don’t believe, you will not have your vision materialized.
November 6, 2009
According to Sir Michael Atiyah, the most important quality for a working mathematician is the ability to maintain concentration for a long time. Via Konstantin Zuev (original post in Russian).
June 5, 2009
While writing the research papers one quite often needs to get back to the full texts of old (pre-Internet or at least pre-arXiv) references. Of course, having access to a good library and/or the interlibrary loan usually solves the problem but can be somewhat time- and cost-consuming.
It is not that well known, however, that there is a fair chance to find the old paper or preprint you need online for free. Of course, the first thing to try is Google or perhaps another search engine of your choosing. However, if this does not work, you still have a fighting chance, at least as far physics and mathematics are concerned. The places to try are:
- the KISS preprint server (you can also try the umbrella interface at SPIRES) allows you to search in (and get to the full text of) a huge database of scanned preprints going back to the 1970s at least. The database covers mostly high-energy physics and related areas, including a fair share of mathematical physics and mathematics. For instance, you can find there a number of preprints by Richard Feynman, including the unpublished ones.
- the Digital Mathematics Library
- NUMDAM and CEDRAM (French mathematical journals)
- The Project Euclid
- MathNet.Ru (Russian mathematical journals)
All items but KISS are purely mathematical databases (to be precise, MathNet.Ru includes several physics, mechanics and mathematical physics journals as well).
If you know of other similar databases (be it in physics, mathematics, life sciences,…), please feel free to drop a comment with the relevant link(s).