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).
January 4, 2010
They say it’s some form of recognition but I respectfully doubt. In my case some [your favorite expletive goes here] have just copied my post on choosing a research topic and put it here: (no clickable link folks, I am not going to improve their search engine ratings!!!). No slightest shade of fun in this for me, unlike, say, in the case of Scott Aaronson.
And that should have been a scientific blog (judging by the domain name)… Talk about plagiarism in science after that. Pathetic, isn’t it? Any suggestions as to what can one do about all this (remember, it’s a Chinese site, so I doubt that the standard things like writing them and asking to remove the content or contacting their ISP would be helpful)?
December 24, 2009
The Princeton Companion to Mathematics was extensively reviewed, and often praised, all over the mathematical and scientific blogosphere, see e.g. here, here, here and here. Most of this praise is probably well deserved. But where should an interested student (or even a professional mathematician who wants to extend her or his professional range, for that matter) go in order to deepen the knowledge acquired from PCM without getting bogged down into the details of the proofs and other such subtleties that abound in the specialized literature?
Of course, there is plenty of possible answers to this one, and you are welcome to share yours in the comments. However as far as “classical” (basically more or less up to the early XXth century level) mathematics goes, the Oxford User’s Guide to Mathematics appears to provide, at least for me, a reasonable, if not quite perfect, enhancement for PCM.
OUGM has many omissions of its own and certainly could use more editing and proofreading — in particular, in order to make it somewhat more self-contained, but nevertheless this book provides a fairly broad and reasonably deep (for the beginner) panorama of the “classical” mathematics as defined above. For instance, it does not cover category theory and related stuff. However, by and large, OUGM does a quite decent job in helping the beginner to advance her/his understanding of a great number of mathematical disciplines from abstract algebra to probability theory, and I certainly recommend to have a serious look into this book if you really want to deepen your knowledge of the “classical” subjects beyond the PCM level.
P.S. I just cannot miss this opportunity to wish merry Christmas and happy New year to the readers of this blog 🙂
November 13, 2009
An important addition to my earlier post on writing grant applications: there is a series of articles on science funding in Science Careers. The author is someone going — quite appropriately 🙂 — by the name of Grant Doctor.
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.