だんだんと寒くなってきて冬を感じるようになってきた．読んでいた論文も読み終わり，来週のゼミで考えたい物理現象のモデリングが終わる（モデリングのチェックが終了する）ので，ようやく研究がスタートする．

さて，テレンス・タオ先生のブログ内のキャリアアドバイスの記事が非常に役にたつという記事を見た．そこで私は英語の勉強も兼ねてタオ先生の記事を読んでみることにした．

Career adviceterrytao.wordpress.com

以下，備忘録として書き留めておく．英訳は各自に任せる．

タオ先生は学部と大学院での勉強の違いについて次のように述べている：

When learning mathematics as an undergraduate student, there is often a heavy emphasis on grade averages, and on exams which often emphasize memorisation of techniques and theory than on actual conceptual understanding, or on either intellectual or intuitive thought. There are good reasons for this; there is a certain amount of theory and technique that must be practiced before one can really get anywhere in mathematics (much as there is a certain amount of drill required before one can play a musical instrument well). It doesn’t matter how much innate mathematical talent and intuition you have; if you are unable to, say, compute a multidimensional integral, manipulate matrix equations, understand abstract definitions, or correctly set up a proof by induction, then it is unlikely that you will be able to work effectively with higher mathematics.

However, as you transition to graduate school you will see that there is a higher level of learning (and more importantly, doing) mathematics, which requires more of your intellectual faculties than merely the ability to memorise and study, or to copy an existing argument or worked example. This often necessitates that one discards (or at least revises) many undergraduate study habits; there is a much greater need for self-motivated study and experimentation to advance your own understanding, than to simply focus on artificial benchmarks such as examinations.

……

Whereas at the undergraduate level and below one is mostly taught highly developed and polished theories of mathematics, which were mostly worked out decades or even centuries ago, at the graduate level you will begin to see the cutting-edge, “live” stuff – and it may be significantly different (and more fun) to what you are used to as an undergraduate! (But you can’t skip the undergraduate step – you have to learn to walk before attempting to fly.)

タオ先生は紛れもない天才だが，次のように，タオ先生は数学には必ずしも才能は必要ではないと強く主張している：

Does one have to be a genius to do mathematics?

The answer is an emphatic NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the “big picture”. And yes, a reasonable amount of intelligence, patience, and maturity is also required. But one does not need some sort of magic “genius gene” that spontaneously generates ex nihilo deep insights, unexpected solutions to problems, or other supernatural abilities.

その理由として次のように述べている：

The number of interesting mathematical research areas and problems to work on is vast – far more than can be covered in detail just by the “best” mathematicians, and sometimes the set of tools or ideas that you have will find something that other good mathematicians have overlooked, especially given that even the greatest mathematicians still have weaknesses in some aspects of mathematical research. As long as you have education, interest, and a reasonable amount of talent, there will be some part of mathematics where you can make a solid and useful contribution. It might not be the most glamorous part of mathematics, but actually this tends to be a healthy thing; in many cases the mundane nuts-and-bolts of a subject turn out to actually be more important than any fancy applications.

数学を研究・勉強していて退屈になってしまったときについて：

There will of course be times when one is too frustrated, fatigued, or otherwise not motivated to work on one’s current project. This is perfectly normal, and trying to force oneself to keep at that project can become counterproductive after a while. I find that it helps to have a number of smaller projects (or perhaps some non-mathematical errands) to have at hand when I am unwilling for whatever reason to work on my major projects; conversely, if I get bored with these smaller tasks, I can often convince myself to then tackle one of my bigger ones.

先生から「卒業」することについて：

While you should talk to your advisor, you should not be completely reliant on him or her; after all, you are going to have to do mathematics primarily on your own once you graduate!

If you feel like you want to learn something, do something, or write something, you don’t have to clear it with your advisor – just go ahead and do it (though in some cases other priorities, such as writing your thesis, may be temporarily more important, and you should of course keep your advisor updated as to what you’re doing mathematically).Research your library or the internet, talk with other graduate students or faculty, read papers and books on your own (both in your field and in nearby fields), attend conferences, and so forth. (See also “ask yourself dumb questions”.)

One specific suggestion I have is to subscribe (either by RSS, or by email) to be notified of new papers which appear on the arXiv in the subject areas that you are interested in.

自分の分野とは「関係のない」研究集会に参加することについて：

Modern mathematics is very much a collaborative activity rather than an individual one. You need to know what’s going on elsewhere in mathematics, and what other mathematicians find interesting; this will often give valuable perspectives on your own work. This is true not just for talks in your immediate field, but also in nearby fields. (For much the same reason, I recommend studying at different places.) An inspiring talk can also increase your motivation in your own work and in the field of mathematics in general.

タオ先生のブログにはたくさんの有益な記事が載っているが，特に次の記事は参照したほうがいいと思う．

There’s more to mathematics than grades and exams and methodsterrytao.wordpress.com

Does one have to be a genius to do maths?terrytao.wordpress.com

Work hardterrytao.wordpress.com

Attend talks and conferences, even those not directly related to your workterrytao.wordpress.com

Take the initiativeterrytao.wordpress.com

では．