若いときに落ち込むな． 数学者の多くは大器晩成型である． 大切なのは努力を続けること．
基礎となる例を集めよ． 抽象的な分野を勉強するときには，一つの定理について少なくとも一つの例を持っていること． 休暇からの脳みそのリハビリにも役に立つ．
下手くそに書かれた論文は無視される． 良い論文を書くことは重要である． 焦ってはならない． まずは論文の構造を練る． 全部を書き直すことを恐れてはならない．
- Always ask yourself questions. Atiyah says that one of the secrets of his success is to always be curious.
- Don’t get disheartened in your early years. Most mathematicians have a slow start. It’s important to keep on pushing and eventually a breakthrough will come. Indeed, Michael mentioned that when he was young he had a slow year, and so he sought advice from a great mathematician of the day. It turns out that this great mathematician had also had a disheartening year when he had started!
- Collaboration is important. There are benefits, mathematical and personal, when one collaborates in research with another. On the technical level, you get a second opinion and a fresh set of ideas to add to your own. On the personal level, you can make good friends while you work.
- Manage how much you get sidetracked. When you’re working on a big problem, you can often get sidetracked on a smaller problem. It’s important to realise that 9 out of 10 such sidetracks will lead nowhere, so it’s a good practice to know when to abandon your deviations. Of course, this management is tricky, as 1 out of 10 times you might just be on to something!
- Attending a bad lecture can be useful. There are many times when you go to a talk where the presenter will state a beautiful theorem. However, they make a complete mess of things when they go to the proof. If it looks like the talk is heading in this direction, have a go at understanding intuitively the ideas behind the proof. That is, try and nut out the proof yourself on a piece of paper, rather than continue to watch the presenter struggle.
- Keep a collection of grounding examples in your area. If you are working your way through a very theoretical area, it’s important to have at least one example of every theorem you come across. This helps to keep things both intuitive and connected, and makes for an easy catch-up if you ever have a few weeks away from work.
- The most important thing for your PhD is to have a good supervisor. In particular, try to get supervision by a world expert. Second rate mathematicians won’t be able to give you clear answers to high calibre questions. By having a sharp supervisor, you get a front row seat to the thinking that happens on the top floor of maths.
- Make the introduction to your papers readable to any mathematician. As it stands, about 99\% of papers in mathematics are unreadable to the non-specialist. You should aim to make the introduction accessible to all mathematicians.
- A badly written paper is ignored. Writing a good paper is important. You should never rush it. A good practice is to start by first laying out the logical structure of the paper, before having a go at writing. Once you’re done, put the paper to the side for a few weeks before giving it a look over with a fresh set of eyes. Don’t be afraid to rewrite the entire paper, especially if it would improve it.
- Use rough ideas in your papers. Before you launch into your formal proof, give the reader a rough outline as to what is about to happen. You’d do this in a talk, right?