Manuel Blum has some unique and helpful advice for graduate students starting a PhD. A lot of the advice applies to anyone, however.
- Blum advises to not use books like scrolls and read everything from the start to finish. In mathematical sciences especially, it is better to identify and read relevant material and try to understand something in depth.
- He also advises to write down whatever one reads. This is definitely useful when reading complex material.
- Another piece of advice is to have the belief that there must be some easy way to overturn a hard problem. This helps one to look for innovative solutions and remain optimistic.
- In case someone solves a problem before you, one should take it as an opportunity to analyze how one could have approached the problem better. Moreover, one should read the other person’s solution carefully to learn the method.
- Blum emphasizes the importance of contradiction and paradoxes in mathematical reasoning.
- There is also a suggestion to list all possible ways to solve a problem, by, for example, looking at small examples in order to get a handle on the problem.
- Blum writes that thesis advisors could be of two types. One is someone who gives you a problem from his or her own research area. This might be solvable or very hard but it is advisable to look for a non-standard answer. The second type of supervisor may encourage or let you work in an area that the supervisor is familiar with, which means that you will have to do a lot of literature research. Both types of supervisors have their pros and cons.
- The most important advice in the whole article is that whatever one does, one must enjoy doing it. This is critical in maintaining sufficient enthusiasm to solve problems.