Fermat’s Last Theorem

I know it’s a rare privilege, but if you can tackle something in adult life that means that much to you, then it’s more rewarding than anything imaginable.

Basically it’s just a matter of thinking. Often you write something down to clarify your thoughts, but not necessarily. In particular when you’ve reached a real impasse, when there’s a real problem that you want to overcome, then the routine kind of mathematical thinking is of no use to you. Leading up to that kind of new idea there has to be a long period of tremendous focus on the problem without any distraction. You have to really think about nothing but that problem – just concentrate on it. Then you stop. Afterwards there seems to be a kind of period of relaxation during which the subconscious appears to take over and it’s during that time that some new insight comes.’

An expert problem solver must be endowed with two incompatible qualities – a restless imagination and a patient pertinacity.

An expert problem solver must be endowed with two incompatible qualities – a restless imagination and a patient pertinacity.
Howard W. Eves

Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove this conjecture.’

Any insoluble problem in one area of mathematics could be transformed into an analogous problem in another area, where a whole new arsenal of techniques could be brought to bear on it. If a solution was still elusive, the problem could be transformed and transported to yet another area of mathematics, and so on, until it was solved.

There are dozens of such islands, each one with its own unique language, incomprehensible to the inhabitants of other islands. The language of geometry is quite different to the language of probability, and the slang of calculus is meaningless to those who speak only statistics.

For example, there is the island occupied by geometers who study shape and form, and then there is the island of probability where mathematicians discuss risk and chance.

Mathematics consists of islands of knowledge in a sea of ignorance.

Just as the E-series is the DNA for elliptic equations, the M-series is the DNA for modular forms. The amount of each ingredient listed in the M-series is critical. Depending how you change the amount of, say, the first ingredient you might generate a completely different, but equally symmetrical, modular form, or you might destroy the symmetry altogether and generate a new object which is not a modular form. If the quantity of each ingredient is arbitrarily chosen, then the result will probably be an object with little or no symmetry.