When a tool maker designs a drill she does not expect it to also be useful as a hammer or a torch. You might be able to use it as either at a pinch but it would not be very effective. If you want a universal tool that can do lots of different things you need to design it with those things in mind from the start and the result may be just as complex as a box of individual tools.
The strangest thing is that this is not the case with mathematical tools. A concept or method used to solve one mathematical problem often turns out to be just as useful in solving others that are completely different and seemingly unrelated. To give a simple example, the number pi was first defined to quantify the ratio of the circumference to diameter of a circle, yet it appears in a whole host of mathematical and physical equations that have nothing to do with circles. The same is true for a few other special numbers such as e, the base of the natural logarithms. Why do the same few numbers keep coming up in mathematics instead of different ones for every problem? Other examples abound. Special functions, groups, algorithms and many more mathematical structures prove useful over and over again. Why does mathematics have this natural universality?
People who don’t know mathematics well think that mathematicians invent the methods they use in the same way that people invent stories or fashions. Most mathematicians say that this is not the case. Their experience of developing new mathematics feels more like a process of discovery rather than invention. It is as if the mathematical structures were already there before any human was aware of them. You can also invent mathematical structures like the game of chess, but chess is not regarded as important in mathematics because it is not useful for solving unrelated problems. It does not have universality and it is this universality that distinguishes the interesting mathematics from the uninteresting.
Why then is such universality so easy to find in mathematics without trying to look for it? Why does it even extend into physics where deep mathematical ideas originally used to solve problems in pure mathematics turn out to be important for understanding the laws of nature (complex numbers, differential geometry, topology, group theory etc.) ? It is even more surprising to pure mathematicians when theory developed by physicists turns to be useful in mathematics, yet even string theory has already proved useful for solving a whole host of mathematical problems that seemed otherwise intractable. It remains a huge and deep mystery why this happens.
Yesterday the Abel committee in Norway announced that it was awarding its annual prize of 6 million Kroner to Yakov Sinai for his fundamental contributions to dynamical systems, ergodic theory, and mathematical physics. Sinai’s work covers complex dynamical non-linear systems with many variables. Naively our prior expectation for such systems would be that they are going to behave in complex unpredictable ways and the only thing they are likely to have in common is going to be randomness, but that is not what happens. Systems in statistical physics have entropy and temperature. These are macroscopic emergent quantities that follow derived laws which are common to different systems irrespective of the microscopic description of the dynamics. They have phase transitions and near these transitions you get critical phenomena that obey universal laws.
Sinai looked in particular at chaotic systems of non-linear dynamical equations where more universal emergent behavior is found and described in terms of certain Feigenbaum constants. Another area he worked in was algorithmic complexity of binary sequences and dynamical systems. All this work is highly applicable to practical problems but it is also important as a tool for understanding universality and why it arises. Perhaps one day by building on the work of Sinai we will learn much more about the unity of mathematics, the laws of physics and why these things are so beautifully connected by universality.
Congratulations to Yokov Sinai for this well deserved award which will raise the profile of such important work.