Play it for real (in a discrete way)

We all like to play. Or at least we liked to play at a certain stage of our life. Is there a better game than the one in which you create your own universe with only a few basic building blocks? Does this universe have to exist out there in order to call it real? Is the description of an object the object itself? Does this created or idealised universe approximates somehow to the real one, the one most people is used to, or the one in which we think we live?

Plenty of games and plenty of time to play. As for me, I have decided to continue playing for a longer time. I’ve chosen mathematics, a game with simple rules and lots of possible creations. The possibilities are so extent that, once one decides to get serious, which means pursuing a career as a researcher, one has to chose a narrower path, with a fewer rules that could lead to the possibility of new creations or universes which no one else has envisioned yet.

However, the game is not an isolated one. It has been played for thousands of years and it’s being shared amongst people from different times. That is some of the magic of the game. Imagine you could gather your all-time favourite soccer players and not only watch a match but participate in it. That is how we play in mathematics: a bit here and a bit from then and voilà, the match is won.

The sport which I’ve decided to play is called “discrete integrable systems” and, after a few words on it, I’m sure you will agree with me that it is one of the most natural choices with respect to the times we live. By discrete we refer to the possibility of counting one by one, opposed to the continuous. Imagine you are at your local market. Buying 5 apples is a discrete buy, whereas buying a kilo of sugar is a continuous one. But this is not the main attraction to study the discrete world. Nowadays, with more and more computers in our life and the need to communicate with them, there is a need to translate a possible command in terms or ones and zeros, a discrete language. Being a bit more daring, one could think of time as discrete since, isn’t it the moments and not the whole life which constitute our memories? Even space might be thought as discrete, since all matter can be decomposed in terms of elementary, quantified particles.

As for integrable systems, they go back to the 18th century with the theory of the elliptic orbits of the planets. An integrable system is one which is well behaved, that is, which can be predicted to a certain degree and be solved if necessary. One of the main characteristics of integrable systems is that they possess conserved quantities or symmetries, as we sometimes like to think of them. In a few words, conserved quantities give us a way to trust in the world we live in. We know that gravity is constant on earth and that we go around the Sun and will not escape from the orbit the Earth has been following for millions of years. That gives some peace of mind, doesn’t it?

But I was telling you how the game is shared amongst mathematicians from different times. A hundred years passed and integrable systems where the object of attention again. Now with a different object of study: solitons, which are waves that travel without loosing its shape. Just like a tsunami or a nerve impulse. Nowadays, the applications of integrable systems are numerous. An interesting one is used in the implementation of particle accelerators. The reason for it is that when shooting a particle one would like it to hit a certain target, or at least don’t dissipate or run unpredictably. Hopefully some of this particle shooting will give some answers as of how the universe was created. And integrable systems will be a part of this search.