Integrability and M-Theory

In the current physical picture of the world, nature is split into two domains. On the one hand, one has Einstein's theory of general relativity describing gravity, which governs the universe on a macroscopic scale: it allows us to predict the motion of the planets and galaxies, and it predicts the expansion of the universe and the possible existence of black holes. On the other hand, we have quantum mechanics and the standard model of elementary particles, governing physics on the scale of atoms. Bringing both theories consistently together in a unified picture is currently only possible within a framework of ideas called string theory. In string theory, elementary particles are replaced by excitations on vibrating strings, quite similar to the excitations of the strings of a violin producing various notes. String theory has many extraordinary properties and it has led to quite a few new discoveries in both mathematics and physics. Over the last ten years, however, more and more hints of a theory lurking in the background have appeared, and this theory was called M-theory. M-theory might in fact be the long-sought world-formula as it potentially provides a unified description of all the natural phenomena in our universe.In this research programme, some crucial questions about M-theory are asked. In M-theory, the basic building blocks are no longer strings, but 2- and 5-dimensional objects called M2- and M5-branes. One of the great current mysteries is how to describe the dynamics of the M2-branes, even if gravity was ''turned off''. Recently, there has been a proposal for such a theory of M2-branes and it received much attention over the last year. It turned out, however, that this candidate theory is not general enough; for example, it allows only for the simultaneous description of two M2-branes. It is therefore necessary to explore possible extensions and modifications of this theory, and this is the main purpose of this research programme. Mostly, mathematical aesthetics will be used as a guideline for constructing these extensions by identifying broader mathematical frameworks, into which the original theory can be fitted. Of course at the end of the day, it is necessary to check our constructions against physical requirements. Also, I will aim at obtaining an effective description even for the dynamics of M5-branes. This should be possible because in situations when M2-branes end on M5-branes an interesting symmetry is expected to appear, which relates both theories. Finding both descriptions would greatly improve our understanding of the very nature of the universe we are living in.