The focus of the project GALOP was to study certain numbers which are ubiquitous in mathematics and physics and are known as `periods'. Examples include the number Pi, log 2, the period of a pendulum, values of zeta functions, and many other transcendental quantities given by integration. The idea of the project was to study these from a completely new perspective, namely the theory of groups. It is a very recently understood and highly surprising fact that there is a huge group of symmetries which transforms all (`motivic') periods in a consistent manner, in other words, in such a way that all equations between them are preserved. This changes the way to think about mathematical equations: the number pi for example is no longer seen as a static quantity, but one which is allowed to be rescaled by multiplying it by any rational number. The upshot is that periods acquire new features which are not possessed by arbitrary numbers, including a weight (e.g. pi has weight 2), a dimension (pi has dimension 1, but log 2 is 2-dimensional), and so on. In short, the Galois theory of periods allows one to classify periods according to different types, and to deduce new equations from old. It also allows one to solve seemingly intractable algebraic and combinatorial problems using completely different transcendental methods.
A major application of this theory is to high-energy physics, where one studies interaction of fundamental particles. It is a slowly emerging fact that all the different theories of fundamental particles (quantum field theories, or string theories, etc) all admit a unifying underlying mathematical framework, which is the focus of much of this project. In order to make physical predictions, physicists must assign a probability or `amplitude' to certain types of interactions between elementary particles, and this probability is invariably a period or more specifically a period of a moduli space (which is a number or function assigned to the space of all possible particle configurations). These special geometric spaces not only generate the important periods relevant for particle collider experiments, but also generate entire families of periods of given `types' in mathematics. Much of the project was devoted to studying these spaces and the structure of the periods they generate.
Combining the methods of GALOP the PI was led to deduce the existence of a completely new and vast group of symmetries which acts on amplitudes for particle interactions. This is the `cosmic' Galois group whose existence was posited many years ago but which remained elusive until very recently. It leads to a vast over-arching principle which organises the amplitudes in many different physical theories. It can be thought of as a new universal law which dictates how particle interactions constrain one another. It has very recently been verified by different research groups, in a wide variety of situations including: string theory, N=4 super Yang Mills theory, massless phi^4 theory, and even for the anomalous dipole moment of the electron.
The purpose of the GALOP project was to develop ideas from the Galois theory of periods to deepen our understanding of mathematical structures (applications include: the classification of mixed `motives', construction of invariants in geometric group theory, generate new classes of mathematical objects such as modular forms) and to apply these ideas to solve problems in high-energy physics. Conversely, new ideas arising from examples from physics problems were reformulated as general mathematical principles which in future may inspire and develop new mathematical ideas.
