Final Report Summary - PAGAP (Periods in Algebraic Geometry and Physics) The PAGAP project concerned the study of periods. These are a certain class of numbers which include some of the most frequently-occurring quantities in mathematics, such as pi. A fundamental class of periods, called multiple zeta values, were discovered by Euler in the 18th century, and were subsequently forgotten for several centuries. They resurfaced simultaneously in many branches of mathematics in the 1990's, as mathematicians started to realise that they are of central importance, connecting apparently unrelated domains in mathematics and physics.The PAGAP project also concerned high-energy physics. Theoretical predictions in quantum field theory form the basis for all particle collider experiments, such as those taking place at the Large Hadron Collider. Physicists must derive precise observational predictions from the raw laws of physics, as they are currently understood, and compare them with experimental data coming from particle collisions. Discrepancies between the two correspond to the discovery of new hitherto unaccounted for physical phenomena. The calculation of these predictions is an enormously complicated and highly intensive task which pushes the boundaries of modern mathematical techniques. It occupies many teams of physicists the world over.The starting point for the PAGAP project was the observation that the quantities required for high-energy physics, which are called Feynman amplitudes, are in fact periods. Thus the accumulated experience of mathematics can be brought to bear on the difficult task of understanding their structure, and the practical problem of computing them. To summarise the main conclusions of the project, one can say that, before the advent of the PAGAP project, there were three classes of numbers (which are all periods) known to mathematicians and physicists:1). Multiple zeta values, first introduced by Euler2). Numbers called the periods of mixed Tate motives3). Feynman amplitudes in a physical theory called massless phi4A difficult conjecture in mathematics claimed that the first two classes were in fact the same. On the other hand, a folklore belief in physics stated that the first and third classes should also be the same, based on a huge amount of numerical data. The received wisdom, therefore, was that there was a single fundamental class of periods occurring in mathematics and physics.The main conclusion of the project is that:1) and 2) are indeed the same, which led to the resolution of two well-known and important conjectures in mathematics (including the Deligne-Ihara conjecture which had been open for 25 years), but, surprisingly, 3) is in fact quite distinct, and hence the folklore beliefs were in fact false. This showed that the quantities occurring as amplitudes in high-energy physics are much more subtle and mathematically complex than previously imagined. In the process of this work, several other long-standing theoretical questions about periods in both mathematics and physics were resolved by the PAGAP project. The PAGAP project also sought to develop and implement new methods for the practical calculation of Feynman amplitudes in high-energy physics, using new techniques from algebraic geometry. This led to some extremely efficient new techniques which broke several records in the field, and which, for certain questions, are an order of magnitude more powerful than alternative approaches. These have now become part of the standard arsenal of methods available to physicists for making predictions for particle collider experiments.
