## Final Activity Report Summary - PHD (Phenomena in high dimensions)

This project brought together internationally leading researchers and young scientists from several mathematical disciplines - probability theory, high-dimensional geometry, combinatorics and mathematical physics - to focus jointly on the remarkable phenomena that arise in high-dimensional systems. The analysis of such complex systems plays, and will continue to play, a major role in the development of many scientific areas in the coming decades, including physics, biology and economics. The areas of mathematics represented in the project have, until recently, been considered quite separate: but they have developed a commonality of methods and viewpoints because of the appearance in all of them of pseudo-random phenomena, created by the high-dimensionality. Once a system becomes sufficiently complex, it is impossible (and probably not very useful) to predict the detailed nature of small-scale features of it. On the other hand, crucial features of its large-scale behaviour often become predictable because its complexity forces it to mimic random behaviour: behaviour generated by the interaction of many independent effects.

The project combined ground-breaking scientific research with the training of a new generation of young mathematicians who have acquired a familiarity with all or many of the different strands of high-dimensional mathematics. The research outcomes include:

- a rigorous derivation of the formula for the energy of spin-systems of sub-atomic particles;

- faster algorithms (based on carefully designed random sampling) for the calculation of parameters associated to high-dimensional objects;

- a rigorous demonstration of the existence of a universal pattern in the distribution of the characteristic frequencies of matrix models;

- a new formula for the measure of disorder in random systems that can be used for effective computation;

- a more detailed explanation of how high-dimensional geometry automatically produces characteristically random effects; and

- a subtle new way to describe large networks by encoding their local properties in a single 'landscape'.

The project combined ground-breaking scientific research with the training of a new generation of young mathematicians who have acquired a familiarity with all or many of the different strands of high-dimensional mathematics. The research outcomes include:

- a rigorous derivation of the formula for the energy of spin-systems of sub-atomic particles;

- faster algorithms (based on carefully designed random sampling) for the calculation of parameters associated to high-dimensional objects;

- a rigorous demonstration of the existence of a universal pattern in the distribution of the characteristic frequencies of matrix models;

- a new formula for the measure of disorder in random systems that can be used for effective computation;

- a more detailed explanation of how high-dimensional geometry automatically produces characteristically random effects; and

- a subtle new way to describe large networks by encoding their local properties in a single 'landscape'.