## Final Activity Report Summary - ENIGMA (European Network In Geometry, Mathematical Physics, and Applications)

Integrable systems and random matrices arise in various problems of pure and applied mathematics and statistics. In particular, integrable systems with finite degrees of freedom have been the object of intense investigations by mathematicians and physicists for many centuries because of their applications in mechanics and geometry. The trajectories of integrable mechanical systems display a regular behaviour as opposed to a chaotic one.

The discovery of integrable behaviour in classical and quantum physical systems with infinite degrees of freedom is one of the most exciting developments of mathematical physics in the second half of the 20th century. Such systems are present in fluid dynamics, classical and quantum field theory, statistical physics and optical communications. Moreover, algebraic and differential geometry plays a prominent role in the mathematical foundations of the theory of integrability.

The theory of random matrices originates from the study of energy levels of heavy nuclei. It was also proved to be efficient in statistics, combinatorics and the study of growth processes. Recent developments of random matrix theory become intimately connected with the theory of integrable systems.

By the moment of the organisation of the ENIGMA network, European scientists already played an important role in these fields, even though they were scattered all over Europe. However, they were usually established on an individual basis and did not belong to any formal European network.

One of the visible results of the ENIGMA network activities was in joining the efforts of European researchers actively involved in different aspects of the theory of integrable systems and random matrices. The ENIGMA network was estimated as being successful in joining their work by stimulating the exchange of ideas, ranging from theory to applications, and in improving cross-fertilisation of researchers and students across boundaries. In this respect, we noticed that all of the 17 early-stage ENIGMA hired researchers successfully completed their PhD projects, or were on the verge of doing that by the time of the project completion.

The scientific topics addressed by ENIGMA researchers were highly related, and, at the same time, had an impact on many different areas of geometry, combinatorics, probability, statistics and physics. We also started to transform the developed mathematical techniques into working tools for applied mathematics. The significant boost to this domain of European research certainly contributed to the spread of these new ideas in a large number of related fields. We were therefore confident that ENIGMA provided the European scientific community with the necessary impetus to improve communications between researchers, not only in different countries, but also in different areas of science.

The discovery of integrable behaviour in classical and quantum physical systems with infinite degrees of freedom is one of the most exciting developments of mathematical physics in the second half of the 20th century. Such systems are present in fluid dynamics, classical and quantum field theory, statistical physics and optical communications. Moreover, algebraic and differential geometry plays a prominent role in the mathematical foundations of the theory of integrability.

The theory of random matrices originates from the study of energy levels of heavy nuclei. It was also proved to be efficient in statistics, combinatorics and the study of growth processes. Recent developments of random matrix theory become intimately connected with the theory of integrable systems.

By the moment of the organisation of the ENIGMA network, European scientists already played an important role in these fields, even though they were scattered all over Europe. However, they were usually established on an individual basis and did not belong to any formal European network.

One of the visible results of the ENIGMA network activities was in joining the efforts of European researchers actively involved in different aspects of the theory of integrable systems and random matrices. The ENIGMA network was estimated as being successful in joining their work by stimulating the exchange of ideas, ranging from theory to applications, and in improving cross-fertilisation of researchers and students across boundaries. In this respect, we noticed that all of the 17 early-stage ENIGMA hired researchers successfully completed their PhD projects, or were on the verge of doing that by the time of the project completion.

The scientific topics addressed by ENIGMA researchers were highly related, and, at the same time, had an impact on many different areas of geometry, combinatorics, probability, statistics and physics. We also started to transform the developed mathematical techniques into working tools for applied mathematics. The significant boost to this domain of European research certainly contributed to the spread of these new ideas in a large number of related fields. We were therefore confident that ENIGMA provided the European scientific community with the necessary impetus to improve communications between researchers, not only in different countries, but also in different areas of science.