Integrability and Complexity are often viewed as antinomic concepts. The notion of Integrability is associated to systems that exhibit some form of regularity and therefore are "simple enough" to be solved analytically. The idea of Complexity arises in a variety of contexts where a system shows features such as chaos, disorder, instability, randomness.
The modern theory of integrable systems, that is the method of the inverse scattering transform (IST), finite gap theory, integrable perturbations and normal forms, demonstrates that it is indeed possible to use analytical techniques to capture and describe complex behaviours in nonlinear and many-body systems.
For example, finite gap solutions of soliton equations can be effectively used to construct accurate asymptotic descriptions of shock waves arising from the dispersive regularisation of gradient catastrophe singularities in hydrodynamic type systems. In a completely different context, it was observed that the partition function of a variety of random matrix models can be obtained from a particular solution of nonlinear integrable dynamical systems, as for instance the Toda chain.
The method of differential identities arises as a unified approach to the solution of a general class of statistical mechanical systems, from classical "simple" systems, which exhibit Ising type phase transitions, to systems where complex behaviours emerge, e.g. in the dispersive regularisation of the order parameters in Hermitian matrix models. Some key universal features of a variety statistical mechanical models, e.g. singularities, critical exponents, scaling effects, are interpreted and described in terms of solutions of nonlinear integrable systems and their normal forms; critical properties, collective behaviours, phase transitions, lack of equilibrium are therefore understood and explained in terms of scaling properties and regularisation mechanisms of singularities of hydrodynamic systems.
Neural Network theory, developed as an attempt to model information processing in the brain, has been rapidly evolving in recent years as a paradigm to construct artificial intelligence devices, understand biological complexity, classify and extract information from images or more general types of unstructured data sets.
Based on the mathematical connection between statistical mechanical properties of networks, the mathematical structure of integrable systems and the phenomenology of nonlinear waves, the project focuses on the development of an iterative approach to the solution of network models and the definition of a alternative protocols for information processing.