General description of the project

The MMNLP project has been renewed for the three years 2024, 2025, 2026. The main changes are in the number of units of the project: alltogether the units of the project for next three years are Lecce, Milano Bicocca, Milano, Roma, Torino and Trieste.

The aim of the expansion is to consolidate the MMNLP project and to make it the reference project in the field of (infinite-dimensional) Integrable Systems in Italy. At the moment, to our knowledge, MMNLP gathers almost all Italian researchers whose interests are in this field.

The list of research subjects that are covered by MMNLP also grew to encompass almost all Italian researchers who are active in the field of infinite-dimensional Integrable Systems.

Research topics of MMNLP Units

Lecce Unit

• Hamiltonian geometry of PDEs
• Spectral problems and graph theory
• Integrable sectors in effective Field Theory
• Integrable systems in nonlinear elasticity

Milano-Bicocca Unit

• Dubrovin-Frobenius manifolds, flat F-manifolds and integrable systems of topological type
• Stratified fluids and integrable and near-to-integrable models
• Poisson quasi-Nijenhuis manifold
• Linear ODEs and quantum integrable systems

Milano Unit

• Solvable models in General Relativity, and other topics
• Stochastic differential equations and symmetries
• Classification of discrete-time integrable systems
• Differential-geometric Poisson structures

Rome Unit

• Algebraic structures related to integrable systems
• Identification of explicitly integrable systems of nonlinear differential equations; Cosmic Origin of Quantization conjecture
• Towards a theory of anomalous (rogue) waves in multidimensions
• Exact solutions of physically relevant models

Torino Unit

• Homogeneous Euler equations
• Motion of interfaces between fluids
• Semigeostrophic equations
• Focusing NLS equation

Trieste Unit

• Symplectic geometry of moduli spaces and character varieties
• Inverse problems and spectral theory
• Integrable systems in statistical Mechanics and Random Matrices
• (Multi)-Orthogonal polynomials and functions
• Nonlinear waves
• Isomonodromic deformations and Frobenius manifold
• Cohomology of moduli spaces of curves, Gromov-Witten theory, integrable hierarchies and topological recursion.

Abstract

 Many fundamental non-linear models in Classical and Quantum Physics (Fluid,
Nuclear, Condensed Matter and Plasma Physics, Optics, Gravity, Statistical, Quantum
Field and String Theories) are mathematically characterized by their integrability. The
Integrable models, described by PDEs, ODEs, or discrete difference equations (DDEs),
have regular stable solutions with respect to large classes of initial data, characteristic
parameters and, possibly, external perturbations. The identification of integrable
systems and the investigation of their properties is a major area of Theoretical and
Mathematical Physics. The MMNLP research group covers the following topics:

1. classification/construction of integrable models by algebraic/geometric methods:
   integrable PDEs in enumerative geometry (cohomological field theory and topological
   recursion), their Hamiltonian description, Poisson vertex algebras, representation
   theory; 
2. constructive methods for exact, asymptotic or approximate solutions of
   initial/boundary value problems of nonlinear PDE/ODE/DDEs;
3. analytic study of extreme nonlinear phenomena, like development of singularities, gradient
   catastrophes, transitions from/to elliptic/hyperbolic regimes, the dispersive/dissipative
   regularizations;
4. algebras of symmetries and conservation laws, generalized symmetry and W algebras, classical/quantum superintegrability, Frobenius algebras, topological field theories, random matrix models, symmetry preserving discretization;
5. textures and waves in complex classical and quantum fluids, hydrodynamical models in
   higher dimensions, shock and rogue waves, nonlinear gravity and general relativity
   regimes, phase transitions in real gases.