## Research group

Several research groups operate in the field of Mathematical Physics.

### Research topics

### Mathematical Models for Condensed Matter

- Methods and Models for Applied Sciences (Kinetic and hydrodynamic equations of complex collisional systems);
- Geometric and algebraic methods in field theories and data science.

### Kinetic and hydrodynamic equations of complex collisional systems

*(Giuseppe Toscani, Ada Pulvirenti, Francesco Salvarani)*

The group investigate theoretical and numerical problems related to:

- Kinetic theory of rarefied gases, the kinetic theory of dissipative systems with application to granular gases;
- Asymptotic problems deriving from the transition from kinetic to macroscopic models in hyperbolic and diffusive rescaling;
- Asymptotic problems related to grazing collisions, and passage to Fokker-Planck type equations; asymptotic behaviour of nonlinear diffusion equations by means of entropy methods.

Moreover, applications of kinetic theory are applied to the study of **multi-agent systems**, with particular regard to **socio-economic and biological systems**. In this context, models have been introduced and studied for the distribution of wealth and for the formation of opinion which are to be further developed shortly.

### Mathematical models for soft matter and applications

*( Epifanio G. Virga, Fulvio Bisi, Andrea Pedrini)*

Mathematical models capable of describing **self-ordering** and other cooperative behaviours occurring in systems consisting of **molecules** and **colloidal particles**. The characterization of homogeneous systems is developed through the study of phase formation mechanisms, phase transitions and other critical phenomena and is extended to partially ordered systems that support inhomogeneities, typically in the form of ‘defects’. The length scales of these systems vary from nanometric to macroscopic and therefore the greatest challenge is to build, from first principles, truly consistent and reliable ‘multi-scale’ models.

The mathematical methods used come from both **statistical mechanics** and **continuum mechanics **and the spectrum of phenomenology of the systems analyzed ranges from condensed matter, through **physical chemistry**, up to **engineering** (for example liquid crystals, complex fluids, lipid membranes, spin systems, ferrofluids)

### Geometric and algebraic methods in field theories and data science

*(Annalisa Marzuoli)*

Using **geometric and algebraic methods** (from differential geometry and geometric topology to Lie group theory), various problems related to many-body **Hamiltonian quantum systems** and field theories of geometric nature are studied.

In the study of various types of complex systems, it is appropriate to employ **graph theory**, where the nodes of the network represent elementary units and the sides translate the connections (structural or functional) between units. The study of ‘**big data**‘ is one of the most current interdisciplinary research fields and is practiced using various techniques, among which the numerical methods predominate.

In recent years, methods of algebraic topology and combinatorial geometry have been introduced to characterise ‘global’ properties of data spaces, such as the evaluation of specific topological invariants in the category of simplicial complexes.