Series of Thematic Mini-Courses

This intensive course provides an introduction to the mathematical theory and applications of parabolic partial differential equations, with a particular focus on reaction--diffusion systems arising in the life sciences. The course begins with a review of the fundamental concepts of partial differential equations, including classification, well-posedness, and the specific features of parabolic problems. Particular attention is devoted to reaction--diffusion models, incorporating nonlinear effects such as advection and chemotaxis, which play a central role in the modelling of biological and ecological processes. The second part of the course focuses on the qualitative analysis of solutions. We discuss the stability of homogeneous equilibria and introduce the mechanism of Turing instability as a fundamental tool for understanding pattern formation. A weakly nonlinear analysis is then presented to describe the emergence and selection of spatial structures near critical thresholds. Throughout the course, theoretical results are complemented by applications to relevant problems in life sciences, including population dynamics and transport phenomena in biological systems. The aim of the course is to provide participants with both a rigorous theoretical framework and practical analytical tools for the study of nonlinear phenomena in parabolic PDEs.

This mini-course offers a mathematical introduction to the Ericksen–Leslie theory and its extension to active nematics. In the first part, we study the fluid dynamics of nematic liquid crystals, and we derive the Ericksen–Leslie model within the framework of rational continuum mechanics. In the second part, we extend the passive theory to describe the dynamics of active nematic gels, in which energy is continuously injected at the microscopic scale through an additional active term. In particular, we focus on the onset of instabilities in active nematic systems. We analyse the mathematical structure of the resulting bifurcations by means of the Lyapunov–Schmidt method and present the main ideas of this reduction technique in a form suitable for applications to active matter.

Nematic liquid crystals are aggregates of rodlike molecules that tend to align parallel to each other along a given direction. Due to their easy response to externally applied electric, magnetic, optical and surface fields, liquid crystals are of greatest potential for scientific and technological applications. In the last few decades, there has been an increasing interest in soft matter physics on small spherical colloidal particles or droplets coated with a thin layer of nematic liquid crystal. The hope is to build mesoatoms with controllable valence. These coating layers are referred to as nematic shells. When nematic liquid crystals are constrained to a curved surface, the geometry induces a distortion in the molecular orientation. The possibility to have an in-plane order rather than a spatial distribution of the molecules depends on the shell thickness. In ultra-thin shells, the interaction with the colloid surface enforces a sort of degenerate anchoring, i.e., the tendency of the molecules to align parallel to the surface. Thus, unavoidable defects arise when nematic order is established on a surface with the same topology as that of the sphere. Most theoretical studies on nematic liquid crystals are framed within the classical director theory. In this setting, the local properties of the nematic liquid crystals are described through a unit vector, the director, parallel to the local average molecular direction.  However, the director description of a nematic configuration misses a relevant information at the mesoscopic level: the dispersion of the molecules around the average molecular orientation. The order-tensor theory, developed by the Nobel laureate Pierre-Gilles de Gennes, overcomes this gap by introducing a richer kinematic description. Within the framework of both the order-vector and order-tensor theories, these lectures aim to derive models for the free energy density of nematic shells from well-established three-dimensional theories for liquid crystals, to introduce suitable dissipation potentials and to develop a rigorous variational approach for studying the statics and dynamics of nematic shells.