Summer School

Summer School "Mathematics of Liquid Crystals"

Geometry, topology, and analysis of liquid crystal models

June 15-18, 2026Aula multimediale, Department of Structures for Engineering and Architecture - Naples, Italy
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Overview

Three main lecture series (8 hours each) and two invited junior one-hour seminars.

Includes focused mini-courses on nematic shells and active liquid crystal systems.

You can reach us at: naplespir2026.liqcryst@gmail.com

Registration is now open

Deadline: May 25, 2026

Please fill out the following registration form for the Summer School "Mathematics of Liquid Crystals"

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Speakers

Gareth Alexander
Department of Physics, University of Warwick
Geometric Topology in Liquid Crystals
Paolo Biscari
Dipartimento di Fisica, Politecnico di Milano
Relaxation Dynamics and Elastic Coupling in Nematic Liquid Crystals
Apala Majumdar
FRSE, FIMA, University of Manchester
Continuum Theories of Liquid Crystals and their Applications
Federico Luigi Dipasquale
Università degli Studi di Napoli Federico II (Invited junior researcher)
Ferronematics: asymptotics for critical points
Silvia Paparini
Università di Padova (Invited junior researcher)
Topological Defects as Morphogenetic Factors

Timetable (TBA)

Abstract

Gareth Alexander
Department of Physics, University of Warwick
Geometric Topology in Liquid Crystals
Liquid crystals display an enormous variety of topological textures, typically imbued with strong geometric characteristics. These include a wide range of point and line defects or singularities, non-singular topological textures such as skyrmions and hopfions, and other solitons, particularly in chiral materials such as cholesterics. I will give a self-contained introduction to modern topics in the geometric topology of liquid crystal textures. I will aim to discuss the geometric decomposition of director gradients, topological defects, textures including skyrmions and hopfions, and geometric methods in chiral materials. I will also cover both mathematical methods for characterising topology and practical methods for constructing examples in theory.
Paolo Biscari
Dipartimento di Fisica, Politecnico di Milano
Relaxation Dynamics and Elastic Coupling in Nematic Liquid Crystals
Within the Oseen–Frank framework, nematic liquid crystals are continuum media characterised by orientational order described by a unit director field. Equilibrium configurations arise from the minimisation of anisotropic elastic energies under geometric and topological constraints. Many physically relevant processes, however, are intrinsically dynamical and governed by dissipation. This lecture course develops continuum models for the relaxation dynamics of nematic liquid crystals in the director setting. Starting from energetic and dissipative principles, we derive evolution equations as constrained gradient flows of the Oseen–Frank energy and discuss their mathematical structure. Particular attention is given to the motion and interaction of defects as emergent features of the relaxation dynamics.
The second part of the course introduces nematic elastomers, where orientational order is coupled to mechanical deformation. We show how elastic strains and director reorientation interact within a variational framework, leading to spontaneous distortions, soft modes, and complex relaxation phenomena. Selected numerical examples will be presented to illustrate relaxation processes and coupled elastic effects.
Apala Majumdar
FRSE, FIMA, University of Manchester
Continuum Theories of Liquid Crystals and their Applications
Liquid crystals are paradigm examples of partially ordered materials that combine fluidity with the order of solids. Liquid crystal phases typically have distinguished material directions and this intrinsic directionality naturally lends itself to multiple applications across the display industry, energy applications, built-in environments and healthcare technologies. There are multiple liquid crystal phases - nematic phases, smectic phases and cholesteric phases and many more, with varying degrees of orientational and positional ordering. In this lecture course, we review the celebrated Landau-de Gennes theory for nematic liquid crystals and its generalisations to smectic and cholesteric phases. This continuum theory is the building block of many mathematical theories for liquid crystal phases. We also discuss the applications of these continuum Landau-de Gennes type theories to planar systems, three-dimensional cuboids and droplets, to model experimentally observable liquid crystal equilibria.
Federico Luigi Dipasquale
Università degli Studi di Napoli Federico II (Invited junior researcher)
Ferronematics: asymptotics for critical points
We consider a variational model for ferronematics --- composite materials formed by dispersing magnetic nanoparticles into a liquid crystal matrix. The model features two coupled order parameters: a Landau-de Gennes Q-tensor for the liquid crystal component and a magnetisation vector field M, both of them governed by a Ginzburg-Landau-type energy. The energy includes a singular coupling term favouring alignment between Q and M. We report on some recent results on the asymptotic behaviour of (not necessarily minimising) critical points as a small parameter \eps tends to zero. Our main results show that the energy concentrates along distinct singular sets: the (rescaled) energy density for the Q-component concentrates, to leading order, on a finite number of singular points, while the energy density for the M-component concentrates along a one-dimensional rectifiable set. Moreover, we will see that the curvature of the singular set for the M-component (technically, the first variation of the associated varifold) is concentrated on a finite number of points, i.e. the singular set for the Q-component.
Silvia Paparini
Università di Padova (Invited junior researcher)
Topological Defects as Morphogenetic Factors
In biological tissues, elongated cells often exhibit nematic orientational order described by a macroscopic director field. Topological defects, singularities in this field, act as morphogenetic organizing centers, mediating stress and driving shape formation. This is exemplified in Hydra, where these defects align with morphological features (e.g., a +1 defect at the tentacle apex and two -1/2 defects at its base).
This talk establishes a mathematical framework to model the coupling between nematic disclinations and polymer networks. Employing a variational approach, we describe out-of-plane deformations in initially flat nematic polymer network sheets. The equilibrium morphology is determined by minimizing a two-part free energy functional: a nematic elastic energy penalizing director field distortions, and a strain energy characterizing mechanical stiffness.
Building on this framework, we introduce a continuum mechanical model incorporating the theory of solid relaxation and growth to capture the quasi-static evolution of a non-equilibrium system: a folded spheroid with localized topological defects representing regenerating Hydra tissue. We aim to understand how these defects drive the formation of morphological features.