Summer School "Mathematics of Liquid Crystals"

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.