In my talk I will review some recent results, obtained in collaboration with Pierangelo Marcati, Lars Eric Hientzsch and Hao Zheng, on the mathematical analysis of a class of quantum fluid models. They are compressible, inviscid fluid dynamical systems describing physical phenomena such as Bose-Einstein or superfluidity. In general they arise when quantum mechanical effects are evident even at a macroscopic scale.

The quantum hydrodynamics (QHD) system, which is the prototypical system within this class, may be formally derived from the nonlinear Schroedinger (NLS) equation by means of the so-called Madelung transformations, expressing the complex-valued wave function in terms of its amplitude and phase. This description encounters major mathematical problems in the vacuum region, namely where the particle density vanishes and the phase cannot be well-defined. The vacuum problem is of particular physical relevance as quantized vortices, which are one of the main features of quantum fluid models, are located precisely in the vacuum region. Thus, developing a consistent mathematical theory for arbitrary solutions to QHD allowing vacuum is necessary in order to provide a robust framework to study those coherent phenomena.

I will present a polar factorization method that, given an arbitrary (finite energy) wave function, uniquely defines a hydrodynamic state and is stable in the space of energy. By exploiting those properties, it is then possible to show the existence of finite energy weak solutions to the QHD system, by establishing a correspondence with strong H^1 solutions to some NLS equation.

Conversely, it is not clear whether in general arbitrary hydrodynamic states may determine an associated wave function. I will discuss some framework where such wave function lifting is available, by also presenting some related instabilities. This approach will allow us to establish a more general existence result for finite energy weak solutions to QHD systems. More importantly, in the one dimensional case, it will also enable us to determine a class of weak solutions for which a (sequential) stability result is available.

I will conclude my talk by mentioning some recent developments of this theory, especially in the direction of quantum vortex dynamics.

I will outline some recent work on the rigidity of actions of higher-rank lattices. The main result discussed will be the following: for n≥3, an action of a lattice in SL(n,R) on a manifold of dimension at most n−2 factors through a finite action. I’ll review some classical rigidity results for linear representations that motivate such a result and outline some tools used to establish our results.

Nematic liquid crystals are ubiquitous examples of partially orde red materials that combine the directionality of solids with fluidity. Nematics are anisotropic materials with distinguished material directions, referred to as "nematic directors". Nematics have long been the working material of choice for the flourishing liquid crystal display industry, because nematics have anisotropic optical and electromagnetic responses. Nematics are now being used for innovative applications in sensors, actuators, diagnostic devices and nano-scale technologies. In this talk, we review the powerful continuum Landau-de Gennes theory for nematic liquid crystals. We review the essential mathematical theories and present some case studies on the mathematical modelling of nematic-based multistable liquid crystal devices and compare theory with experimental results. In particular, we present some recent exciting results on solution landscapes for nematic liquid crystals in confined geometries, show how stable and unstable solutions are connected to each other and how this connectivity steers the dynamics of a system and the selection of a stable solution for multistable systems. All collaborations will be acknowledged throughout the talk.