09:15 to 09:50 Registration 09:50 to 10:00 Introduction 10:00 to 11:00 D Kinderlehrer (Carnegie Mellon University)Remarks about the Janossy effect Light can change the orientation of a liquid crystal. This is the optical Freedericksz transition, discovered by Saupe. In the Janossy effect, the threshold intensity of the Freedericksz transition is dramatically resuced by the addition of a small amount of dye to the sample. We investigate the theory for this effect derived by E, Kosa, and Palffy-Muhoray. Several themes come together, including molecular motors and Monge-Kantorovich mass transport. This is joint work with Michal Kowalczyk. 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Analysis of Disclination-Line Defects in Liquid Crytals We describe mathematical results and techniques of analysis on the structure of defects in thin nematic liquid crystals described by minimizers of the Landau-de Gennes energy involving a tensor-valued order parameter with Dirichlet boundary conditions of nonzero degree. We prove that as the coefficient of the elasticity term tends to zero, a limiting uniaxial texture forms with a finite number of defects, all of degree 1/2 or -1/2. We also describe the location of defects and the limiting energy. 12:30 to 14:00 Lunch 14:00 to 15:00 Modeling the Dynamics of Liquid Crystalline Systems A key characteristic of liquid crystals, exploited by many device applications, is their symmetry mandated responsivity. Describing the dynamics of the response of liquid crystalline systems to excitations is therefore of considerable interest and importance. Since orientational order is usually characterized in terms of an order parameter, the dynamic response of liquid crystals is often described in terms of the time evolution of the order parameter. A more general description, which gives more information yet is often simpler that the traditional approach, is the time evolution of the generalized density function. We describe a general procedure to obtain such a description, discuss its implementation and give several illustrative examples. 15:00 to 15:30 Afternoon Tea 15:30 to 16:00 Eigenvalue Constraints and Regularity of Q-tensor Navier-Stokes Dynamics If the Q-tensor order parameter is interpreted as a normalised matrix of second moments of a probability measure on the unit sphere, its eigenvalues are bounded below by -1/3 and above by 2/3. This constraint raises questions regarding the physical predictions of theories which employ the Q-tensor; it also raises analytical issues in both static and dynamic Q-tensor theories of nematic liquid crystals. John Ball and Apala Majumdar recently constructed a singular map on traceless, symmetric matrices that penalises unphysical Q-tensors by giving them an infinite energy cost. In this talk, I shall discuss some mathematical results for a modified Beris-Edwards model of nematic dynamics into which this map is built, including the existence, regularity and so-called strict physicality' of its weak solutions. 16:00 to 16:30 Field instabilities of Smectic A liquid crystals in 2D and 3D We study the de Gennes free energy to describe the undulations instability in smectic A liquid crystals subjected to magnetic fields. If a magnetic field is applied in the direction parallel to the smectic layers, an instability occurs above a threshold magnetic field. When the magnetic field reaches this critical threshold, periodic layer undulations are observed. We prove the existence and stability of the solution to the nonlinear system of de Gennes model using $\Gamma$-convergence method and bifurcation theory. Numerical simulations will be given near and well above the threshold. An efficient numerical scheme for some free energy containing the second order gradient will be presented. Undulation instabilities on three dimensional systems will be also discussed. Co-author: Carlos J. Garcia-Cervera (UCSB) 16:30 to 17:00 Discussion chaired by F-H Lin 17:00 to 18:00 Drinks Reception
 09:00 to 10:00 P Zhang (Peking University)The Small Deborah Number Limit of the Doi-Onsager Equation to the Ericksen-Leslie Equation We present a rigorous derivation of the Ericksen-Leslie equation starting from the Doi-Onsager equation. As in the fluid dynamic limit of the Boltzmann equation, we first make the Hilbert expansion for the solution of the Doi-Onsager equation. The existence of the Hilbert expansion is connected to an open question whether the energy of the Ericksen-Leslie equation is dissipated. We show that the energy is dissipated for the Ericksen-Leslie equation derived from the Doi-Onsager equation. The most difficult step is to prove a uniform bound for the remainder in the Hilbert expansion. This question is connected to the spectral stability of the linearized Doi-Onsager operator around a critical point. By introducing two important auxiliary operators, the detailed spectral information is obtained for the linearized operator around all critical points. However, these are not enough to justify the small Deborah number limit for the inhomogeneous Doi-Onsager equation, since the elastic stress in the velocity equation is also strongly singular. For this, we need to establish a precise lower bound for a bilinear form associated with the linearized operator. In the bilinear form, the interactions between the part inside the kernel and the part outside the kernel of the linearized operator are very complicated. We find a coordinate transform and introduce a five dimensional space called the Maier-Saupe space such that the interactions between two parts can been seen explicitly by a delicate argument of completing the square. However, the lower bound is very weak for the part inside the Maier-Saupe space. In order to apply them to the error estimates, we have to analyze the structure of the singular terms and introduce a suitable energy functional. Furthermore, we prove the local well-posedness of the Ericksen-Leslie system, and the global well-posednss for small initial data under the physical constrain condition on the Leslie coefficients, which ensures that the energy of the system is dissipated. Instead of the Ginzburg-Landau approximation, we construct an approximate system with the dissipated energy based on a new formulation of the system. 10:00 to 11:00 Tangent unit-vector fields: nonabelian homotopy invariants, the Dirichlet energy and their applications in liquid crystal devices We compute the infimum Dirichlet energy, E(H), of unit-vector fields defined on an octant of the unit sphere, subject to tangent boundary conditions on the octant edges and of arbitrary homotopy type denoted by H. The expression for E(H) involves a new topological invariant – the spelling length – associated with the (nonabelian) fundamental group of the n-times punctured two-sphere. These results are then used for the modelling of the Post Aligned Bistable Nematic (PABN) device, designed by Hewlett Packard Laboratories. We provide analytic approximations for the experimentally observed stable equilibria in the PABN device and propose novel topological and geometrical mechanisms for bistability or multistability in prototype liquid crystal device geometries. This is joint work with Jonathan Robbins, Maxim Zyskin and Chris Newton. 11:00 to 11:30 Morning Coffee 11:30 to 12:30 On stability of radial hedgehog in Landau - de Gennes model We investigate stability of radially symmetric solutions in the context of Landau - de Gennes theory. It is well known that radial hedgehog is an unstable solution for low enough temperatures. We show that radial hedgehog is locally stable solution for temperatures close to isotropic-nematic phase transition temperature. 12:30 to 14:00 Lunch 14:00 to 15:00 AD Zarnescu (University of Sussex)Eigenframe discontinuities, commutators and nematic defects In the framework of De Gennes' Q-tensor theory of nematics one can interpret defects as eigenframe discontinuities. A significant analytical difficulty related to understanding these discontinuities is due to the rather complicated relation between the multiparameter dependent matrices and their parametrized eigenvectors. We present necessary and sufficient criteria for determining eigenframe discontinuities (criteria expressed in terms of suitable commutators) as well as some consequences. This is joint work with Jonathan Robbins and Valeriy Slastikov. 15:00 to 15:30 Afternoon Tea 15:30 to 16:00 Wellposedness of a Coupled Navier-Stokes/Q-tensor System In this work, we show the existence and uniqueness of local strong solution for a coupled Navier-Stokes/Q-tensor system on a bounded domain $\Omega\subset\mathbb{R}^3$ with Dirichlet boundary condition. One of the novelties brought in with respect to the existing literature consists in the fact that we deal with Navier-Stokes equation with variable viscosity. Concerning the methodology, we use an approximation method to handle the linearized system and the existence of solution to the nonlinear system is proved via a Banach's fixed point argument, based on the estimates on the lower order terms. 16:00 to 16:30 Radial symmetry and biaxiality in nematic liquid crystals We study the model problem of a nematic liquid crystal confined to a spherical droplet subject to radial anchoring conditions, in the context of the Landau-de Gennes continuum theory. Based on the recent radial symmetry result by Millot & Pisante (J. Eur. Math. Soc. 2010) and Pisante (J. Funct. Anal. 2011) for the vector-valued Ginzburg-Landau equations in three-dimensional superconductivity theory, we prove that global Landau-de Gennes minimizers in the class of uniaxial Q-tensors converge, in the low-temperature limit, to the radial-hegdehog solution of the tensor-valued Ginzburg-Landau equations. Combining this with the result by Majumdar (Eur. J. App. Math. 2012) and by Gartland & Mkaddem (Phys. Rev. E. 1999) that the radial-hedgehog equilibrium is unstable under biaxial perturbations, we obtain the non-purely uniaxial character of global minimizers for sufficiently low temperatures. 16:30 to 17:00 Discussion chaired by OD Lavrentovich
 09:00 to 10:00 Mechanical response and microstructures in liquid crystal elastomers: small vs large strain theories In this talk, we will present recent theoretical and numerical results for models of nematic elastomers within the small strain approach. While strains exhibited by nematic elastomers are usually large, there are cases where this is not so, and the early modeling approaches were using this framework. In fact, the main reason for the developing small strain theories for nematic elastomers is the clear geometric structure of the resulting energy landscape. We will exploit this structure to discuss material instabilities and stress-strain diagrams, and to suggest possible generalizations to more realistic models. 10:00 to 11:00 Polar Active Liquid Crystals : microscopics, hydrodynamics and rheology Colonies of swimming bacteria, mixtures of cytoskeletal protein filaments and motor proteins, and vibrated granular rods are examples of active systems composed of interacting units that consume energy and collectively generate motion and mechanical stresses. Due to their elongated shape, active particles can exhibit orientational order at high concentration and have been likened to `living liquid crystals". Their rich collective behavior includes nonequilibrium phase transitions and pattern formation on mesoscopic scales. I will describe and summarise recent theoretical results characterising the behaviour of such soft active systems. 11:00 to 11:30 Morning Coffee 11:30 to 12:30 D Phillips (Purdue University)Analysis of defects in minimizers for a planar Frank energy Smectic C* liquid crystal films are modeled with a relaxed Frank energy, \begin{equation*} \int_\Omega\Big( k_s(\text{div}\, u)^2 + k_b(\text{curl}\, u)^2 + \frac{1}{2\epsilon^2}(1 - |u|^2)^2 \Big)\, dx . \end{equation*} Here $k_s$ and $k_b$ represent the two dimensional splay and bend moduli for the film respectively with $k_s, k_b > 0$, $\Omega$ is a planar domain, and $u$ is an $\mathbb{R}^2$-valued vector field with fixed boundary data having degree $d>0$. We study the limiting pattern for a sequence of minimizers $\{u_\epsilon\}$ as $\epsilon\to 0$. We prove that the pattern contains $d$, degree one defects and that it has a either a radial or circular asymptotic form near each defect depending on the relative values of $k_s$ and $k_b$. We further characterize a renormalized energy for the problem and show that it is minimized by the limit. This is joint work with Sean Colbert-Kelly. 12:30 to 12:45 Discussion chaired by J Ball