My research interests focus on the geometry and topology of soft materials, in particular the effects of nonlinear elasticity on emergent structural and mechanical properties in complex systems. This encompasses a broad class of systems in several fields, from soft condensed matter physics to materials science to mechanical and biomedical engineering, with problems including programmable matter, pattern formation and elastic instabilities and the structure of membranes and interfaces.
Biomimetic 4D Printing and Programmable Matter
The nascent technique of 4D printing has the potential to revolutionize manufacturing in fields ranging from organsonachip to architecture to soft robotics. By expanding the pallet of 3D printable materials to include the use stimuli responsive inks, 4D printing promises precise control over patterned shape transformations. My collaboration with the Lewis Lab turned a unique material and printing method – which enable the direct writing of local anisotropy into both the elastic moduli and the swelling response of the ink – into a predictive manufacturing technique for arbitrarily complex, morphable structures. By implementing a fully anisotropic model of thin film mechanics, not only can we predict the final transformed geometry from a given design, but my model generates the specific pattern of anisotropies needed to produce a target structure. 
Wrinkles in thin film diblock copolymers

The ability to create any curved surface merely by imprinting its metric into a flat sheet has great appeal for use in materials, manufacturing, and diagnostics applications. Freestanding smectic membranes provide new insight into the coupling between inplane topological defects and surface curvature. The molecular splay distortions associated with disclinations act as sources of Gaussian curvature. Yet remarkably, positive disclinations are sources of negative Gaussian curvature and vice versa. The elastic anisotropy coming from the modulated molecular composition results in a pattern of wrinkles in the membrane which form perpendicular to the underlying smectic layers. The wavelength of the wrinkles is dictated by the interplay of the elastic constants. Thus, by dictating the distribution of topological defects, it should possible to control the specific nonEuclidean geometry of the membrane. 
Elastic Instabilities and LongRanged Pattern Formation
Under mechanical compression – both uniaxial and hydrostatic – circular holes in thin elastic sheets undergo a buckling instability and snap shut, generating patterns with longranged orientational order. Of particular experimental interest, the diamond plate pattern generated by a square lattice of holes achieves longranged order due to the broken symmetry of the underlying lattice. Despite the extreme nonlinearities in this system, a linear theory modeling each of the collapsed holes as a pair of dislocations captures the interactions between holes and accurately predicts the pattern transformation they undergo. The limited number of two dimensional lattices restricts the classes of patterns that can be produced by a flat sheet. By changing the topology of the membrane to a cylinder, the types of accessible patterns vastly increases. The key lies in the periodicity of the cylinder not the curvature. Because the interaction between two holes is longranged, simple implementation of periodic boundary conditions does not suffice, and the interaction on a cylinder becomes an infinite sum of terms. Varying the relative orientation of the periodic direction of the cylinder and the principal lattice vectors allows for the creation of a host of new patters. The periodicity of the cylinder enables the continuous transfer of the pattern to a flat surface of any size. Because cylinders are isometric to the plane, Gaussian curvature does not hinder the ability to transfer patterns onto a flat substrate. Using an ink containing, for instance, nanoparticles, quantum dots, or polymers, adds functionality to the patterned surface. 
Homotopy Theory of Nematic Defects
Topological defects pervade a wide range of condensed matter systems. The behavior and inter actions of such singularities impart many materials with a wealth of rich behavior. The set of all topological defects in a system determines its singularities and thus dominates the behavior of sys tems, making even nonlinear theories tractable. Due to the ease of their experimental accessibility and the robustness of their theoretical frameworks in field, Landau, and homotopy theories, liquid crystals prove an archetypal system to study problems of interest to the condensed matter theory community as a whole: how the interactions of topological defects and the means by which they couple to fields determine or constrain the mechanics and dynamics of the underlying system. The advent of experimentally realized knotted disclination lines in nematic and cholesteric liquid crystals precipitated the need to characterize the topological charge of a collection of point defects and disclination loops. It is a common misconception to treat topological charges like static electrical charges, where like charges repel and opposites attract and can annihilate each other. It turns out to be a coincidence that this method works for spins in the Heisenberg model. Nematic liquid crystals are described by a unit line field or equivalently by a unit vector field n where n and −n are identified. The ambiguity between n and −n results in inconsistencies in the measurements of point defects. Homotopy theory is the natural language for describing topological defects in ordered matter. The goal of this approach is to generate a set of invariants that innumerate and classify all possible types of topological defects in a system (not merely isolated defects), up to smooth deformations. In field theories, the ordered phase can be distinguished from a higher symmetry phase by an order parameter field, which is the product of a scalar field, nonzero values of which indicate the presence of the ordered phase, and a field that encodes the local configuration. Here, we consider an order parameter that varies continuously except at isolated singular points or lines. The space of all allowed values of the order parameter field is known as the groundstate manifold (GSM). The local configuration at any point in real space corresponds to a point in the GSM. 
Chiral Smectics and the Helical Nanofilament Phase
Smectic liquid crystals are a one dimensional crystal of two dimensional fluid layers. In the Landaude Gennes free energy, chirality in the smectic acts as a frustrating boundary term analogous to the presence of an external magnetic field in the LandauGinzburg theory of superconductivity. Type II variants of both materials admit a mixed phase. Flux vortices in the Abrikosov phase allow magnetic field to penetrate the superconductor. Likewise the twistgrainboundary (TGB) phase utilizes screw dislocations to admit chirality into a system of equally spaced layers. Both systems exhibit thermodynamic behavior, yet the discovery of the helical nanofilament (HN) phase brought this analogy into question. The HN phase is a complex hierarchical texture characterized by coherently rotating helical bundles (57 layers thick) arranged into a hexagonal lattice. Unlike the TGB phase, where grain boundaries admit chirality by allowing regions of smectic A to slowly rotate, the chirality in the HN phase is expressed along the centers of the bundles. The HN phase is a new solution to the Landaude Gennes free energy. This solution attempts to grow a smectic in a highly chiral cholesteric background texture, n = {cos(q₀z),sin(q₀z),0}, where q₀=2π/pitch. A single bundle of fixed radius R depending on the elastic constants of the material has lower free energy than an ideal smectic or cholesteric occupying the same volume. The problem now consists of packing the helical bundles into a bulk texture. The phase field that describes a bulk texture is given by Φ = Re[Θ(w)e^{iq₀z}], where Θ(w) is an analytic function of w = x + iy. Simple zeros of Θ(w) give the location of bundles. Such a phase field has two pleasing qualities: it guarantees coherently rotating bundles, and it penalizes bundles of the opposite handedness as the underlying cholesteric. Although the HN phase has no known superconducting analog, an equivalent theory has been recently shown to describe the A phase of helimagnets, comprised of a hexagonal lattice of skyrmions. 
Riemann's Minimal Surface and Sums of Helicoids
In order to minimize bending energy, smectic layers assume the morphology of minimal surfaces. This method is particularly useful in situations in which boundary conditions force defects or where defects are prescribed. Given a specific smectic topology, a minimal surface with the same topology will have the same compression divergences. Previous examples include the helicoid as a model for a screw dislocation and Scherk’s first surface as a model for a single twist grain boundary. Here we extend this formalism to study a very different smectic texture – a series of pores. Riemann’s minimal surface is composed of a nonlinear sum of two oppositely handed screw dislocations and has the morphology of a pore, as shown in Fig. 5. Because it has an infinite number of planar ends, Riemann’s minimal surface describes a smectic far away from the pores. Previous studies have only been able to show either the energetics of a single dislocation or an infinite lattice of screw dislocations. Thus, Riemann’s minimal surface gives insight into the interaction between two screw dislocations while maintaining flat, equally spaced layers at the boundary. A single free parameter varies the size of the neck of each pore and the separation between neighboring pores. This allows for a rigorous study of the energetics and dynamics of a pathway by which pores form in a smectic or a relaxation mechanism by which two regions of smectic A, joined by pores, rejoin to a single domain. 
Focal Conic Domains and Equally Spaced Smectics
Finally, we turn to the ubiquitous smectic texture – the focal conic domain. While the exact locations of the layers within a single focal conic are known, joining several such regions together to form a bulk texture without introducing additional focal lines is an extremely daunting task. In order to gain insight into equally spaced smectic tex tures, we developed a novel approach to the study of focal sets in smectics which exploits a hidden Poincaré symmetry revealed only by viewing the smectic layers as projections from onehigher dimension. Consider a null hypersurface N in d + 1−dimensional Minkowski space R^{d,1}. Projections of N taken at times t_{n }= t_{0 }+ n dt, for integer values of n, with constant dt, creates a family of equally spaced surfaces in ddimensional Euclidean space, E^{d}. 