Projects

Differential geometry is a branch of mathematics that uses techniques from analysis and algebra to study geometric structures and forms. Outside of mathematics, it has historically been used with dramatic success as a language in the formulation of fundamental theories in physics. More recently, it has grown to play a fundamental role in applied mathematics and engineering including statistics, numerical integration, robotics, computer vision, medical imaging, shape analysis, and machine learning. A key reason for this development is the prevalence of symmetries, nonlinear spaces, and non-Euclidean data in all of these applications.

The aim of this project is to advance research in optimization on manifolds and develop innovative statistical tools and algorithms for applications involving data in non-Euclidean spaces, with a particular focus on spaces of symmetric positive definite (SPD) matrices and associated non-Euclidean geometries, which feature prominently across engineering applications. The project aims include fundamental theoretical contributions such as the rigorous development of new algorithms, as well as concrete applications of the techniques to real-world data including multivariate-time series and imaging data in close coordination with leading practitioners and experimental colleagues in neuroscience and robotics.

Complex shape transitions driven by differential growth patterns are ubiquitous in nature and are observed at a variety of length scales, ranging from the cell walls of plants and bacteria to macrostructures such as plant leaves. Modern responsive materials can be preprogrammed to undergo spatially inhomogeneous expansions and contractions in response to external stimuli such as heat and light. Examples include thermally responsive hydrogels that can be designed to undergo inhomogeneous isotropic deformations and nematic liquid crystal elastomers (LCE) that experience anisotropic deformations dictated by a nematic director field whose spatial variation can be exploited to induce intricate global shape changes in response to stimuli including light, heat, and pH. Shape-programming in these materials has been the subject of intense research in recent years, including various attempts to address the inverse design problem of discovering and practically encoding the pattern of local deformations in a flat 2D sheet that yields a given target geometry upon actuation. The project aims to use mathematical and computational models, mechanical principles, and experiments to build upon these advances in several important ways:

1. The utilization of multiple designable features to simultaneously encode multiple target geometries for specific tasks in soft robotics.

2. The programming of controllable shape transitions between curved geometries and geometric boundary value problems relevant for the integration of soft robotic components in traditional robotic designs.

3. The optimal design of the trajectory of the evolving geometry in shape space.

4. The successful integration of sensory capabilities in shape-morphing soft robots and machines.