In modern geometry, the notion of “shape” has expanded far beyond familiar visible objects, allowing us to regard a wide variety of mathematical objects as geometric forms. Moreover, by combining methods from analysis, algebra, and other areas of mathematics, we have developed many powerful theories and approaches for studying such shapes. Our laboratory focuses particularly on curves, surfaces, and their higher-dimensional generalizations, known as submanifolds. We study these objects using mathematical techniques such as Lie theory and geometric analysis. We are also interested in discrete analogues of these geometric objects and theories, with the aim of developing applications to other areas.