How to deduce shape from shadow? How to design a 3-sided fair coin? How to create printing protocols to print a face? These are all inverse problems reversing the typical formulation of questions, making them ill-posed in the absence of additional information, such as symmetry, smoothness, robustness to noise in data, adherence to physical law, etc. Our interests in this area are pragmatic and practical, and tightly linked to science and technology.
Our all-too-human drive to optimize suggests many mathematical questions associated with the development of general-purpose tools to solve problems of design, control and optimization, and their deployment on questions of engineering and scientific relevance. Our work in this area is primarily in applications of deterministic and stochastic control theory, and their cousins such as reinforcement learning to a variety of practical questions in physical and biological settings.
Art and artists have inspired mathematics and science for aeons, usually by prefiguring many ideas and artifacts at an intuitive level long before we realized that there was even a scientific question to which art had already provided an answer. A natural challenge is to ask if mathematics can help quantify aspects of the underlying principles behind art forms? And can engineering help realize and up-scale them, combining aesthetics with function?
Our interests in data analysis are experimental and exploratory with a particular focus on geometrical and topological aspects. One aim is to quantify the variability in the patterns of shape and connectivity in physical and biological systems. In particular, we aim to link statistical approaches with generative mechanisms using low-dimensional representations.