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 all-too-human drive to optimize suggests many mathematical questions associated with design, control and optimization, and their deployment in engineering and scientific problems. Our work in this area is primarily in applications of deterministic and stochastic control theory, and their cousins such as reinforcement learning to a range of questions such as shape and motion control, navigation, drug dosage etc.
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 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.
Our interests in cognition and ethology are a natural extension of studying the everyday world, but moving from “how it works” to “how it behaves” and “how it learns.” At the individual level, the questions range from material examples such as the geometry, dynamics and planning of the ball throw, to ethereal examples such as the perceptual psychophysics of space. At the collective level, the questions include the functional dynamics of cell aggregates, and the behavior of super-organisms exemplified by social insects.
Morphogenesis, concerned with the origins of form in biology, is an old question. Shape arises because cells change in number, size, shape, and position; understanding how this actually happens in space and time, and how it is regulated is a natural goal. Our focus has been on the larger scale questions of self-organized cellular and tissue shape, in asking how we should describe it, how we may predict it, and finally, how we may control it.
One of the pleasures of physics at the everyday scale is the challenge that every observation brings to the table. There are many such curiosities … Similarly, the everyday biological world is full of wonders .. triggered by a walk in the garden, along a river, or watching David Attenborough!
Soft materials have a number of emergent properties that arise from the combination of geometry and softness. Our work in this area has focused on understanding the role of geometry and disorder in soft matter systems, explored by solving a variety of problems across scales ranging from the microscopic to the geologic, using experiments and theory to guide each other.
Characterizing the shape of a solid is inherently a geometrical problem in that one is interested in defining the distances, angles (and changes therein) of material points near and far. How these change in response to external forces and internal activity leads to a rich experimental phenomenology seen in simple tabletop experiments combined with the host of mathematical questions…
How can one not be fascinated by the flow of fluids that are in us and around us, at every imaginable scale. Our work in this area is deliberately unfocused and driven by everyday observations and random ruminations – the rolling of a drop of mercury, the aggregation of cheerios on milk, the swimming of fish and the flutter of a flag, the viscous catenary, the origin of controlled gliding, …