Biology offers many beautiful examples of the interconversion of matter, energy and information in non-equilibrium systems. Our interests in biology are driven by simple questions about “how things work” and “why things are” which leads naturally to morphology, physiology, ethology and ecology, couched when possible in an evolutionary framework. We believe that a practical approach requires a comparative study of extremes, using experiments and theory. In morphology and physiology this is naturally manifest in the evolutionary diversity of form and function seen in organisms, locomotory dynamics on land and in water, the physical limits on physiology in such instances as plant movements, swimming etc. In ecology and ethology, we take a pragmatic comparative approach that links experiments and theory in such instances as the behavior of individual organisms, from insects to humans, and the collective behavior of large aggregates, from bacteria to social insects.
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 in the life of social insects.
How can we make sense of biomacromolecular structure, dynamics and cellular function in light of the ability to image and manipulate them? How can we construct effective theories for cellular sensing, motility and behavior that do not drown in molecular details, and are yet are experimentally testable (and falsifiable)? We have explored simple aspects of these questions in such instances as a framework for dynamic instability of microtubules, immunological synapse patterning, cell spreading, cell motion and navigation etc.
Morphogenesis is one of the grand challenges in biology. 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. 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.
Understanding how living systems work opens a window into how life harnesses the physical and chemical world to achieve biological function. A comparative view of function across species and genera is often suggestive of optimal ways of processing energy and information at the molecular, cellular, organ and organismal level. Of particular interest are questions associated with the physiology and ethology of autonomous movement. We are also interested in the breakdown of normal physiology associated with diseases, e.g. sickle cell anemia.
“I wonder why?” is the response of a child when confronted by the bewildering beauty and complexity of the everyday world. Our approach is to use this as inspiration to study the patterns of shape and motion in solids and fluids. Observations are plentiful, if we choose to pay attention to them, and they lead to questions at the intersection of classical geometry and physics, subjects with enormous explanatory power and veins that run deep. Our exploratory approach is also classical, and helps remind us continually of why we do what we do: because we are a curious species that continually wonders what, when, why and how, one humble observation at a time!
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, …
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.
Mathematics and Engineering
While we aim to understand the world of matter and motion as it was, and as it is, it is but natural to dream – of a world that never was, or one that could be. The power of mathematics and the allure of engineering serve to both inspire and guide these reveries. Our approach is experimental and exploratory – probing geometrical and topological descriptions of complexity (literally fold+ together) in physical and biological systems (e.g. quantifying shape information, and network connectivity), and the design and control of engineered systems (e.g. origami, kirigami, autonomous locomotion, protein misaggregation etc.).
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.
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 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.