Biology offers many beautiful examples of the interconversion of matter, energy and information in non-equilibrium systems. Our interests in biology are driven by the quest to understand “how things work” which leads naturally to morphology and physiology since form both enables and constrains function. We believe that a practical approach requires a comparative study of extremes and 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. Almost all our work in biology involves both theory and experiments, with the latter done both in our own lab, and through collaboration. We have recently also become interested in aspects of ethology and ecology. Here too, 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.”Areas of current interest include understanding the nature of geometry and the nature of dynamics, and aspects of behavior in animals and humans at both the individual and collective level. We have studied aspects of both these in the context of object recognition via shape and size discrimination, navigational tasks in insects, collective physiology enabled by the ethology of social insect groups such as termites and bees, the statistical nature of geometric reasoning, detecting motion using the early warning signals embedded in fluctuating motion, etc. A subject of particular recent interest is linking geometry, dynamics, and probability, in such material examples as the coin toss and such ethereal examples as the perceptual psychophysics of space-time.
A recurrent theme in current biology is that of collective behavior at all scales. We have a growing interest in exploring aspects of these collective dynamics in extreme situations, e.g. collective dynamics of cell aggregates, and the behavior of super-organisms exemplified in the life of social insects. A basic question here is not so much the plethora of patterns that they exhibit, but why they do so, and how they are regulated to achieve a modicum of functional efficiency.
Like the everyday physical world, the everyday biological world is full of wonders – a walk through the garden full of flowers, a hike through the woods, a visit to the zoo, or many hours watching nature documentaries, particularly those hosted by the most widely traveled human in history, David Attenborough. We have worked on and played with a range of problems in this arena that include understanding how to create birdsong using a rubber tube that flutters when air is forced through it, to figuring out how Pollock might have intuitively learned fluid dynamics to paint, to explorations of our knowledge of Euclidean geometry, or how we respond to each other by being acutely sensitive to early-warning fluctuations that often precede actual motor signals etc.
Over the last few decades, structural biologists have delivered a treasure-trove of data on the shapes and sizes of large biomacromolecules and their assemblies. When this information is combined with the biochemistry and genetics to understand aspects of the kinetics and manipulate them, we can begin to ask questions about how structure impacts function dynamically at both the level of the individual molecule and in large aggregates of molecules such as polymeric filaments and membranes.
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 movement; understanding how this actually happens and how it is controlled 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 processing of energy and information, and the exquisite morphologies used to do so, at the molecular, cellular, organ and organismal level. There are enough questions here to keep us occupied for a while. Of particular interest are questions associated with the physiology of locomotion, since biology is almost synonymous with autonomous movement. Movement requires force production, and its spatiotemporal coordination and control in light of sensory feedback from the environment. The questions any study of coordinated movement raises thus impinge on molecular, cellular and tissue dynamics and neuroscience, and from a mathematical perspective involves ideas from continuum dynamics, control theory, optimization etc. and impinge on questions of sensory physiology, behavior, ecology etc.
“I wonder why?” is the response of every 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, particularly those that are soft either by virtue of their constituents or as an elementary consequence of geometry. 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 not driven by grand questions but humble observations, 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.
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, relative to each other. In the past, we have studied aspects of this problem in the context of the low-dimensional behavior of filaments and membranes in the context of their equilibrium, and nonequilibrium behavior to understand how these objects fold, wrinkle, and pattern often driven by simple packing constraints. The rich experimental phenomenology combined with the host of mathematical questions that these lead to have kept us occupied for longer than one might want! We continue to work on variants of these problems with applications to a range of problems, from origami to tectonics.
How can one not be fascinated by the flow of fluids ? From the dripping faucet to meandering rivers, from a viscous thread of honey to the flow of blood, fluids 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 (when one could still play with the fragments that remain from a broken thermometer), the aggregation of cheerios on milk, the swimming of fish and the flutter of a flag (one converts AC to DC, and other reverses the effect), 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 …
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
As we aim to understand the world of matter and motion as it was, and as it is, it is hard not 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. We have have recently become interested in quantitative 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 help quantify aspects of the design principles behind art forms? And can engineering help realize and functionalize these?
Our interests in data analysis is associated with quantifying the variability in the geometrical and topological aspects of 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 for the 4d printing of a face? How to walk on a slackline or a tightrope? These are all inverse problems that reverse the typical formulation of questions, that make them hard to solve 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 – we have helped determine optimal protocols for 3d and 4d printing, control-theoretic approaches to drug dosing etc.
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 uncover strategies for such mundane questions as how to balance on a tightrope or throw a failed calculation scribed on a crumpled sheet into a wastebasket, to optimal control in space and time of protein misfolding diseases, the possible control mechanisms involved in the evolution of gliding flight in animals, to learning how to crawl, navigate, swim etc.