“I am fond of mathematics of every description,” says Dr. Gautam Bharali, one of this year’s 11 recipients of the prestigious Swarnajayanti Fellowship. Awarded by the Department of Science and Technology on behalf of the Government of India, the fellowship recognises outstanding contributions to scientific research by Indian scientists. An Associate Professor in the Department of Mathematics at the Indian Institute of Science (IISc), Dr. Bharali specialises in the field of Several Complex Variables. Six years before receiving the Swarnajayanti Fellowship, he was also felicitated with a Young Scientist Medal by the Indian National Science Academy (INSA).
Several Complex Variables is a catch-all term for the study of any mathematical object or relationship that involves multiple variables which are complex numbers. Most of Dr. Bharali’s current work focuses on complex-analytic dynamics. Iterative dynamics is the study of dynamical systems defined by iteration of functions on some space. Complex-analytic dynamics is the study of the dynamics of analytic functions on spaces described by complex numbers. Dr. Bharali remarks, “Many of us have some acquaintance with this subject having seen pretty pictures of objects like the Mandelbrot Set. These pictures are associated with non-linear dynamical systems obtained by iterating a polynomial over and over or, more generally, iterating a rational function, which is an analytic map from the sphere to itself.”
Dr. Bharali’s research interests include Riemann surfaces. Riemann surfaces can be thought of as modified versions of the complex plane. Locally, near every point, they look like patches of the complex plane, but the global topology can be quite different. If one tries to iterate an analytic map defined on a Riemann surface more complicated than the sphere―and which maps this surface into itself―then the results are disappointing: no complex behaviour emerges. Dr. Bharali remarks, “My interest these days is to show that there is complexity lurking around in these Riemann-surface settings.”
A function or map is a mathematical object that we intuitively understand as a ‘relation’ between a set of inputs and a set of permissible outputs: a map is a special type of relation with the property that each input is related to exactly one output. A general relation can also be iterated over and over, and as any Riemann surface is rich in interesting, non-linear relations, this gives rise to interesting emergent behaviour that researchers know very little about. “I am working on learning more about such behaviour and building a theory, together with other mathematicians, that parallels what we have for the polynomials,” Dr. Bharali says.
Dr. Bharali has also published several peer-reviewed papers in other areas of mathematics which are outside of his primary area of interest. “Those papers came about because the problems addressed in them were simply attractive and intriguing to me,” he says.
Dr. Bharali has studied and worked at premier research institutes in India and abroad. He obtained a master’s degree in mathematics from the Indian Institute of Technology (IIT), Kanpur. He then completed his Ph.D. at the University of Wisconsin, Madison, in 2002. Following this, he was appointed as Assistant Professor in the Department of Mathematics at the University of Michigan. Three years later, he joined the Department of Mathematics at IISc. He was a Young Associate at the Indian Academy of Sciencesfrom 2006 to 2009, and an Associate of the International Centre for Theoretical Physics, Trieste, from 2007 to 2013. During the 2012-13 academic year, he was a Guest Associate Professorat the Norwegian University of Science and Technology in Trondheim, Norway.
Motivation is the driving force behind all meaningful work and Dr. Bharali feels that the quest to unearth logical links between simple ideas and profound ideas is what motivates those working in the area of pure mathematics. He explains, “While all of us have a core research goal to work towards, most of us carry in our minds several other problems whose unifying theme is that they are, by some personal standards, aesthetically appealing. On days when the challenges of one's primary research goal seem insurmountable, the mind turns to these other problems for distraction, solace, and possibly, inspiration.”
Dr. Kaushal Verma, an Associate Professor in Mathematics at IISc, has maintained a strong scientific association with Dr. Bharali for almost ten years. “I knew him from the time we both worked at the University of Michigan and now we are colleagues at IISc,” says Dr. Verma. He appreciates Dr. Bharali’s contributions to the field and says, “Several Complex Variables is a field that lies at the interface of several disciplines. This means that there are many possible approaches to it and I think his approaches have always proved very effective. I must add that I really admire his motivation to explore new problems and not be restricted by a narrow field of research.” On Dr. Bharali’s recent nomination for the Swarnajayanti Fellowship, Dr. Verma says, “I am delighted to hear such good news. It is a well-deserved honour.”
Dr. Bharali expressed his gratitude on receiving the award by stating, “I am most grateful to my entire department at IISc, my senior colleagues in particular. It’s on their encouragement that I competed for this Fellowship.” He modestly adds, “It is an honour and it feels good to be recognised but, very soon, the mind turns to the research problems that I am thinking of, and to all the work that is waiting to be done.”
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