## Beyond Just Shapes: A Professor’s Insights into the Complex World of Geometry

Geometry is more than the study of shapes—it is the blueprint of the world around us. From engineering marvels like kilometre-spanning bridges, to minute details such as the whorls of a snail’s shell, to leading-edge advances in image analysis through machine learning, geometry is woven into the fabric of our world. And at the Department of Mathematics at the Shiv Nadar Institution of Eminence, Prof. Indranil Biswas is uncovering new depths to the field, every day, pushing the boundaries of how we understand and describe complex geometrical structures.

A Fellow of the Indian Academy of Sciences and three-time recipient of the prestigious J. C. Bose Fellowship (2013, 2018, and 2023), Prof. Biswas has over 200 publications in his field. He is also featured in the Stanford/Elsevier Top 2% Scientist Rankings 2024. Speaking of what drew him to the subject, he reminisces, “*When I was in school, I was interested in Euclidean geometry. During my master’s degree, I tried many different kinds of mathematics, but in the end, that love of geometry persisted. That’s when I got deeper into the field*.”

The word ‘geometry’ conjures up the mental image of shapes and figures, but much of research into geometry deals with equations. “*Consider the simplest curve: a circle. It can be described by an equation, x**2** + y**2** = 1. There are many other simple curves that can be described by equations like this: parabolas, ellipses, hyperbolas, and so on. But you can also have much more complex objects, like a figure 8. You can’t describe that with a simple 2-degree polynomial (one where the highest power a variable is raised to is 2), you need a 3- or 4-degree polynomial. And in the opposite way, suppose you have a complicated figure, you can try to approximate it by polynomial equations. And to describe more and more complicated objects, you can play with the degree of the equation or even the coefficients of the variables*,” explains Prof. Biswas.

It was this understanding that geometric shapes can be described by equations that kicked off the modern geometric field as we know it back in the 17th century. As objects and structures get more complicated, describing them visually might no longer be intuitive, but describing them mathematically is*. “Differential geometry is the study of geometry using calculus, while algebraic geometry is the study of objects using algebraic equations,”* clarifies Prof. Biswas, adding,* “Suppose you want to study a complicated 3-D object. Say a statue made of clay. You shift its shape around to make it smoother, and therefore simpler. This turns the original statue into a ‘deformed object’ (A deformed object in geometry is an object that has changed shape due to the application of force). This means it still retains some of the properties of its original shape, such as volume, weight etc. Essentially, you deform the object into a simpler object which retains the properties you are interested in, to make it easier to study.”*

Of the many ways to describe a geometric object mathematically, one is through a vector bundle. A vector is a quantity that has magnitude, as well as direction. A set of these vectors forms what is called a vector space. “*A vector space is a very simple object. You can combine vector spaces to get a complicated object*,” notes Prof. Biswas. “*Take a table for example. I can assume a one-dimensional vector space (a line) for each point on the table. Then when I take all these lines together, it becomes a 3-D object. A vector bundle is a family of these vector spaces. Many real objects are approximated by these bundles, and they arise again and again in various types of mathematics.*” For his extensive research contributions to algebraic geometry and the moduli problems of vector bundles (which relate to the classification of vector bundles), Prof. Biswas was awarded the prestigious Shanti Swarup Bhatnagar Prize in mathematical sciences in 2006.

Currently, Prof. Biswas’ research focuses on complex geometry, which deals with the description of objects using complex equations—equations in which the coefficients of the variables are described by complex numbers. By definition, a complex number is a combination of a real number (what we consider measurable numbers) and imaginary numbers (a mathematical concept describing the square roots of negative numbers). “*If you have a complex number, then visualising the object becomes much more complicated. But it is still an equation that describes an object. Real geometric objects can be contained in complex geometric equations*,” says Prof. Biswas. “*Right now, my research interest is in some special geometric structures, such as projective structures or structures that can be described by holomorphic functions. They have some special properties in complex geometry, but are still general enough that many objects can be modelled on them*.”

As science becomes more global and collaborative in nature, Prof. Biswas emphasises the importance of staying connected to the latest research. “*You can no longer just sit in your office and be disconnected from everything,” *he remarks. “*You need to interact with other mathematicians and get to know what is the current state of the research. Many students who join Ph.D. courses in India are not aware of the latest research, so they end up working on what was already done a few years back. This does not apply so much to the Shiv Nadar Institution of Eminence or other top research institutions, but we need more ways for Indian students to get exposed to the current research in mathematics and better connect them to the rest of the world*.” However, he is also keen to highlight the positive aspects of the research landscape. “*The funding that exists for mathematics is quite enough, for pure mathematicians at least,” *he emphasises. “*For applied mathematicians maybe they might need a powerful computer or something, but for pure mathematicians, at least in my opinion, the funding is good*.”

Mathematics, especially higher mathematics, tends to carry a reputation of being ‘too difficult,’ but Prof. Biswas is eager to encourage young researchers to test that stereotype for themselves. Speaking on a good starting point for newcomers to the field, he says, “*Exposure to mathematics from experts is what is needed. Exposure can help get rid of bias or fear. It is the job of an expert to explain to a student that a subject is not difficult.*” Outside of that, he also underscores the need for students to attend lectures and conference presentations even if they think the subjects discussed are beyond their level of comprehension. “*You may look at the abstract of a paper and think, ‘I did not understand most of this, so I should not attend the lecture,’ but that’s not correct,”* he insists. “*It doesn’t matter. You might learn some new words, or ideas. You might not learn the entire subject, but you might learn some directions that will help you understand the subject. And this will make it easier to understand subsequent lectures as well. So, don’t go by the preconception or reputation that something is difficult, try it for yourself and see*.”

As Prof. Biswas continues his pioneering work in geometry, the field is set to evolve in new and exciting ways, and his insights will undoubtedly inspire the next generation of mathematical explorers. We are excited to see where his research takes him next!

*This blog is written by Editage Digital Media Solutions, the research promotion division of Cactus Communications.*