School of Mathematics
Tata Institute of Fundamental Research
AsianScientist (Dec. 5, 2017) – Numbers have been an enduring fascination for Professor Amalendu Krishna, a mathematician at the Tata Institute of Fundamental Research (TIFR). The prolific researcher started off his academic career by studying statistics at the Indian Statistical Institute in Kolkata. However, he found the approximate nature of statistics to be unsatisfying, eventually leading him to turn his keen intellect to solving mathematical problems instead, drawn by the level of precision required.
After completing his PhD at the TIFR under the supervision of Shanti Swarup Bhatnagar Prize (2003) winner Professor Vasudevan Srinivas, Krishna spent five years as a researcher in the US before returning to TIFR as a faculty member in 2005. Krishna is renowned for his work on algebraic cycles and K-theory, dealing with complex formulae and theorems that have important applications in fundamental mathematics and theoretical physics.
In 2016, Krishna became a Shanti Swarup Bhatnagar Prize winner himself, adding to his list of accomplishments which include the prestigious Ramanujan Prize which he won in 2015.
- What is the biggest misconception people have about mathematics and mathematicians?
- What made you decide to switch to mathematics from statistics?
- Could you explain algebraic cycles and K-theory in layman’s terms?
- What is the significance of your work on 0-cycles and what can the findings be used for?
- What are you working on at the moment?
- Who are your mathematical heroes and why?
- What role do awards such as the Shanti Swarup Bhatnagar award and the Ramanujan Prize play in shaping the direction of science?
- How should developing countries like India balance between funding basic and applied research?
- What can be done to make mathematics more interesting and less intimidating for students?
The biggest misconception I have encountered about mathematics is that it is probably the hardest subject and requires special abilities to learn. This fear often drives children and senior students away from the subject.
There may be some truth in the belief that mathematics is a difficult subject, but the misconception that only people with special abilities can learn it is probably misplaced. As I see it, compared to most other subjects, mathematics requires a very good understanding of fundamentals because what one learns at level one is carried over to level two and so on. When people do not follow this sequence, gaps in understanding keep adding up and it eventually becomes too difficult to understand at senior levels.
The root of this problem is that unlike other subjects, mathematics requires an extremely high level of precision. In mathematics, there can be only one right answer, whereas in other subjects, especially in the social sciences, there is no single correct answer. People tend to give up after repeated encounters with this requirement.
I think that mathematics teachers and book authors have a crucial role in removing this misconception. Like doctors, they need special training in tackling this problem.
I studied statistics for five years but could not convince myself that I should continue with it. The reason for this was that I found that statistics was a lot about approximation and estimation and less about precision.
I could not enjoy this sort of study as much as I enjoyed the precision that mathematics provides. It is possible that I did not explore various specializations that statistics offers, some of which are highly mathematical in nature.
K-theory is a subject in the field of mathematics which was discovered by Grothendieck and Quillen. This theory provides a sort of universal invariant attached to various mathematical objects that we want to understand. It is therefore used in almost all fields of pure mathematics, such as algebra, geometry, topology and analysis.
For example, to understand a very complex geometrical pattern in any dimension, one would need certain parameters whose values contain definite information about the pattern. This is what K-theory provides.
Algebraic cycles provide geometric information about the given pattern that we use in order to compute the K-theoretic parameters associated with the pattern.
Think of the pattern that you want to understand as a building. To know how the building was constructed, we need the blueprint, and algebraic cycles are analogous to blueprints, specifying the shape, size and composition of the building. Meanwhile, all blueprints fall within a larger framework of architectural concepts. In the same way, algebraic cycles can be considered essential components of K-theory.
Together with my many collaborators, I was able to make significant breakthroughs in using algebraic cycles to compute K-theory of various geometric objects. 0-cycles are special algebraic cycles which can be used to compute a certain significant piece of K-theory.
My findings on 0-cycles have led to complete solutions to some very long-standing conjectures and open questions in the study of K-theory. In particular, it led to the discovery of the theory of additive higher Chow groups and algebraic cycles with modulus, which are geometric techniques to compute K-theory in the relative setting, when we have to compare two or more geometric patterns simultaneously.
My current research is focused mainly on two independent things. Firstly, I aim to provide a theory which allows us to directly connect algebraic cycles with modulus to K-theory. Here, my interest is to find a formula which will allow one to write down a comparative K-theory of two different geometric patterns in terms of some cohomology groups, which are obtained by means of algebraic cycles. This is challenging problem at the moment, but I have made some crucial progress in this direction.
Secondly, I am developing geometric tools to compute equivariant K-theory. In equivariant K-theory, we use geometric as well as group theoretic inputs to compute the K-theory. This is especially applied in the situation where a geometric pattern is equipped with a collection of automorphisms. These automorphisms allow us to use some representation theory to compute the K-theory.
I have proven several results in the past on this subject and my hope is to use representation theory ideas to write down simple formulas to compute equivariant K-theory.
I am not sure if there is one individual who is my mathematical hero. In my area of specialization, mathematicians like Alexander Grothendieck, Daniel Quillen, Pierre Deligne, Andre Suslin, Vladimir Voevodsky and Spencer Bloch made a very serious impact on me during my early research days.
I have not written a mathematical research paper where at least one of these has not featured in the citation list. Their contribution to my subject of study is just so enormous that it is impossible to write a research paper without reading their work.
Apart from these, I highly admire those who work hard with complete honesty to achieve their goal irrespective of their ability.
In any field, an award or prize provides recognition for past work and stimulus for future work. The same holds for mathematics as well. As I said before, research in pure science in general, and mathematics in particular, is a very challenging profession. It requires a huge amount of mental preparedness to face the possibility of failure.
In this situation, an award provides a significant amount of encouragement. The Ramanujan prize is the most prestigious prize given to a mathematician from a developing country by UNESCO through its research organization ICTP at Trieste, Italy. It is extremely competitive and is decided by a jury consisting of the very top mathematicians in the world. The Shanti Swarup Bhatnagar prize, on the other hand, is the most prestigious award given to a scientist in India.
These awards are so famous that they inspire people to pursue higher study and research in science, even though there are many alternative careers that are much more financially attractive. Personally, these awards give me a sense of satisfaction about my research and motivates me to do better research in future.
It is true that a developing country like India needs to balance between funding basic and applied research. However, one should be mindful that basic research is as much important as the applied research, apart from being much harder to achieve. Fundamental research in basic science contributes significantly in applied research.
I personally believe that the Indian government needs to be more sensitive towards basic research in science, and it needs to invest more, especially in establishing top-class research and teaching universities. It is pity that a huge country like India does not a have university which has a respectable ranking in the world.
The government needs to address this immediately. Basic research in science should also be made more financially lucrative so as to attract eminent researchers from abroad while retaining local research talents.
People who teach mathematics have to first acknowledge that this is indeed a somewhat difficult subject at the high school level. By the time a student begins college, he or she has already lost interest in mathematics.
Hence, we need to make changes in our style of teaching at the school level. Except in some very few schools, most of the teachers simply go through the usual paces and somehow finish their teaching job. These teachers need to be specially trained on the techniques of mathematics teaching.
Instead of burdening the students with huge lists of problems to solve, they need to explain why it is important to learn mathematics. They need to be taught more through figures and patterns than through numbers. The authors of mathematics textbooks also have to make certain changes in the way they explain various concepts.
At the end of day, we need to understand that a child does not know which subject is difficult and which is not; it is the teacher who inculcates the feeling of either. So, more than the students, it is the teachers like us who need to learn.
This article is from a monthly series called Asia’s Scientific Trailblazers. Click here to read other articles in the series.
Copyright: Asian Scientist Magazine; Photo: International Centre for Theoretical Physics.
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