‘Today, mathematics is not only necessary in daily life but pervasive’

Apoorva Khare is an associate professor of mathematics at the Indian Institute of Science, Bengaluru. He is one of the winners of the recently announced 2022 Shanti Swarup Bhatnagar Prizes (now remodelled as the Vigyan Yuva-Shanti Swarup Bhatnagar Award). He spoke to Mohan R., a mathematician at Azim Premji University, Bengaluru, for *The Hindu*.

*The questions are in bold. Post-interview additions are in square brackets. The transcript has been edited for style.*

**What was your immediate reaction upon learning that you had won a Shanti Swarup Bhatnagar prize, which is currently the highest national science award?**

I was caught by surprise. The CSIR Director General usually announces the Bhatnagar prizes on the 26th of September, but this time it happened 15 days early. Also, the prizes were not announced at all last year, so I was very pleasantly surprised to find that I had got the prize.

It was my student who first told me that I had received a Bhatnagar prize. Within a few minutes of appearing on the official website, it was already reported on various news websites. It was quite surreal.

**What was your childhood like? Was maths a big part of it?**

I grew up in Bhubaneswar [Odisha]. My parents Pushpa and Avinash Khare are both physics professors and researchers and so all my life I’ve grown up in a science environment, hearing about Einstein or Edison, atoms and galaxies, and so on. But otherwise I would go to school, come home, play table tennis, and read lots of storybooks – Enid Blyton, Sherlock Holmes, Agatha Christie, the usual. I listened to lots of music – both Indian and Western classical, and old Hindi film songs. I also learned Hindustani vocal for seven years, completing my Sangeet Visharad while in school.

My sister Anupama and I grew up solving problems for fun. My parents would get maths books from book fairs. I remember that the old Soviet publishing house called Mir would publish these small and wonderful books under the title ‘Little Mathematics Library’*. *These books asked lots of interesting questions and whether or not I could solve them all, I was very intrigued by the questions. I also used to try solving the logical puzzles that appeared in some newspapers. I enjoyed reading Shakuntala Devi’s books.

On a lighter note, I started to read science fiction in high school, and came across Isaac Asimov’s *Foundation* Series in class IX. There is a fictional mathematician who develops the field of psychohistory, where the future of human civilization as a whole can be mathematically predicted with very high probability. I thought then that maybe someday I will do this [laughs]. Of course, I am not doing anything close to this today, but it was certainly inspirational.

**When did you realise that you want to pursue mathematics? **

I think the first time I realised that I might want to pursue maths as a career came during the Maths Olympiads. It was the Regional Maths Olympiads in Odisha and my mother had got me some question books. I read through them and learned about AM-GM inequality, the Pigeonhole principle, and some geometry. I didn’t have much intuition for geometry, though. I was as surprised as anybody that I stood first in that exam. I think I knew then that I wanted to be more serious about maths. I took the exam again in class XI, stood first again and this time I was able to clear the national exam. At the training camp for the International Maths Olympiad, I came across these incredibly clever people with really sharp minds. It really inspired me to see people think so fast.

That summer [in 1996], I wrote my first paper on divisibility tests. I had seen a note in one of Shakuntala Devi’s books where she had described the divisibility test for the number 19. I knew the tests for 3, 9 and 11, as we study them in school. I wondered, “What about a test for 23 or 37, or any number? And why just for decimal base? Why not for every base?” So I came up with a test for any number written in any base and I wrote that up. It’s very simple “congruence modulo” arithmetic, but then I could get it published in an undergraduate journal. Having done this, and having seen my parents lead an academic life, I decided that I definitely wanted to pursue mathematics research as a career.

**Would you call the publication of that paper when you were still in Class XI the turning point for you?**

Not the publication, but the experience! There was no answer that I had seen written down to the question about the divisibility test for any number. There was no internet at that time. So I was just doodling in a notebook, dividing numbers and seeing whether something works. It’s like tinkering until something works and the bulb lights up. The process of discovering, formulating a hypothesis, and then actually coming up with a proof felt so exhilarating to the 16-year-old me, that I was sure I wanted to keep doing this as a career. And writing up the paper was an experience in itself.

**Some people have noted that both the mathematicians who won Bhatnagar prizes this year were participants of the Math Olympiad. As one of the regional coordinators, I was very interested in your thoughts on the role of the Olympiads in nurturing a culture of problem solving.**

The Math Olympiad Camps encourage critical thinking and tackling problems outside the box, meaning outside the regular school curriculum. Even the tools and tricks we learned to solve such questions were not taught in school. They definitely helped me gain more skills to approach problems with confidence.

The situation is different in schools, where I have seen students get scared of maths exams. It shouldn’t have to be that way. You need to make mathematics, or any subject for that matter, fun. One should be looking forward to questions that challenge them. Since my days as a teaching assistant and then a teacher, I have always tried to involve students in approaching problems independently. They should look at a difficult problem with curiosity, not fear.

The general feeling of phobia of maths should be addressed. There are lots of problems that even I don’t get anywhere with and that’s part of research. It doesn’t mean I don’t like to try and solve them. Research is about solving problems that nobody has before, right? I’ve always felt that way and this, I guess, is also reflected in the Olympiad camps.

**After school, you headed to the Indian Statistical Institute (ISI) for your undergraduate degree. A bachelor’s degree at ISI is one of the most popular programmes for anyone who wants to pursue a career in mathematics. What was that experience like?**

It was fabulous. Back then, there were very few places where one could pursue mathematics. Chennai Mathematical Institute wasn’t founded yet, nor did the B.Math. programme at ISI Bangalore or the BSc Mathematics programme at IIT Bombay exist. None of the IISERs existed. So there were very few options for me – either B.Stat. at ISI or the Integrated M.Sc. in an IIT.

I had heard about two ISI students who had left for the U.S. for a maths PhD right after their Bachelor’s – Siddharth Gadgil who is now my colleague here at IISc, and Amritanshu Prasad who is now at MatScience (IMSc) in Chennai. I wanted to follow their example.

And so I came to ISI. They started the very first day at such a high level! We studied groups, rings, fields, vector spaces, and so on. The Olympiads had put me on a mental high so I really loved starting out at that level. I made it through the JEE but I did not attend the counselling at IIT, as by then I was sure I wanted to stay on in ISI.

**I heard from one of your seniors at ISI that you used to literally run all the time. What was that about?**

[Laughs] ISI at that time was very small. As the B.Stat. curriculum progressively got less mathematical, I wanted to attend advanced maths courses. I requested my classmates and my seniors to adjust their timetables so that I could attend classes with my seniors in parallel to attending my own classes. They agreed, and so did our teachers – an amazing gesture to make just for my request, and one for which I have always remained grateful to the friendly culture of my classmates, seniors, and teachers at ISI Calcutta. Thus, I was able to attend five MSc-level maths courses in my last three semesters at ISI. This really helped me while applying abroad for a mathematics PhD.

I remember I would get caught up in something, realise that lunch break is ending and I have to come back to class. So I would run to the hostel campus, which was seven minutes away, quickly eat lunch, and run back. Unfortunately life is still busy, and my running habit has not stopped. Even at IISc, people say I still run, and it’s true!

**You then moved to the University of Chicago. Could you recall for us how you arrived at this decision?**

At that point, the internet had just come to ISI and in our undergraduate hostel, there were four desktop computers that we would take turns to use. I think I had one Yahoo email and one Mauimail account. Even universities in the U.S. only had rudimentary websites and department websites weren’t always the most informative. But what I knew was that there were very few places that admitted students after three years of undergraduate study from India. Now the situation is much better because of places like ISI and CMI, which are well known abroad. This is because good students have been going abroad to pursue their PhDs for decades now. Luckily, ISI was well-known during and even before my undergraduate days because of people like V.S. Varadarajan, S.R.S. Varadhan, C.R. Rao, and so on.

I applied to four universities, thinking that if nothing worked, I would do a master’s at ISI and then apply again more widely. Luckily I got into two places, and chose Chicago as it was better ranked for algebra, which I thought I wanted to study.

In my second year, my advisor Victor A. Ginzburg gave me a book to read on the basics of Lie algebras, by [James E.] Humphreys. I had to read that book inside out and do most of the problems. Now I ask the same of many of my summer internship students. At the end of that year, Ginzburg gave me a research problem and then I wrote my first paper in representation theory.

**How did you end up co-authoring Beautiful, Simple, Exact, Crazy: Mathematics in the Real World (2015)? It’s quite uncommon for a research mathematician to attempt to write such a book before they have a permanent position.**

While I was in Yale, I met Steven Orszag. We had no research interests in common, but he told me about something he felt passionately about. He said there is a lot of maths phobia in the U.S. Students who enter college are often required to take rigorous calculus courses, but they cannot handle epsilons and deltas. Steve said he wished somebody like me would come up with a course that explained the beauty, power, and applicability of maths.

If one tells people that something will help solve money problems, that would make them very happy. The simplest money problem is that of mortgage – this is a big thing even here in India. Whether for a car, a house, or an expensive iPad, what you’re doing is adding up a finite geometric series. This does not need knowledge of calculus or trigonometry or linear algebra. At the same time, if you know how to add geometric series, you can also talk about fractals, a theoretical concept that appears in mathematics, but also in art. On one side there is beautiful art and on the other side there is the practical idea of mortgage payments – both governed by the same mathematical equation.

I had the idea to collect a bunch of such equations that have actual real-world applications and teach them to students. Then I proposed it to Steve and he helped me submit this idea as a course to the Yale academic council. The course got approved. Teaching this course was quite a memorable experience.

By American standards, Yale is a small school. The class size was typically 25 or less. Steve advised me to restrict the registration to those who have not taken calculus or trigonometry in high school. The idea was to attract students who were not comfortable with these subjects in school. I was given a small classroom with a capacity of 25, but 70 people showed up! The students were overflowing and some of them had to sit outside. I felt so bad. So then I worked with the dean of the college to create a list of the top 35 and bottom 35 in the class. This was based on what they had learned, how comfortable they felt about the assignments and so on. We then rejected the top 35 and selected the bottom 35 for the course.

It was very rewarding to see these kids learning an idea and getting the confidence to solve problems on their own. Learning how to do their mortgages might be one of the most important things they take away from college. There were also topics on probability, conditional probability, standard statistics, basic linear regression, and so on.

This happened during my last semester at Yale and I soon moved to Stanford. Anna Lachowska took over the course from me and she added her own set of topics; for example, she introduced the logarithmic scale of music. She suggested that we write a book combining both our topics.

As far as I know, the ‘Mathematics in the Real World’ course has been offered annually at Yale for more than 10 years now. The book is a resource that can be used by anybody. You can be a high school student, or somebody who has lost touch with maths, or a working professional going to night school. It’s fairly readable if one remembers some class-X mathematics.

**It sounds as if teaching is a particularly rewarding experience for you.**

Indeed it is. I have taught more than 50 courses in the past 20 years, and I feel joy and fulfilment in passing along knowledge to the next generation. And in teaching, I feel it is about that “eureka” moment, the moment a student gets the concept. As a researcher I live for getting the key breakthrough idea that will complete a proof of a theorem, and as a teacher I live for students understanding the key idea behind solving a problem or a class of problems. If I can say that a student has learned something concrete in my class, then I’ve succeeded in the teaching problem, just like how I’ve succeeded in the research problem.

When I think of it that way, the fulfilment one gets from teaching is not very different from the fulfilment one gets from research. Research and teaching are more alike than I thought.

**How did you move from representation theory to other areas of mathematics?**

Initially I was focused only on algebra, and primarily representation theory, my PhD area. Then my wife Amruta Joshi, who did her master’s in computer science at Stanford and later her PhD from UCLA, encouraged me to follow Stanford’s spirit of interdisciplinary research. Thus I started to explore other areas.

In the process, I met Bala Rajaratnam, a statistics faculty at Stanford. We wrote a grant which enabled me to move to Stanford. Bala encouraged me to continue pursuing my research in representation theory, but he also wanted me to work with him in statistics. He was interested in data analysis, so he cared about analysing sample covariance matrices. Because nowadays there is “big data”, the covariance matrices one would obtain are enormous in size and they have poor properties. One would want to improve their properties without losing the covariance structure.

In the language of mathematics, covariance matrices can be called positive semi-definite matrices, and what one wants is to find what kind of operations can be performed on them that would still preserve the positive semi-definiteness. Bala was interested in the applied aspects of the problem and I was interested in the theoretical aspects. Later, we were joined by a postdoc, Dominique Guillot. The three of us soon got good research momentum and wrote several papers together. Bala was very supportive and he said that we would write pure maths papers and he would do his statistics research on the side with his other collaborators. That is how I slowly got into matrix analysis.

**How did you get into combinatorics? Was it through representation theory?**

It came from both, actually. With every matrix we can associate a list of numbers called its eigenvalues. Irrespective of however large the matrix is, we can understand some important properties of the matrix using its eigenvalues. If we take *N* matrices at a time, instead of getting a list of numbers, we get a shape in *N*-dimensions known as a convex hull. This naturally led me from representation theory to the study of geometric combinatorics.

With a former postdoc here (Projesh Nath Choudhury), who is now a faculty member at IIT Gandhinagar, I also studied pure combinatorics. One can look at the combinatorial objects called graphs and study them using the eigenvalues of their distance matrices. So one can combine matrix theory and combinatorics.

One can also enter into combinatorics from mathematical analysis. I think one of the things that I have contributed to the research community is to systematically show how Schur functions, which are objects in representation theory, show up in the study of smooth functions. I discovered this with three other co-authors in 2016, and finally by 2022, I was able to isolate how they occur inside any smooth function. This means you can reach algebraic combinatorics and symmetric function theory from analysis. That was one novel bridge that people had not realised before.

**One of your most popular projects is PolyMath14. The story of how the project unfolded itself is an intriguing one. And what you could establish in that paper was a bridge between group theory, geometry, analysis, probability and computer assisted proofs. Could you recount the story? Also what is it like to collaborate with Terence Tao, who is considered one of the greatest mathematicians of our time?**

While working on a probability paper with Bala, I was led to think about a property that holds for every abelian group in which we can talk about the distance between any two elements of the group. The two-dimensional planes we study in school and the three-dimensional space we live in are some examples of abelian groups together with the notion of a distance.

I wondered if the property held for a non-abelian group equipped with the notion of distance. But I could not prove anything. Then I started wondering if I could even find an example of a non-abelian group equipped with a distance. I emailed a lot of people and asked if they knew of any such structure. I wanted to either find an example or prove that if there is a notion of distance on a group, it must be abelian. But nobody could provide an answer.

Before returning to India, I had given my last talk at University of California, Los Angeles, (UCLA) where Terence Tao works. We met and even wrote a paper together, on a topic I mentioned above: Schur polynomials from smooth polynomial functions, and applications to preserving covariance matrices. The next time I went to UCLA, from India, I discussed this question with Tao and somehow it got his attention. We tried a few things, but made very little progress. He asked if he could share the problem on his blog, which was super popular.

So he put it on his blog and meanwhile, I boarded my flight back to India. By the time I landed, the blog had exploded with mathematical activity. So many comments, so many approaches, so many people trying things out! The next two days were spent in what we eventually realised was a wild goose chase, but in five days the problem was solved, by six people in India, the U.S.A., Canada, and Germany (with contributions from several others) – and all of this progress and research was carried out in comments on Tao’s blog, by mathematicians most of whom had never met each other physically! And yes, we ended up proving that indeed such examples of non-abelian groups with distances cannot exist.

The original question came out of curiosity, it was a thought experiment. I was lucky that Tao got interested and put it on his blog, and the rest is history. People in numerous time zones were working on the problem and there was round-the-clock progress. Even now, after knowing the proof inside out, and having given so many talks on it, I still don’t know at a deeper level exactly why such a characterisation holds.

Abelian groups and distances are among the first and most basic concepts any undergraduate student in mathematics is introduced to in college. So this may well end up being the most fundamental result in any paper of mine! I have been lucky to be part of the story.

**For the general public interested in your work, could you briefly explain the areas you work in and the types of problems you focus on?**

I would say if you look at Bhatnagar awardees in mathematics, I am someone whose research interests are relatively spread out, in a sense. Of course, my primary area is matrix analysis. In addition, I work in combinatorics and in representation theory.

Right now my coauthors and I are finishing up a paper on operations that preserve matrices with prescribed number of negative eigenvalues. A matrix with no negative eigenvalues is said to be positive semi-definite. Now we are trying to find what kind of operations on matrices will preserve a given number of negative eigenvalues or less. I work on such preserver problems, and these have led me to other kinds of problems.

I can tell you the technical details of the work that I do; but on a broader level, let me say that what thrills me is this idea of connections or bridges. For instance, PolyMath14 was a bridge that connected analysis and group theory. Or look at the functions that preserve positivity in analysis: you take determinants of matrices whose entries have such a function, and suddenly you get Schur polynomials and algebra out of them. This is a connection that was not known before, and it has led to several other mathematical discoveries since – to mention a technical example, new results connecting Schur polynomials in algebra and combinatorics, to weak majorisation in analysis.

I feel I was lucky that I was originally trained in algebra and representation theory, and then went to Stanford and started working on the analysis of covariance matrices, and so I was able to spot this connection since I had the technical knowhow from both sides of the bridge.

**How is mathematics perceived in popular culture? It is very common for us to see people proudly saying that they are bad at maths. Mathematics is seen as something that is forced on people in the school. Should every citizen worry about maths?**

First of all, you should not be *worried* about mathematics. This is what I was trying to achieve in my Yale course. Anna Lachowska and I wrote in the preface to our book that it is fashionable (in the context of U.S. colleges) to say that I am bad at maths, even though the same people wouldn’t say that they are bad at other subjects like English, say.

I think mathematics is necessary and a powerful tool. Mortgages, basic banking and household accounting all require mathematics – you don’t need trigonometry or calculus to do all these things, but you should do some maths. I feel everybody should be quantitatively literate at a basic level.

In fact, in today’s world, mathematics is not only necessary in daily life, but all-pervasive. This is the era of big data. Data science and analysis and their applications are in every field. If one decides to do statistics and mathematics, there are wonderful jobs in places like Silicon Valley and Wall Street. These jobs are very stable and highly paid. Even within India, in my college (ISI), I would see companies come for campus placements every year and hire my seniors for high-paying jobs in data analytics, because they were good at statistics.

**In your view, do awards in scientific fields significantly impact the development of those fields?**

If you look at lab-based sciences like biology, awards that come with grants, such as the Swarnajayanti Fellowship, would certainly help the scientists enhance their lab capabilities and hire more students. Government fellowships like NPDF, PMRF or NBHM, provided at the doctoral and postdoctoral levels, are a source of income. We certainly need labs to be flourishing and talented personnel to come in. So in those kinds of sciences certainly these recognitions would help.

Coming to my research in a theoretical field like mathematics: funding helps me travel to conferences, meet old and new experts to form collaborations, and also financially support students and postdocs as well as visitors coming to our department. These are important for research progress.

For example, when I worked at Stanford University, I co-organised a conference with Bala and Dominique. We could gather a bunch of people and from that came collaborations. We found people with matching wavelengths and we have proved many results and published many papers together. Funding also allowed me to visit Terence Tao and that led to our collaboration on matrix positivity, as well as our Polymath project. So funding is also important in theoretical research like in mathematics. It comes in the form of research grants or research awards; some of the grants are also called awards!

In addition to these practical aspects, awards like the Nobel Prizes and Fields Medals also catch the attention of the general public and students, and hopefully inspire students to get into research and young researchers to do more creative science. Certainly I was inspired by reading about these prizes and personalities and their contributions. And hopefully, in the Indian context, the Bhatnagar Prizes do the same.

**We have been observing a trend especially after the pandemic that few students are taking up a BSc in mathematics. In some places, mathematics departments are shutting down or merging with data science departments. Do you see this as a short-term trend or is this an indication of something that we are not yet aware of?**

It is difficult to predict what kind of paradigm shifts happen or for what reason. The pandemic came and changed the way in which we work. The internet is another example. Pre-smartphone and post-smartphone experiences are completely different. We would not have imagined groceries getting delivered at home via the internet even 10 years ago.

It is certainly true that college programmes and streams keep changing and evolving with the times. And this is the age of big data and analytics, machine learning, artificial intelligence, neural networks… So it would be natural for mathematics departments and programmes to evolve with the times, especially at a time when enrollments are dwindling. Maybe it is good, given the opportunities available.

**But what happens to pure mathematics that is close to our hearts?**

There are lots of places that still offer pure mathematics programmes. The IISERs were established precisely for pure sciences, just like the IITs were established for engineering. IISc is one of the places where sciences and engineering are treated on equal footing. The IITs also have very good maths faculty, and some of them have maths major programmes which see very good students too. In fact, the other mathematics Bhatnagar Prize winner this year was also in the Math Olympiads and then went to IIT Kanpur. And then there are places that offer maths/stats undergraduate degrees, like ISI and CMI. And these are just a few of the places that offer pure mathematics coursework and degrees.

But moreover, it will take time to find enough teachers who are trained to teach the newer courses in data science, and maybe AI and ML. So I am sure that not every university/college is changing their BSc. mathematics to a BSc. data science degree just yet!

**Speaking of mathematics programmes and faculties in India: when you worked at Stanford University, you organised a Young Researchers Meet. Can you tell us what that was about?**

In 2012, Professor G. Rangarajan, who is now my department colleague and the Director of IISc, and I organised a ‘Young Researchers’ Meeting’ with the goal of attracting world class talent into Indian academia. We invited PhD students and postdocs of Indian origin, working in mathematics and computer science in the U.S.A., to Stanford in May 2012. A group of delegates from India (1-2 delegates per institution, including IISc, IITs, IIITs, IISERs, CMI, ISI, IMSc) flew in to tell the assembled participants about the research scene, funding sources, and other details of Indian academia.

I think the meeting went very well. Out of the 74 participants, 26 returned to India, and I believe 20+ are still in faculty positions in India. So we were very happy that the meeting ended up serving Indian academia well. And of course I myself was one of these 20+ returnees. I came back and joined IISc. Prof. Rangarajan but also Prof. Gadadhar Misra were very helpful in my moving and settling down here.

**There is a lot of concern about the gender ratio among Bhatnagar awardees. Very few women or scientists of other genders are recognised for their work, compared to men. What are your thoughts on this?**

As far as specific awards go, I am too junior to be on any committees to know how it all works. But of course there is clearly a gender imbalance and this is reflective of the broader gender imbalance in sciences and mathematics going all the way down. Even now, if you look at the maths majors in IISc, there are fewer girls than boys in classes. This has to be addressed at a fundamental level. Several things need to be done. Of course, one should start at the grassroots and encourage girls to pursue science at all levels, but maybe one should also try to incentivise higher education through scholarships and other opportunities.

The SERB – POWER Research Grant is a recent government scheme specifically for women researchers, not just mathematicians but across all disciplines. Maybe in this day and age, incentivising is the way to address the issue, from the entry level to the faculty level. Another measure could be to reserve seats for female students in higher education in science and technology.

**What advice would you give a school student who is finding mathematics interesting and wants to take up mathematics as a career?**

First of all, if someone is interested in maths, they should be solving problems – here, there or anywhere. I found problems not just in books but also in the newspaper. Additionally, if students want to seriously think about mathematics as a career, then they have to be able to understand what advanced mathematics looks like. Attending camps can help here.

When you are in middle school, you should try to get an exposure to high school or Olympiad-level mathematics. When you’re in college, you should try to do more rigorous courses or advanced courses. That is the practice here at IISc, for instance. For younger students, some of the summer camps I know are Bhaskaracharya Pratishthana in Pune as well as ‘maths circles’ in Mumbai and Bengaluru.

**That brings us to the end of the interview. It was really really fantastic meeting after a long time and listening to you open up about various things. Congratulations once again!**

Thank you too and bye.

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