Education

# A Mathematical Researcher is Navigating New India with Number Theories

### Kaneenika Sinha studies questions about distribution patterns of special numbers like prime numbers.

Kaneenika Sinha. Credit: The Life of Science

“This blog is no longer anonymous.”

And that’s how ‘New Prof in New India’, as she was known in the blogosphere, revealed her identity to her followers after two years of anonymous blogging. Kaneenika Sinha admits that her identity was not a very well-kept secret in the relatively close-knit Indian science community, but anonymity gave her greater freedom to write honestly. She continued to do so for years after the revelation, and this time she was free to write about more specific things like her research in number theory.

Articulate, candid, humorous and relatable, Sinha ticks all the boxes for a popular blogger and a great science communicator. Mathematical research can be an awfully tricky topic to simplify and mathematicians often struggle to do so. So it is with considerable trepidation that I walked up the lush winding staircases to IISER Pune’s Math department. I needn’t have worried. Ever ready with her marker and whiteboard, Sinha steered through an hour-long fascinating, and surprisingly non-turbulent, cruise across a river of numbers, Greek symbols and unsolved hypotheses.

Prime numbers, as many of us know by now, are notoriously unpredictable. 2, 3, 5, 7, 11, 13… they pop in and out of the number series infinitely (as proved by Euclid 2,300 years ago) but with no apparent pattern. For centuries, mathematicians have been trying to peg down the distribution patterns of primes. So what have we learnt in all these years, I asked. “A lot, and yet very little,” responded Sinha . She elaborated with the example of the ‘twin prime conjecture’.

A twin prime is a pair of prime numbers that have a gap of two between them, such as three and five or five and seven or 11 and 13. The question that haunted mathematicians for long was whether there are infinitely many such pairs, or if there is a point in the sequence of numbers after which no more twin primes exist. As of September 2016, the current largest twin prime pair known is 388,342 digits long (the exact number is 2996863034895. 21290000 ± 1)! So it does seem like twin primes would go on infinitely, and indeed some of the most celebrated minds in the area like G.H. Hardy and E. M. Wright have provided evidence for it, but none of it has been conclusive.

At its prime

After repeated failures, mathematicians began to tackle this problem more gradually. Instead of twin primes with a gap of two, they began proving different versions of the conjecture. In these versions, the prime pairs had larger gaps between them. Yitang Zhang, a professor in a fairly obscure university in USA, provided a breakthrough in 2013. “That was the first time a finite quantification was given,” said Sinha. The gap between the pairs in his proof was 70 million.

Seventy million is a long way from two, but an even longer way from infinity. Sinha stressed that Zhang’s elegant proof opened the doors for this gap to be shrunk. In the past few years, the gap has been brought down from 70 million to 246 by Australian-American mathematician Terence Tao. “That’s as far as we have come,” summed up Sinha. “It is believed that further reducing the gap will require an infusion of radically new ideas!”

Sinha’s own research doesn’t deal directly with prime numbers, but her goals are similar. “I study questions about distribution patterns of special numbers like prime numbers,” she said. She elaborates in this video.

Patterns in nature

“We construct mathematical functions and series. What’s interesting is that the patterns we see match patterns in other phenomena, such as energy levels in atoms.” How can this be? That is a very deep question, she answered.

But so stark are these commonalities between mathematical patterns and natural phenomena, that there is a very strong indication that the solution to another century-old unsolved math problem called the Reimann Hypothesis will come from physics.

In math circles, Bernhard Riemann is famous as “the mathematician who made the single biggest breakthrough in prime number theory”. In this video, Sinha beautifully describes to me the essence of his hypothesis.

The multitude of such ‘deep’ problems is exactly what enamours mathematicians like Sinha. The thought of possible application may be far away from a mathematician’s mind, but she pointed out to me that fundamental sciences have an uncanny way of throwing up unintended applications. A classic case of how prime number theory has crept into our daily life is the Rivest–Shamir–Adleman (RSA) algorithm, an encryption method based on prime numbers that forms the foundation of how all our information online is kept secure, including bank transactions made through the internet. “All of this is fundamentally based on how hard it is to factorise a large number,” said Sinha, and she knows only too well that it’s not easy at all.

The key idea behind the RSA algorithm comes from a note made by a French lawyer and mathematician Pierre Fermat in a letter to his friend. And this fact, for Sinha, is testament that fundamental research must be encouraged, not just ‘translational’ or application-oriented research. “Fermat made this simple observation while writing a letter to his friend, maybe during tea time. It took 100 years to write a full proof and applications came so much later. Now all our digital security is based on that. Basic research must not be ignored!”

She recalled with some amusement, recently delivering a talk at her alma mater Lady Shriram College in Delhi. There, she shared her fascination with her favourite math theorem – Lagrange’s four-square theorem. “It’s a very cute theorem that states that if you give me any number (positive integer), it can be written as a sum of at most four squares (i.e. it will either be a square, or a sum of two, three or four squares).” After her talk, a student posed a question about the application of this theorem. Slightly miffed, Sinha retorted that such theorems cannot be looked at from that angle. “I gave her a long lecture about appreciating these deep facts and trying to understand the maths behind it. She wasn’t very convinced!”

Lagrange’s four-square theorem examples. Credit: The Life of Science

While preparing for a class at IISER recently, Sinha found herself unintentionally vindicated. “I realised that the [Lagrange’s four-square theorem] also has an application! It lies at the heart of another theorem which is today used to create ‘error correcting codes’. These are used to send and receive messages from Earth to, say, a spacecraft near Mars. You take the message, encrypt it into a bigger message, and send it. Any errors during transmission are fixed during decryption,” she described. “Of course this is not the reason the theorem was discovered. The next time I give a talk and get this question, I am ready!” she laughed.

With a little help from mentors

When Sinha began her bachelor’s in mathematics in 1997, she had no intention to continue in the area. “Like in any Bihari family, I was also expected to become a doctor, engineer or join the civil services. I convinced my parents that I was taking maths because it was a subject where I could score well. Math is used everywhere so I was sure I could take up any field of my choice later,” she said.

It was in her final year of undergraduate studies that a passion for math came. “I attended some workshops, including the famous Mathematics Training and Talent Search Programme (MTTS). They do an incredible job there. For three weeks they teach college students math as a fun subject, not as something that is dry and done just for the sake of doing. That changed things for me.”

Delivering a conference lecture. Credit: The Life of Science

Newly instilled with a drive for the subject, Sinha kept attending workshops, including a three-month programme at TIFR Mumbai. She was in the middle of breakfast one day there when she struck up a conversation with a man in the next table. “He was very nice and kind and turned out to be a number theory professor called M. Ram Murty. He gifted me his latest book and I stayed in touch with him.” The following year, Sinha moved to Canada where she completed her MSc and PhD under Professor Murty.

From 2006 to 2010, Kaneenika did postdoctoral studies in University of Toronto and University of Alberta in Canada respectively. After nine years in the country, when the time came to look for a job, she kept her options open. “There was a crunch in North America at the time because of which good jobs were hard to obtain. Meanwhile in India, the IISERS were established, so I did a reality check. I could get a job somewhere in middle of nowhere in North America and remain obscure, or return to India and join one of these new institutions which were giving a good salary and had good facilities. And that would give me a feeling that I was building up things from scratch. From a career point-of-view, I decided that it was better to come back,” she said.

Making a home at home

IISER Kolkata made Sinha an attractive offer so she joined them in 2010. She reminisced the feeling of ownership that came with bringing up the math department in a swanky new institution. It was this sentiment that led her to young mathematician Abhishek Banerjee. “I’d met him at a conference. Noticing his Bengali surname, I thought it would be good to ask him to join IISER Kolkata. With this intention, I invited him to give a talk.” By the time, Dr. Banerjee came, he already had an offer from IISc, Bengaluru, so Sinha’s plans did not work out. However, the two became friends and two years later, in 2013, they got married.

Sinha left IISER Kolkata two-and-a-half years into the job. After a promising beginning, she admits to getting somewhat disillusioned. Some problems arose with an administration change and an offer from IISER Pune was difficult to resist. “By then, Pune already had a growing number theory group. I knew that if I went to a place like that, I would have more people to talk to, students interested in working with me… At that point of time, it was more important for me to build my own research career than building up the institute.”

The move proved a wonderful decision for Kaneenika, who has worked here for five years and is looking forward to many more. Her husband still works at IISc in Bengaluru – “yes, the two-body problem!” she laughed, referring to the oft-stated difficulty couples in academia face in getting positions in the same institution. But for the time being, the couple are in no hurry to change the status quo. “Bengaluru is so well connected to Pune and both our institutes have very flexible leave policies so it’s fine for the time being,” she said, adding that “maybe, at some point, it will be good to stay together!”

Everyday sexism

With her current research students.  Credit: The Life of Science

Despite being in a male-dominated STEM field, Sinha has never been made to feel inadequate. IISER Pune’s math department currently comprises seven women, out of 26 faculty members. Both of Sinha’s current research students (one MSc and another postdoc) are women.

Unfortunately, everyday sexism doesn’t escape even someone like her who has had such an impressive career. She bitterly recalled being at the end of a classic case of mansplaining she encountered while visiting an institute: “The Director asked me how I would solve my two-body problem, since “IISc does not hire couples”. I replied that I don’t care, since I already have a very good job. I then got a long lecture from him about how “Indian infrastructure” does not support these kind of relationships and how it would be difficult to raise children. He then looked at another colleague sitting next to me and asked her to convince me that I should be serious about this issue, thereby embarrassing her as well. I made the mistake of not changing the topic immediately.”

These kinds of attitudes are especially toxic when coming from people at high positions. Sinha brought up another episode when her unmarried scientist-friend approached the director of her institute to talk about an issue with housing. She told him that her present quarters were too remote and she wanted a bigger house on campus so that her family could visit. The director retorted in a particularly distasteful manner, “May I ask why you have not got married till now if you feel so isolated?”

“Was it fair to bring marital status into consideration for something for which she was already eligible?” questioned Sinha. In this case, however, she is grateful that things worked out for her friend. “Her departmental chairperson took up her case and pushed the administration to come up with a proper policy regarding housing. She finally got allotted the house she wanted.”

Note: Sinha has quite a repertoire of popular science articles under her belt. Besides her blog, she has written about prime number theory here. Another article that delves deeper into her research can be found in Indian Express. She also writes on and off for Swarajya magazine.

This piece was originally published by The Life of Science. The Wire is happy to support this project by Aashima Dogra and Nandita Jayaraj, who are travelling across India to meet unsung women scientists.

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