JSPS Quarterly
No.53 2015 Autumn

Research and Life in Japan by a JSPS Fellow (36)

Dr. Soma Purkai
Dr. Soma Purkait

JSPS Postdoctoral Fellow, Faculty of Mathematics, Kyushu University, 2014-present
Research Fellow, Warwick Mathematics Institute, University of Warwick, 2012-2014
Ph.D. (Mathematics), Warwick Mathematics Institute, University of Warwick, 2008-2012
Junior Research Fellow, Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore, 2007-2008
M. Math., Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore, 2005-2007

Hailing from India, Dr. Soma Purkait has been conducting research with her host professor Dr. Masanobu Kaneko at Kyushu University under a JSPS postdoctoral fellowship. We asked her about her research and life in Japan.

What are you currently researching under the JSPS fellowship?

Dr. Soma Purkai

   I work in the area of Number Theory in Mathematics. One of its basic problems is to find solutions of a given polynomial with rational coefficients in rational numbers, if they exist. For example, consider the polynomial Y2 = X3 - X in the two variables X and Y, it has precisely three rational solutions, namely, (X,Y) = (0,0), (1,0), (-1,0). In some special cases, this arithmetic question can be related to analytic objects called modular forms from which you can sometimes deduce whether or not a given polynomial, like the example above, has infinitely many rational solutions. I am studying modular forms of integral and halfintegral weight and questions regarding the infinitude of rational solutions for elliptic curves, which are defined by rational polynomials of the form Y2 = X3 + aX + b where a, b are rational numbers. (The above polynomial Y2 = X3 - X is an example of an elliptic curve.)

When did you find your research subject and why did you choose to pursue it?

   I spent my childhood in Delhi, India, where I did all my schooling and received my undergraduate degree. While I was in school, mathematics was my favorite subject, mainly because mathematics is logical and rule-based and unlike history, for example, I didn’t need to memorize a lot of facts! Also, scoring high in math tests was easy for me. So, it was natural to pursue mathematics as an undergraduate and continue on to a master’s study, which I did in Bangalore, India. I particularly liked algebra and number theory. For me, there are two things that make number theory special. First, problems are usually very easy to state so that even school students can understand them. At the same time, number theory problems also constitute some of the oldest unsolved problems in mathematics, for example the Congruent Number Problem, which asks if a positive integer N can be realized as the area of a right-angled triangle with all sides having rational lengths. This problem is still unsolved, making it a 1,000 year-old unsolved problem. Second, most of modern mathematics has been advanced while solving this kind of problems in number theory. So, it is absolutely fascinating!

How did you get to know your Japanese host researcher?

   Before coming to Japan, I had decided to pursue my PhD in the UK like some of my batch mates. During my PhD studies in Warwick, I visited Japan and really liked the place and culture. At that time, I visited Kyushu University to give a talk and made some friends here. After completing my PhD, I wanted to come back to Japan to continue my research. I wrote to Prof. Masanobu Kaneko, who specializes in number theory and modular forms, and later met him at Kyushu University. He kindly agreed to be my host researcher.

Is that why you choose Japan to pursue your research?

   The Japanese contribution t o the mathematical area of number theory and particularly modular forms is well known. There are, for example, the works of Yutaka Taniyama and Goro Shimura including their famous Taniyama–Shimura conjecture, which is now a theorem thanks to Andrew Wiles et al. There are also several fundamental lifting results like the Saito–Kurokawa lift constructed by Hiroshi Saito and Nobushige Kurokawa, the Doi–Naganuma lifting, and the Ikeda lifts. When I visited Kyushu University to give that talk, I had a good chance to interact with people in its number theory group. I was also fascinated by Japan’s beauty and culture. Besides, as my husband is Japanese and is currently working in Tokyo, I was also looking for a work opportunity in Japan.

What is your impression of your host institution?

   I feel that Kyushu University is one of the leading and ever-growing universities in Japan. We have a lot of international students as well as several international faculty members, which makes it extremely vibrant. I was pleasantly surprised to see so many people of different nationalities and hear different languages being spoken when I first visited the university’s Ito campus. The campus is very big and surrounded by nature. It provides a very good environment for studying and doing research. Currently, the Hakozaki campus is moving to Ito campus, expanding it even further. This will definitely accelerate the growth of Kyushu University. Most particularly, the faculty, staff and other members of the mathematics department are extremely helpful and accommodating to me. Language has never been a problem. I enjoy my work and life in Japan thanks to them.

What is your impression of Japan’s research environment compared to India and the UK?

   I feel that the mathematics research environments in Japan, India and the UK are quite similar. However, the laboratory culture in Japan is rather unique. Professors have their own labs and put a lot of effort into supporting PhD students by, for example, organizing regular study groups for them and reviewing their presentations. The lab feels more like a family. In India and UK, I had not experienced this kind of labs, which hold regular seminars involving several groups under different professors. The research environment in the UK feels more independent; students tend to work on their own, occasionally meeting their supervisors. In my lab here in Japan, I really don’t feel any difficulties; there is quite a lot of freedom and no hierarchy at all. Regarding the language, most of my Japanese lab mates present their work in written English and they help me with translation when I do not understand something in spoken Japanese. I would like to see Japanese academic societies organize more seminars and conferences in the English language so that more foreigners can participate.

How have you advanced your research under the JSPS fellowship so far?

   In the classical theory of modular forms of integral weight, we have a very special subspace called “newspace.” Its definition involves using a certain inner product called “Petersson inner product.” In my research, we have been able to define a new family of operators on the space of modular forms and to characterize the newspace in terms of eigenspaces with specific eigenvalues of finitely many of these operators. Essentially, we were able to remove the Petersson inner product from the characterization. We would like to similarly define newspaces in the space of halfintegral weight modular forms in terms of the eigenspaces of certain operators. I am currently working towards accomplishing this. So far, I have had several opportunities to present my results, including four research meetings in Japan and a conference in Beijing.
   Also, my host researcher Prof. Kaneko is an expert in the theory of Multiple Zeta values, so besides my own research, I am also learning about MZVs. In mathematics one often finds quite similar results in different branches. There appears to be a certain unifying framework among them. Difficult problems have been solved using connections between theories in different fields. For example, the proof of Fermat’s last theorem relies on Taniyama– Shimura conjecture, which connects the world of modular forms to that of rational elliptic curves. As a mathematician, I want to be part of this mathematical unification process, so I am trying to understand many different theories. Mathematics has several applications in our everyday life, in particular number theory is applied to the field of cryptography; for example, elliptic curves have recently been used for encryption in mobile devices.

What do you think of life in Japan - its culture and customs?

   Japanese culture is a fantastic fusion of tradition and modernity. On one hand, you can see leading technology all around, such as a Softbank robot. On the other, Japan continues to maintain age-old traditions and customs like the admonishment “Hi no youjin” (Be careful not to cause a fire). Japanese culture seems to me to be conservative and full of intricacies, but it doesn’t take long to appreciate and adapt to this cultural blend. Another thing I really like about the Japanese culture is its harmonious way of living. Japanese people are very polite and accommodating. I also enjoy the cuisine here, the fruits and vegetables are extremely delicious. My life in Japan has been quite wonderful so far.

Before coming to Japan, what kind of image did you have of the country? Has it changed?

   When I first came to Japan, I had imagined that modern bustling cities like Tokyo were in almost every place, so I was captivated by the country’s beautiful countryside. Also, threefourths of Japan is covered by mountains and two-third by forests, so for me, as a person who likes to explore nature, it was a most pleasant surprise.
   Japan is known to have an extremely demanding work culture. After I started living in Japan, however, I felt that while people do work hard they also enjoy life a lot. People are very collaborative, giving the country a nice atmosphere for working and interacting with various people.

What do you do outside your research work?

   In my free time, I travel and do photography. Also, I am studying Japanese and like to explore the food and culture with my Japanese friends. I like most of the food here and enjoy its differences from Indian cuisine which uses a lot of different spices. Japanese food enhances the natural taste of its ingredients like in sushi and pot-cooked nabe cuisines. I really enjoy eating sushi and Fukuoka’s specialty ramen! Although my mother in India still doesn’t believe that I can eat raw fish! Well, I am not a fan of some foods with a strong smell like dried squid.

What do you plan to do after your fellowship ends?

   I would like to pursue research and teaching in mathematics, so I will be looking for an academic position in Japan or other place.

Please give some adv ice for young researchers who may be thinking about doing research in Japan?

   Japan is one of the world’s leading countries in technology and innovation, so it definitely provides top-class opportunities for research. Japan also has a tradition of beautiful co-existence with nature. The people are very polite and helpful and life is very comfortable. All in all, it will be a lifetime experience for you to stay and do research in Japan.

Throughout our interview with Dr. Soma Purkait, she confounded our perception of a typical mathematician as a person with a contemplative and subdued manner. Dr. Purkait was certainly contemplative but strikingly extroverted and she burst into a laugh whenever she found a chance. We were intrigued by her research, which delves into mathematical problems that seem like mystifying puzzles to us. We are very encouraged that she has been able to advance this perplexing research to the extent that she is being invited to present papers both in Japan and abroad. In continuing her work, Dr. Purkait said that she may seek a position in Japan after her JSPS fellowship ends next year. We hope that this will fit into her career equation, as she has created a wonderful relationship with her Japanese colleagues and has strong affinity for the Japanese culture and people. Not to leave out of the equation the “operator” that her husband Kiminori is Japanese.

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JSPS Quarterly No.54 2015