Becoming a Mathematician
Taketo Sano, April 2023
Taketo Sano, April 2023
This article is a translation of a piece originally contributed to the April 2023 issue of 数学セミナー (Mathematical Seminar), published with permission from the editor.
What makes someone a mathematician? There are many possible criteria: earning a Ph.D. in mathematics, proving some theorem and writing a paper, making mathematical research one’s life’s work, and so on. As for myself, I still hesitate to call myself a mathematician; honestly, it feels more accurate to say that I am one of those who aspire to become one.
The title of this article, “Becoming a Mathematician” was suggested by the editor. On this theme, the only thing I can really write about is my own experience: after taking a very long detour, I found myself aspiring to become a mathematician just before turning forty. It is not a graceful story, but I hope it may be of some help to students who feel uncertain about pursuing research, or to those who are struggling with how to relate to mathematics.
When people hear the word mathematician, they often imagine someone who showed extraordinary mathematical talent from early childhood, or someone with exceptional calculating ability. I was not such a child. I always found it hard to accept things taught at school simply because they were presented as “correct,” and I felt that my teachers did not seriously engage with the questions that arose in my mind. But Mr. Kimura, a math teacher at the juku (a private after-school tutoring school) I attended, dealt sincerely even with a troublesome child like me, and taught me both the ideas behind mathematics and the enjoyment of it. I remember always looking forward to going to his classes.
Even after moving on to junior high and high school, I never came to like school itself, but at the science-focused juku I began attending in my third year of junior high, I always enjoyed the mathematics lectures. If I went to ask questions after class, the teachers would carefully dig into each point I did not understand and resolve it one by one. In mathematics, there is always a reason why something is “correct,” and if you trace things back one step at a time, that correctness is always made understandable. Through experiences like these, my trust in mathematics and in math teachers grew, and I began aiming for university entrance with the thought, “I want to be able to teach mathematics too.”
Once I entered university, I took various liberal arts courses, but mathematics was still what I found most interesting, so I advanced to the mathematics department without hesitation. However, once I entered the specialized program, the level of difficulty of the lectures rose sharply, and little by little I could no longer keep up. Interesting things were being discussed right in front of me, yet I could not understand even the basics. I also had little interaction with my classmates and did not participate in student-led seminars. Even if I were to go on to graduate school, I could not envision either a future in which I lived as a researcher or one in which I took a job making use of mathematics. These anxieties about the future began to wear me down, both mentally and physically.
At the end of the summer of my fourth undergraduate year, I took the graduate school entrance exam and barely managed to pass, but I was not positive about continuing on. Around that same time, a senior from the computer club I had belonged to in junior high approached me about starting a company together. Once I actually got involved, I found it challenging and interesting, so I decided to leave mathematics behind and do my best there instead, and I declined admission to graduate school.
I started a company with friends in 2006, and after that I lived as a software engineer with almost no connection to mathematics. Later I changed jobs twice, got married in 2012 (at age twenty-eight), and in 2014 (at age thirty) my child was born. Having a family gradually helped me come to terms with parts of my past that I had avoided facing, and I became able to think about how I should live from then on.
Around 2013, as AI and machine learning began to attract serious attention in the software industry, I sensed a growing interest among the programmers around me in the mathematics underlying those fields. At the company where I worked at the time, a senior colleague asked me to hold an in-house mathematics study group, and it was well received. I began to think that perhaps, as an software engineer with a background in mathematics, there was more I could contribute.
In January 2015, I began voluntarily organizing a public study group called “Mathematics for Programmers” (Pro-Su). It was a general-interest event in which several speakers would present interesting mathematical topics connected with programming. Each time, the number of people who wanted to attend far exceeded the capacity, and I was surprised to realize just how many people loved mathematics. At the same time, with each meeting my desire to do more mathematics grew stronger. Until then I had had no one around me with whom I could talk about mathematics, so it felt as though I had suddenly been freed from loneliness.
In May, when I held the third Pro-Su meeting, I resolved to enter graduate school once again and applied for that year’s entrance examination. In the gaps between work and childcare, I began reviewing material starting from first-year undergraduate mathematics. This decision and action were driven by the impulse that I could no longer keep suppressing my feelings toward mathematics. At the same time, I was also supported by the conviction that now, when the importance of mathematics was being reassessed so strongly, relearning it would not be wasted effort.
I took the graduate school entrance exam in September, and two weeks later I received notice of my acceptance. The joy I felt then was incomparably greater than the first time. In April 2016, I left the company where I had been working and entered the master’s program at the Graduate School of Mathematical Sciences at the University of Tokyo.
As my supervisor in the master’s program, I hoped to study under Professor Mikio Furuta. I had decided to go on to graduate school simply because I wanted to return to mathematics, so I had not yet decided what I wanted to research. The reason I chose Professor Furuta was that when I spoke with him in an interview before entering, I felt something like overwhelming power from him.
As I had hoped, I was assigned to Professor Furuta’s lab, and the graduate seminars began in April. That first year was harder than anything I had ever experienced in my life. The senior students and my classmates in the lab were doing seminars at a much higher level, and I could not understand the content at all. Week after week, I seemed to be exposing my lack of understanding while still being shaky even on undergraduate-level knowledge, and the small pride I had acquired as a working adult came crashing down. This did not change even in my second year. Realizing that at this rate I would not be able to write a master’s thesis, I decided before summer to extend my enrollment by one year. After making that decision, I was finally able to settle down and devote myself to studying.
At the beginning of 2018, near the end of the second year of the master’s program, my professor told me about a knot homology theory called Khovanov homology. When I looked into it, I realized that by slightly extending a program I had written as a hobby for computing homology groups of simplicial complexes, I might also be able to compute Khovanov homology. I immediately wrote the program and carried out computational experiments on various concrete examples. Then I noticed a mysterious phenomenon: in the component expressions of a certain homology class, powers of 2 always appeared. My research began with investigating this phenomenon in detail.
Around the same time, I met Kouki Sato (now an assistant professor at Meijo University). Sato specialized in knot theory and was scheduled to come to the Furuta lab as a postdoctoral researcher starting in April. Beginning in March, he attended my seminar every week and spent a great deal of time discussing everything with me, from the basics of knot theory to matters directly related to my research.
Once my research topic was set and I had met a senior researcher I could rely on, my research advanced all at once, and by around summer I had obtained one main theorem. By the end of the year I had completed a draft of my master’s thesis, and I was also able to present the results at a workshop on knot theory. In January 2019, I submitted my master’s thesis; in February, I underwent the thesis defense; and in March, I was able to complete the program successfully.
At the point when I entered the master’s program, continuing on to the doctoral program had not seemed realistic to me. But by the time I finished writing my master’s thesis, my desire to deepen the research further had grown stronger, and with my family’s agreement, I decided to advance to the doctoral program.
In my first year of the doctoral program, a paper based on my master’s thesis was accepted for publication, and I was also selected to become a JSPS Research Fellow (DC2) starting the following academic year. In my second year, just as I was excited to finally devote myself fully to research, the global spread of COVID-19 completely changed the situation.
In April 2020, a state of emergency was declared, the university was closed, my wife’s company shifted to remote work, and the kindergarten my child attended was temporarily shut down. Under circumstances where all three of us were at home, it was impossible to make progress on research. Thinking that this situation might continue for a long time, I felt drained by the possibility that I might not be able to complete the doctoral program within the remaining two years. For the time being, I decided that my family’s health would be the top priority, and that even if my research progressed slowly, my goal would be simply not to stop moving forward.
By June, kindergarten had reopened, and our family was gradually regaining its normal life. In July, I renewed my determination: while I was still able to work, I needed to produce results, because I had no idea when I might be halted again. I concentrated on my research. In about a month, I completed a new project based on the results of my master’s thesis, and then I began working on the next project, based on a paper I had been reading since April. This, too, progressed according to the ideas I had in mind, and by around October the broad outline of the next paper had taken shape. These two papers would become the first and second halves of my doctoral dissertation.
For the two years after the pandemic began, until I completed the doctoral program, seminars and research meetings were conducted almost entirely online. The move online greatly reduced constraints of place and time, so for me as someone with a child there were advantages. But even so, the loss of dense, face-to-face exchanges of ideas was a major one.
Even in my third year, my days passed with almost no direct contact with other people, but my research progressed more or less steadily, and in March 2022 I was awarded a Ph.D. in Mathematical Sciences. I owe my having come this far to the support of many people. I am deeply grateful to Professor Furuta, to the teachers who taught me the joy of mathematics, to the members of the lab who accepted me as one of their own, to the friends and companions who supported my return to study, and to my family, who always understand and encourage me.
Since April 2022, I have continued my research in topology as a Special Postdoctoral Researcher at the RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS). My first year as a researcher is about to end, and the days since April have been very fulfilling. At iTHEMS there are not only mathematicians but also researchers from a wide range of fields, including physics and biology, and we enjoy interacting every day while sharing our knowledge with one another. As for research, I worked with Kouki Sato on a project developing my master’s thesis, and we were able to complete the paper in a far more satisfying form than the previous one. In the summer, I traveled to the U.S. for research and was able to meet researchers there who were already aware of my work. This strengthened my desire to continue advancing my research and to connect with the world through mathematics.
Looking back on the path that has brought me to where I am now, I tried once again to think about what it is I like about mathematics.
(1) Certainty
In mathematics, you can think through something you do not understand thoroughly, until you are satisfied. Once you come to understand something clearly, the landscape before you changes dramatically, and you become able to explain it to others almost as if you were speaking about yourself. Truth is equal for everyone; it does not reverse itself depending on a person’s attributes or differences in values.
(2) Creativity and diversity
There are no restrictions on how one engages with mathematics. One may carefully build up logic, work tirelessly by hand through calculations, imagine things in one’s head, draw pictures and contemplate them, or exchange ideas while talking with others. The process is creative, and it is surprisingly diverse, with methods that are characteristic of each field and each researcher.
(3) Exploration of the unknown
I feel a sense of awe toward the lineage of mathematical research—a body of knowledge developed by mathematicians, from before the Common Era to the present day, across times and places, driven by curiosity and the spirit of inquiry. At the same time, I feel a kind of romance in being able, as one researcher among them, to participate at the front line in exploring the unknown. Just as I sometimes gain hints by reading papers written decades ago, perhaps future mathematicians may one day read my papers and receive something from them.
Back when I was an undergraduate and struggling over whether I should continue with mathematics, I suffered under the persistent question, “Do I really love mathematics?” Now I am free from that suffering, but not because I found the answer that “yes, I love mathematics.” Behind that question lay the assumption that “if it is something you truly love, then you should be able to overcome any hardship for it.” But in my case, I could not have returned to mathematics unless three conditions were met: freedom from anxiety about survival, companions with whom I could study mathematics, and a family. I was not strong enough to live on the feeling of loving mathematics alone. I now accept that, in order to understand this from the bottom of my heart, I needed to take a long detour.
Mathematics is difficult and vast, so it is easy to become impatient when your studies do not progress. This is especially true when you are a student, time is limited, and there are many talented students around you. I struggled with this problem for a long time as well, but recently I came up with one solution: make it your goal to continue for a long time. Even if something does not feel like it can be mastered in a year, what if you allow yourself three years, five years, or ten? Of course, there will be situations in which you must produce results within a short and fixed period of time. Even then, please do not burn through all your passion for that alone. Even if it means taking a detour, I hope you will protect your mental and physical health, continue your long relationship with mathematics, and enjoy it along the way.
Thank you very much for reading to the end.