Pure Mathematics
from Keio's Faculty of Science and Technology In quest of truth that mathematical objects have

The fundamental motivation for mathematics lies in “Beauty”

Of the many academic disciplines, mathematics is one of the most indispensable as the foundation of our civilized society but is perhaps the least understood by people due to its breadth and depth. Particularly, the field known as pure mathematics – which is in the opposite end of practical sciences – is a world extremely hard to see and a world that only a handful of specialists can deal with. For this issue, we asked Dr. Takeshi Katsura about the attractiveness of and his approach to pure mathematics, and some of the theory of C*-algebras which is his specialty.

We sometimes hear of news bandy around like this, “A mathematical problem, which has been unsolved for centuries, has been solved at last!” But it is often the case that understanding how such a difficult problem was solved is extremely difficult even for specialists and can take years to examine whether the proof is correct. Such is a hardto-understand aspect of the world of pure mathematics. Then, how are mathematicians solving problems?
“Of course I admit that solving problems is the mainstream of endeavors for studying pure mathematics. On the other hand, in pursuing studies we mathematicians rarely experience solving a problem truly worth solving. This is because many of the problems that remain unsolved by humans are either totally worthless or ones too difficult to solve. It is also true that too difficult a problem to solve is not always well worth solving. Many scientists then scrutinize the value of a solved problem and its results over a period of many years. It can also happen that results of abstract pure mathematics studies are unexpectedly applied to practical uses after a century or two. That’s why I think we, researchers of pure mathematics, must establish and maintain values and an aesthetic sense of our own that are not affected by others and yet sympathized by them,” emphasizes Dr. Katsura.
If solving problems is very difficult as he says, then what are mathematicians advancing their own research for, and how?
“In the study of pure mathematics, I think much of our efforts are directed toward finding curious phenomena and understanding them, rather than simply solving problems and finding answers. To do so, we must thoroughly investigate the object, build a hypothesis and verify it through experiments. This approach seems similar to approaches taken in the engineering field. What characterizes pure mathematics is that our targets are abstract mathematical objects. Another characteristic is that we basically take an approach that we term ‘thought experiments’ although experiments often depend on computers.”
He continues, “No matter how acceptable an answer appears, it cannot be considered to be the ‘result’ unless it is proven properly. This is still another characteristic of mathematics. Suppose you have found a phenomenon to be proven but are unable to prove it. In that case you may publish it in the form of a conjecture, but even so it is regarded as an important research achievement in mathematics. Some day in the future, I’d like to publish a conjecture that would lead mathematical studies of the world. Of course, it would be best if I could prove it.”

How is it possible to investigate into mathematical objects that are both invisible and intangible? The keywords are sets and structures, according to Dr. Katsura.
“When asked about the most familiar mathematical objects, everyone will answer they are numbers such as natural numbers, integers and real numbers. In investigating into numbers, there is an approach: to collect and examine a set of all numbers that satisfy certain properties, instead of dealing with each individual number. In mathematics, we call a collection of such mathematical objects a ‘set.’ For example, a set of all natural numbers is represented by N, which is an infinite set. On the other hand, a set of all real numbers, R, is often expressed as a number line.”
“These sets of numbers naturally have their own ‘structures.’ For example, a number line as an expression of a set of all real numbers, R, is considered to express structures such as a ‘relation’ for two real numbers – which is large and which is small – and a ‘metric’ between the two numbers. Besides these structures, sets of numbers have a structure known as ‘operations’ that involves addition, multiplication and so on,” Dr. Katsura mentions.
He says it is possible to shed light on the very structures by describing these various types of structures in set-theory languages like mapping (*1).
“E ach time we notice a certain structure, the set having that particular structure becomes a research object in mathematics. For example, the structure ‘metric’ leads us to concepts such as metric spaces and topological spaces (*2), which are the objects of geometry. Likewise, the structure ‘operations’ involving addition and multiplication leads us to concepts such as “groups,” “rings” and “fields” (*3), which are the objects of algebra. In today’s pure mathematics study, it is common to look for instances of such mathematical objects and investigate into relationships between properties shared by such objects and those between different objects,” Dr. Katsura continues.
In particular, Dr. Katsura shows a special interest in and focuses on relationships between objects of different structures (ring, metric space, etc.) and objects having multiple structures (group, topological space, etc.) at the same time.
“Up to now I have focused on the object called ‘C*-algebras’ (*4). In addition to structures such as metric spaces and rings mentioned earlier, C*-algebras also have structures called linear spaces and ‘*’ (star) and satisfy various conditions (*5) among these structures. While intriguingly interrelating with other objects such as dynamical systems (*6), topological spaces and fields, C*-algebras are being studied even today. I personally regard C*-algebras as very charming, but I cannot explain it in compact, easyto-understand expressions. Because easy explanations often come with incorrectness and falsehoods, you know,” says Dr. Katsura smilingly.

What meaning does Dr. Katsura find in studying the highly abstract world of mathematics? “For me, the foremost motivation is that I feel ‘beauty’ in there. In fact, while I’m wrestling with a question, there happens to be a moment when I find a truth that I feel truly exquisite. In the 19th century, mathematicians discovered an extremely beautiful area of mathematics called complex analysis (*7). In the 20th century, this complex analysis played vital roles in mathematical description of quantum mechanics by von Neumann and others, the results of which have supported our modern society through semiconductors. In this manner, it is often the case that truly beautiful mathematics can find unexpected applications after many years. I’d like to leave such a significant achievement to posterity myself,” talks Dr. Katsura eagerly. He also expressed prospects that he would like to pioneer yet unexplored fields of mathematics by expanding his research objects beyond C*-algebras to cover boundary areas of dynamical systems, number theory and set theory studies.

Toward the end of my high school second year, I was chosen as one of the twenty Japanese candidates for the International Mathematical Olympiad (IMO). This event turned out to be an opportunity for me to find fun in pure mathematics for the first time in my life. Joining the screening camp, I was surprised to find a number of extremely bright fellows who were as well versed in mathematics as being able to see through to the point of the problem. Until then, I had never met students of my age who were better at mathematics than myself, which was a tremendous stimulus.
At that time, I was left out of the final selection of six representatives to my regret. However, in the summer of my third year of high school, I had an opportunity to participate in a camp mostly for IMO representative candidates. It turned out that some of the fellow students I met there would change the course of my life. They invited me to go to the University of Tokyo to study mathematics together, so I decided to go to the university instead of the nearby Kyoto University. Then I was admitted to the university’s Natural Sciences I, left for Tokyo and began living there alone.

Yes. Again at the invitation of those friends, I challenged the grade-skipping exam, an opportunity allowed only for juniors. All of us passed the exam, so we quit the undergraduate course halfway in the third year and advanced to the master’s course. While two of my friends chose the number theory, I joined the lab of Professor Yasuyuki Kawahigashi who specialize in the operator algebras theory.
At the Kawahigashi lab, we were required to speak without looking at notebooks. At my own lab, we are following suit with this style of learning. It was a tough practice in the beginning, but I soon found that it is a superb method to accurately understand the problem and nurture the ability to restructure the problem – far from compelling us to memorize things. Not limited to mathematics, training yourself in this way will be surely useful even after you go out into the world.
In the second year of my master’s course, I studied abroad at the Mathematical Sciences Research Institute (MSRI) near the University of California, Berkeley for a little less than a year.
Returning to Japan, I completed two years of doctoral course before getting married. Then I continued research activity at the Graduate School of Mathematical Sciences, the University of Tokyo in the capacity of a postdoctoral fellow (PD) of the Japan Society for the Promotion of Science (JSPS). During this period, I also experienced an extended stay in Oregon, U.S.A. Then, I had an opportunity to spend three enjoyable years of research at the Hokkaido University Graduate School of Science (Faculty of Science) as Superlative Postdoctoral Fellow (SPD) of JSPS. Those three years were fruitful as I could fly around the world and cultivate valuable human relationships, which became the foundation supporting my current research activities.
There is one more thing I gained during the years at the Hokkaido University. Inspired by the manga “Hikaru-no-Go,” I was awakened to the fun game of “Go”. Currently, I’m a third grade rank holder of Go.

Because securing a post as a mathematics researcher is highly competitive, one needs good timing and luck in addition to ability. When I completed the doctoral course, an increasing number of young people wanted to become researchers but the number of posts available was on the decrease. Naturally, they were faced with keen competition. In my case, following the service with the Hokkaido University, I had to serve as a PD at the University of Tokyo for half a year then at the University of Toronto. Since I got a preliminary offer for a post from Keio just before leaving for Toronto, I put an end to my six months of stay in Toronto much earlier than the initial plan. In April 2008, I reported to Keio University Department of Mathematics, Faculty of Science and Technology as an assistant professor. Because I was able to find employment with such a highly reputable university as Keio, I feel the five years of my PD period were not useless after all. After assuming a post with Keio, I also had a chance to study at the University of Denmark.
A good thing about Keio University Faculty of Science and Technology lies in that the relation between teachers and students is close, reflecting the Keio policy of “Half Learning, Half Teaching.” When it comes to study we have hot discussions, but enjoy sake together at other times in an at-home atmosphere, which I like very much. Unlike mathematics departments of other universities (Japanese or overseas) I know, Keio’s Department of Mathematics not only pursues pure theories of mathematics but also emphasizes application aspects of this study. Of our students, only a few become mathematics researchers after graduation. Their future courses are diverse, some becoming school teachers and others finding employment with businesses and so on. At Keio, interactions among teachers from various departments, such as labs specializing in statistics and computer sciences, are also active. Speaking of myself, I feel my field of vision has expanded considerably thanks to such an open culture.

Yes. I studied alone in the beginning. But while I was in Toronto, a set theory specialist, who had read my paper, visited my office and offered a joint research on a certain problem. It was a difficult problem left unsolved for 30 to 40 years. By joining forces, however, we could solve that problem. Since that time, I’m willing to call out to other researchers when I feel joint research approach would be better.

Student : I joined the Katsura lab simply because Dr. Katsura’s class lectures were overwhelmingly interesting, where I could image even difficult problems in an understandable way. It’s a tough requirement that we shouldn’t look at notebooks when speaking in seminars. But this rule helps me a lot in understanding things accurately and speaking logically in public.