The Contributions of Japanese Mathematicians since 1950

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The Contributions of Japanese Mathematicians since 1950

Overview

The latter part of the twentieth century has been called the golden age of mathematics. Much of the luster of this period comes from the contributions of European, Chinese, and American mathematicians. However, to dwell entirely on their contributions does a disservice to the mathematicians from Japan because this era has certainly been a golden age for Japanese mathematicians, too. In the latter decades of the twentieth century, Japanese mathematicians made significant contributions to several fields of mathematics and helped launch the field of computational mathematics. In so doing, they not only helped expand knowledge of traditional mathematics, but helped enlarge the very boundaries of their profession by taking advantage of increasingly powerful computer technology. This, in turn, had important effects on mathematics and many aspects of modern society.

Background

Japan has long been accused of being a nation that can innovate but not invent. Critics, pointing to Japan's culture of conformity and traditionalism, suggest that the Japanese lack true originality and creativity, excelling instead at exploiting discoveries made by other nations. While this stereotype carries with it some degree of truth, it must be noted that Japan has produced a great many innovative scientists who have won acclaim for a number of important discoveries. The field of mathematics is no exception.

Japanese mathematicians came into their own in the years following World War II. Kunihiko Kodaira (1915-1997) was awarded the Fields Medal in 1954 for his outstanding contributions in harmonic analysis, algebraic geometry, complex manifolds, and other areas of mathematics. In 1970 Heisuke Hironaka won the Fields Medal for his work on algebraic varieties, and Shigefumi Mori (1951- ) earned the 1990 Fields Medal for his work in algebraic geometry. Other Japanese mathematicians who have won high honors for their original and important work in mathematics include Kiyoshi Ito, Kenkichi Iwasawa (1917-1998), Goro Shimura, Tosio Kato, and others.

Still other Japanese mathematicians made significant contributions to understanding one of the most important problems of the last 300 years: the proof of Fermat's last theorem. In this endeavor, Yutaka Taniyama and Goro Shimura helped lay the groundwork that led to Andrew Wiles's (1953- ) 1994 proof of the validity of Fermat's last theorem. The Shimura-Taniyama conjecture, which describes the relationship between certain families of curves, proved key to solving this theorem.

In addition to pure mathematical research, Japanese mathematicians have helped develop computational techniques that use computer technology to solve a variety of problems. In this endeavor they are building on the many contributions of Japanese computer scientists and electronics corporations. Of particular importance are the exceptionally sophisticated computational techniques to determine airflow around vehicles developed by Kunio Kuwahara and Susumu Shirayama. These simulations have been extremely useful in designing cars that are safer and more fuel efficient. In another application of using computers to solve mathematical problems, Yasusi Kanada has performed very interesting and important work in the areas of cellular automata and has used supercomputers for a wide variety of mathematical research.

Impact

Andrew Wiles's proof of the solution to Fermat's Last Theorem stands as one of the most important mathematical works of the twentieth century. Because it had been unproven for three centuries, its proof won Wiles worldwide acclaim. It is entirely likely that this problem would not have been solved without the work of Shimura and Taniyama on elliptic functions.

These mathematicians showed that a class of curves called elliptic curves were related to another class of curves known as modular curves. When this association was made, other aspects of Fermat's Last Theorem fell into place. It took Wiles seven years of hard work to take these observations and turn them into a rigorous mathematical proof, but he did just that, presenting his proof to the world in 1993. Since that time, work on the Shimura-Taniyama conjecture has continued and seems to indicate that other, deeper aspects of mathematics may be explored using this relationship. If so, then the Shimura-Taniyama conjecture will prove to be much more important than suggested even by its use in proving Fermat's last theorem.

It is also interesting to note that Fermat's last theorem is one of the few mathematics problems to capture widespread attention in the media and among the general public. From this perspective, too, the work of Shimura and Taniyama, while not subject to the same acclaim as that of Wiles, helped to raise public awareness and appreciation of mathematics and mathematicians, even if for only a short time.

Other mathematicians also made singular and lasting contributions to the study of pure mathematics. The work of Shigefumi Mori seems to have been of particular importance because his outstanding mathematical creativity clearly disproves the preconception that Japanese scientists lack creativity. In fact, a fellow mathematician said of Mori's work that "the most profound and exciting development in algebraic geometry during the last decade or so was the minimal model program or Mori's program in connection with the classification problems of algebraic varieties of dimension three. Shigefumi Mori initiated the program with a decisively new and powerful technique, guided the general research direction with some good collaborators along the way, and finally finished up the program by himself overcoming the last difficulty." For this work, Mori was awarded the 1990 Fields Medal, mathematics' highest honor.

Impressive as these accomplishments are, other Japanese mathematicians also made significant contributions in the use of computers to solve mathematical problems. In 1988 Yasumasa Kanada used a computer to calculate the value of π to over 201 million decimal places. While such calculations may not seem terribly important or relevant to everyday life, they are not only an important test of a computer's power, but also help to demonstrate efficiency and effectiveness of new computing techniques. In effect, they may be used as a sort of computational (or programming) test drive.

Kanada also worked in the area of cellular automata (CA or "computer life"), small programs that interact within a computer under certain rules governing their behavior. Researchers, including Kanada, noticed that CA tend to build complexity when left to their own devices, constructing increasingly complex systems of interactions. In this, they seem to mimic what we see in the real world, where life has built a very complex biosphere from the relatively simple building blocks that existed in the early earth. One of Kanada's insights into CA mathematics has been to introduce noise—random variations—into his scenarios. This makes them more realistic, giving deeper insights into the phenomena under study. For example it might be possible in a computer program to stand a baseball bat on top of a baseball because, if the forces are aligned correctly and no outside influences are permitted into the program, the bat will balance indefinitely. However, we know that this does not happen in the real world, so such a model is inaccurate. Kanada's programming would include random fluctuations in such a problem, so that in this analogy the bat would tumble to the ground because, in real life, imperfections in the covering on the ball, small gusts of wind, or spare vibrations all occur and cause the bat to fall. By introducing similar random variations into systems of cellular automata, Kanada is able to make them more closely resemble real-world systems.

Finally, we must mention the contributions of Kunio Kuwahara and Susumu Shirayama as examples of similar work performed by many other Japanese scientists. Kuwahara and Shiryama have taken a leading role in the use of computers to solve mathematical equations governing complex physical phenomena. These equations cannot be solved analytically (that is, by plugging numbers into the equations). Their most impressive achievements have been in the area of fluid flow, specifically around automobiles, in which they examined the various factors that cause drag and reduce fuel efficiency. The basic equations that govern fluid flow, the Navier-Stokes equations, are well known, but are also very difficult (possibly impossible) to solve analytically. Kuwahara and Shiryama have translated mathematical algorithms into computer code that allow these complex equations to be solved numerically at a large number of locations in space. They then program the important parameters of the automobile to be tested (in effect, telling the computer what the car looks like) and then simulate releasing smoke into mathematical streamlines when the car is put into an electronic "wind tunnel." These models, then, are used to verify actual airflow characteristics around the proposed new car. By helping design fuel-efficient cars, much of this work helps to save money and natural resources as well.

P. ANDREW KARAM