Story

Everything about the universe is written in the language of mathematics

　Motoko Kotani advances research in theories of discrete geometric analysis, which deals with structures made of discrete geometric objects. She also studies crystal lattices and conducts research connected to materials science. In 2005, she was awarded the 25th Saruhashi Prize, an award given to female scientists who have achieved outstanding research results in the field of natural sciences. Not confined to the field of pure mathematics, she has collaborated with people in other scientific disciplines, including materials science, and has also been involved in the formulation of national science and technology policies. At RIKEN, she serves as Director of the Pioneering Research Institute, which pursues challenging R&D unconstrained by the boundaries between basic and applied science. This article traces her career as a scientist.

　Since early childhood, I’d had a vague desire to be a scientist, specializing in mathematics, which is related to the basis of the universe. In middle and high school, I was the kind of student who constantly asked teachers questions about anything that seemed unclear to me. At some point, I noticed that if students asked the right questions, math teachers were willing to discuss them logically (whether they were right or wrong) on an equal footing. This was probably made possible by the nature of mathematics, a discipline not based on observations or experiments but on thoughts.

　As the saying goes, if ever there is the book of the universe, it will be written in the language of mathematics. Mathematics is the common language for science. Mathematics is the study that explores the basic principles for understanding natural and social phenomena. I felt that I wanted to engage with the basis of the universe because we can only live once.

　Mathematics is broadly divided into pure math and applied math. Pure mathematics consists of algebra, geometry, and analysis. Geometry is further divided into topology, known as “soft geometry,” and differential geometry, known as “hard geometry.” Topology takes a global perspective, while differential geometry studies shapes precisely from local information. The important point is to clarify the relationship between these two.

　Since the time when I was a master’s student, I have specialized in geometric analysis, which applies analytical techniques to differential geometry. In other words, it describes what an object is shaped like or how it moves using differential equations when studying the shape of an object.

　Think of a flat metal plate, for example. If a heat source is placed somewhere on the plate, the heat spreads through the plate. If the plate is not flat but bent, the heat may spread in a different way. The shape of the plate is related to the diffusion of heat. Conversely, by studying the heat distribution, that is, by solving differential equations, you can find out the shape of the object. This is geometric analysis.

　One of the basic objects of geometric analysis is a soap film. Dip a wire frame into a soap solution and lift it, and a soap film forms across the frame. Take a close look at the film, and you will see that it is stable with the least surface area. However complex the shape of the wire may be, nature immediately finds a surface with the minimum area to form a film. This is the law of nature. Such a surface, formalized mathematically, is called a minimal surface and expressed as an equation.

　There are many beautiful symmetrical shapes in nature, such as honeycombs, snow crystals, water drops, and ripples. However, nature does not necessarily pursue symmetry, but rather the state in which the energy of the whole system is minimized, that is, the shape that minimizes energy. But why do highly symmetrical shapes arise as a result? Furthermore, why do we feel that such shapes are stable or well-balanced?

　Mathematically, an energy-minimizing state can be expressed by an equation called a harmonic map. My research was to determine mathematically what symmetries or geometric properties are exhibited by the solutions to harmonic maps.

　To treat geometric objects, including harmonic maps, as mathematical equations, a local coordinate system needs to be introduced. Such a shape that can be described as Euclidean space locally and that has a smooth structure overall is called a manifold. My research subject up to the 1990s was to abstract the structures of shapes seen in nature as mathematical objects and to investigate the nature of the harmonic maps on a manifold, in particular, the symmetries and geometric characteristics of the solutions.

Even non-smooth shapes have hidden curved surfaces: Toward the emergence of discrete geometric analysis

　Describing a manifold using coordinates resembles drawing a map of an area on the Earth. Points on the map correspond to points on the Earth, and lines of latitude and longitude provide the coordinates of the area drawn on the map, making it possible to examine the surrounding features of the area. The features on the Earth that lack coordinates can be examined in detail by comparing them with a human-made map that has coordinates. However, there are some shapes and substances whose structures are difficult to describe using coordinates. Solving this problem using geometry became Kotani’s next challenge.

　In the late 20th century, geometry based on manifolds made great progress, and geometric analysis significantly deepened our understanding of smooth shapes. In light of these developments, toward the end of the century, the international mathematical community became increasingly interested in “discrete” objects.

　Until then, geometric analysis dealt with smooth shapes that could be described using coordinates. Around 2000, interest began to grow in whether geometric analysis could also be applied to shapes that are not smooth—those with pointed sections or singular points, in other words, shapes for which differential equations cannot be written—or on discrete objects. This movement toward a discrete version of geometric analysis led to discrete geometric analysis.

　I had the opportunity to spend time as a visiting researcher at the Max Planck Institute for Mathematics in Germany from 1993 to 1994 and at the Institut des Hautes Études Scientifiques (IHES) in France in 2001. While being inspired by scientists from around the world, I began to try to add a new dimension to my research. In particular, at IHES, I worked on establishing discrete geometric analysis, initiated with Toshikazu Sunada, Professor Emeritus at Tohoku University, in 2000.

　With Professor Sunada, I studied the question of how geometric structures relate to the long-term behavior of a random walk. A random walk is a fickle movement in which the direction of movement is determined randomly at every step. It seems haphazard, but the global characteristics of the space can be captured without bias in the long term.

　I was immersed in understanding geometric quantities that appeared in the long-term behavior of random walks on discrete spaces in response to Professor Sunada’s questions. I recall being really happy because I could leverage the harmonic map research that I had been doing since I was a master’s student.

　Currently, I am considering developing discrete geometric analysis, which is the discrete version of geometric analysis, into something that is not merely a formal discrete version of geometry, but also something in which discrete objects lead to corresponding continuous objects in the limit.

　For example, suppose a net is placed over a curved surface. The net is a discretization of the curved surface. A discrete geometry, that is, a set of discrete geometric objects, is the net without the curved surface. If the curved surface is approximated by discrete objects, it resembles a curved surface covered with a net. The finer the net is, the closer it is to the original curved surface, and ultimately it becomes the original surface. I want to find a curved surface discretization that results in these objects when only discrete objects are given and there is no curved surface. In other words, if a curved surface is hidden behind discrete objects, I want to find a mathematical technique for identifying that surface. Discrete geometric analysis seeks ways of finding a continuous object that is hidden in discrete data.

　This new scientific field, called discrete geometric analysis, has attracted many scientists. For instance, immediately after returning from France, I had the opportunity to organize the first international research conference of the Japan Association for Mathematical Science, founded by Professor Heisuke Hironaka, on geometric analysis.

　Another opportunity was an international workshop series called the Mathematical Society of Japan’s Seasonal Institute (MSJ-SI). “Geometry meets probability theory” was chosen as the subject of the first MSJ-SI, and I was appointed to chair the organizing committee. Both were held to gather seemingly related scientists from across the world and have deep discussions at a time when little was known about what kind of research discrete geometric analysis involves.

　In the world of mathematics, if there is a bud likely to grow into a new field, it is effective to gather together those interested and discuss things thoroughly before the branches or leaves sprout. Through such discussions, the scope expands and gradually converges into an independent discipline. This is a very exciting experience.

Linking discrete objects with a continuous object led to collaboration with materials science

　Discrete geometric analysis is a new area of geometry. The new field fascinated Kotani, but it was more than just pursuing beauty confined to the world of mathematics. She found that she could do something more interesting by working with other fields and expanding her interests.

　I participate in various committees at the university as a faculty member. In the intervals of meetings, I chat with people sitting next to me, asking about their research interests, and they are often specialized in materials science. This may be a characteristic of Tohoku University.

　As I got to know these people and talked with them, I found that what we explored in math often matched the interests of materials scientists. I realized that state-of-the-art knowledge in math was unexpectedly in demand in other research fields.

　Just then, a collaboration between mathematics and other fields was selected as an area for a Core Research for Evolutionary Science and Technology (CREST) program of the Japan Science and Technology Agency (JST). Mathematics had never been selected as an area of JST programs before, and I think it reflected the expectations for mathematics by other scientific fields. To meet such expectations, I suggested a research topic, “Clarification of materials creation and property emergence proposed by discrete geometry.” Luckily, my suggestion was adopted.

　Materials science focuses on the functions of materials. Materials are considered a hierarchical network. If the relationship between the hierarchical levels is clarified, it will help clarify the structure of a material with a target function. The idea of linking different levels was my key subject of discrete geometric analysis, which is to relate discrete objects to a continuous object. I thought this was the precise moment when mathematics should come in.

　The materials we deal with are made of atoms. Today, we can observe atoms and molecules and manipulate them to create materials. If you consider that atoms are discrete and tangible materials are continuous, then you will see the role of discrete geometry. This was the challenge I decided to pursue.

　Tackling the problem of linking discrete geometry with materials science led to an offer to join Tohoku University’s Advanced Institute for Materials Research (AIMR), which was established in 2007 as part of the World Premier International Research Center Initiative (WPI). The colleagues were all top-level scientists from all around the world in materials science, metallurgical engineering, physics, chemistry, life science, and various other fields. The aim of the institute was to create new materials science by cross-disciplinary efforts of scientists in different fields gathered in one place.

　Since mathematics is a common language for science, it can serve as an interpreter between different fields. For this reason, coupled with the rise of materials informatics, which increases the speed and efficiency of exploring new materials using AI, data analysis, and other techniques of information science, a mathematician was chosen to lead the institute, even though it was a materials science institute, and thus I was appointed director in 2012.

Establishing a system to foster RIKEN’s pioneering spirit

　Kotani became a member of the RIKEN Advisory Council (RAC) in 2016 and served as an Executive Vice President of RIKEN from 2017 to 2020. She has been involved with RIKEN as a Visiting Senior Researcher in the Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS) since 2018, and as Director of the Mathematical Application Research Team since 2024. In April 2025, she assumed the role of Executive Director of Science (Pioneering Science Domain) and Director of the Pioneering Research Institute (PRI). How does Kotani envision further developing RIKEN’s pioneering spirit?

　RIKEN’s spirit is grounded in the principle of pursuing only ambitious and original research, not research that can be conducted elsewhere or research that can be achieved merely by investing sufficient time and funding. This spirit is embodied by the Chief Scientists at the PRI.

　Today’s scientific research, compared with the past, involves a larger number of people, large budgets, and large-scale facilities. There is a limit to the scope of research a scientist can do alone. To pursue large-scale research, scientists must work with other scientists. However, digging deep into subjects based on the curiosity of each individual scientist is still the root of all research. We need an environment in which different individuals gather together, exchange ambitious ideas, and develop innovations. Clusters of small studies emerge, and they eventually grow into a research field—that is the role the PRI is playing.

What should science do to build an ideal society?

　It was once thought that mathematicians were people confined to the world of mathematics. This notion is now changing, as some say that mathematical knowledge helps stimulate other fields and that mathematics could boost the innovation. Kotani has also participated in drawing up the Science and Technology Basic Plans as a member of the Council for Science, Technology and Innovation (CSTI; former Council for Science and Technology Policy).

　I became a member of the CSTI while the 4th Basic Plan (FY2011 to 2015) was underway and participated in drawing up the 5th (FY2016 to 2020) and 6th (FY2021 to 2025) basic plans. Up to the 4th period, the basic plans aimed to promote science and technology, but in the 5th plan, drastic changes were made. The plan was formulated in terms of how science and technology should be developed to build a better society for people.

　Of course, the need to develop science and technology is a self-evident truth for us scientists, but is it really important for Japan? We started by considering what would be necessary not only to grow the economy but also to build a society in which people can live happily. The 5th Basic Plan includes the concept known as Society 5.0.

　There was a debate over whether it would really be beneficial for people to promote digitization and AI. In order to consider how science and technology should be used in practice in society, we must question what an ideal society should be like. What is important for people is well-being—that is, the state of being fulfilled physically, mentally, and socially—and thus we should build a society that ensures our well-being. This is what science and technology are for, and this concept will continue to be important, without doubt.