## Message from the Director

RIMS Director
Takashi Kumagai

The Research Institute for Mathematical Sciences (RIMS) is an institute that conducts comprehensive research in mathematics and the mathematical sciences. Since being established in 1963, RIMS has been recognized as one of the world's leading research institutes in these fields. Mathematics is, of course, an important basic discipline located at the deepest part of human knowledge and forms the basis of science. It is a mystery of mathematics, and at the same time fundamentally accounts for its existence, that the structure of the real world can be captured profoundly and beautifully by pure mathematical thoughts.

In modern times, mathematics plays a central role in supporting science and technology. Especially
since the beginning of the 21st century, vast changes to society and industry have led to mathematics
gaining significant importance in creating new ideas, giving these foundations, and leading them to
realization ^{1} . However, mathematics has not solely been developed to meet the demands of applied
fields. Indeed, looking back in time, one can find many examples where the mathematics that is
widely applied for practical purposes today was created purely from a mathematical perspective. For
example, Galois theory, which was introduced about 200 years ago to address the solvability of
equations, supports the codes and ciphers of modern network communications, and non-Euclidean
geometry is applied to present-day GPS technology through the theory of general relativity.
Moreover, as a newer example, the theory of stochastic analysis, which was founded by Dr. Kiyosi Itô,
a director of RIMS in the mid-20th century, is widely used in mathematical finance. Through such
integration with applications, the cultivation of a deep and broad generality of mathematics has a
strong power to change the world we live in. On the other hand, applications of mathematics have
played a role in evoking new possibilities inherent inside mathematics and motivating further
progress in mathematics from diverse viewpoints. The pursuit of mathematics based purely on
mathematical interests and its applications to science can thus be compared to the roots and branches
of a tree. One cannot be established without the other, and the mutual support of the two leads to
the global development of mathematics.

Since its inception, RIMS has had the mission to comprehensively promote fundamental research in mathematics and research on the applications of mathematics to science. To this end, there are three pillars to the activities of RIMS. The first is research by faculty members. Over the half-century of RIMS existence, this has led to many globally-recognized achievements. Currently, RIMS has around 38 faculty members and is organized into three major divisions (Fundamental Mathematics, Infinite Analysis, and Applied Mathematics), as well as a Computer Laboratory, making it a flexible organization. In addition, RIMS has established the “Center for Research in Next-Generation Geometry” to strengthen research in this area, the “Center for Research Interaction in Mathematical Sciences” to promote international research collaboration, and the “Liaison Center in Mathematics” to promote collaborative research with researchers from a wide range of scientific fields. It has always been a fundamental policy of RIMS to maintain an excellent research environment where members can concentrate on research, with space being given so that researchers may pursue their mathematical endeavors above all else, and the research time of younger members in particular being highly valued.

The second pillar is the implementation of joint usage projects aimed at contributing to the research of a wide range of mathematicians and mathematical scientists. Since its foundation, RIMS has served as a national joint research institute, and since 2010 it has been certified as a Joint Usage/Research Center (JU/RC). At present, RIMS hosts around 80 RIMS Workshops annually, with a combined total of more than 4000 participants (of which about 400 are from abroad). RIMS Kokyuroku, a collection of reports on these projects, has now exceeded 2000 volumes, and the number of accesses of these exceeds 1.6 million annually (a little fewer than 40% being from abroad). Since November 2018, RIMS has been certified as an “International Joint Usage/Research Center”, which has enabled it to expand its joint usage projects and make international open calls. In this direction, RIMS is promoting the internationalization of its core projects, setting up international collaborative research support mechanisms and starting large-scale international collaborative “RIMS Research Projects”. In particular, for RIMS Research Projects, we select specific research themes, offer positions of project fellows, and pursue research progress through collaborations with leading international researchers, who are invited to make mid- to long-term stays at RIMS. Moreover, by establishing new projects that support individual-scale international collaborations, recruiting projects all year round, and establishing a framework to support young researchers, RIMS will continue to play a role in supporting research in mathematics and the mathematical sciences both within Japan and beyond.

The third pillar is the education of graduate students ^{2}. So as to foster young researchers, RIMS has
had a graduate school since 1970. Currently, it accepts 10 Master students and 10 PhD students
every year. Under the guidance of outstanding faculty members, students can learn contemporary
mathematics in an environment where leading researchers visit from all over the world and
collaborate on a daily basis. In this way, RIMS has produced many excellent young researchers.

In the activities of RIMS, the three pillars described above are not separate, rather they are a trinity that strengthens each other, which is extremely important for the development of the institute. From a practical perspective, one of the current challenges of RIMS is to secure building space that fits the variety of its activities in a cohesive manner, and efforts to realize this will continue in the future. The role of the institute is to provide a high-level research environment where researchers can discuss with each other actively and think deeply about mathematics. Through its activities, RIMS is making efforts to contribute to further developments in mathematics and the mathematical sciences in the 21st century.

Footnotes

- In the 5th Science and Technology Basic Plan (FY2016-2020), mathematical science is positioned as a "transverse science and technology that supports basic technology".
- For graduate education, we are conducting the KTGU (Kyoto Top Global University) project in collaboration with the Department of Mathematics, Graduate School of Science.