On November 2018, RIMS was certified as one of the International Joint Usage/Research Centers by the Ministry of Education, Culture, Sports, Science and Technology (MEXT).

Request regarding Research Paper Acknowledgements

Childcare Services during Joint Research Activities
It is possible to set up a temporary nursery on campus during research meetings for the use of researchers with children.
For details, please contact the organizer of the meeting you are attending or the Cooperative Research Service Section at RIMS.

List of RIMS Research Projects


The elucidation of the way in which the additive and multiplicative structure of the integers are intertwined with one another is one of the most important and central themes in number theory. In August 2012, Shinichi Mochizuki (the proposer and chief organizer of the present RIMS Research Project) released preprints of a series of papers concerning "Inter-universal Teichmüller Theory", a theory that constitutes an important advance with regard to elucidating this intertwining. Moreover, the proof of the "ABC Conjecture", which follows as a consequence of the theory, attracted worldwide attention. In the roughly nine years since the release of these preprints:

・The four papers were published in the international mathematical journal PRIMS after undergoing a roughly seven and a half year long review.

・A preprint of a joint paper by five authors in which various numerically effective versions of the inequalities that appear in the theory was recently released.

・The number of researchers who have already acquired a thorough understanding of the theory, as well as advanced learners of the theory, has, as a result of the diligent efforts of many of the researchers involved, increased slowly, but steadily.

・Quite a number of surveys and related expositions of the theory (6 of which have been published or accepted for publication; another survey has been released, but remains unpublished) have been written, not only by the author of the theory, but also by researchers who have already acquired a thorough understanding of the theory.

・Although it is difficult to ascertain the precise number, at least on the order of 30 lectures and small-scale workshops on the theory have been conducted all over the world (in Japan, the UK, Russia, the US, China, Germany, and France).

・At least 4 large-scale workshops (of one to two weeks in length) on the theory have been conducted not only within Japan (in Kyoto, March 2015 and July 2016), but also in China (in Beijing, July 2015) and the UK (in Oxford, December 2015). Moreover, a long-term online (Zoom) workshop involving participants mainly from Japan, France, and the UK was conducted during the period September 2020 ~ April 2021.

As a result of these activities, a sort of "inter-universal Teichmüller theory community", consisting of between ten and twenty researchers, is currently in the process of forming. Moreover, as a result of advances in research, such as combinatorial anabelian geometry, based on ideas closely related to the ideas that underlie inter-universal Teichmüller theory, important links between research on inter-universal Teichmüller theory and research concerning the Grothendieck-Teichmüller group and the absolute Galois group of the rational numbers have begun to form.

In light of these developments, the present RIMS Research Project seeks to bring together various researchers not only from the "inter-universal Teichmüller theory community", but also researchers interested in various forms of mathematics related to inter-universal Teichmüller theory, and to provide all such researchers an opportunity to engage in lively discussions concerning the various developments discussed above in an environment in which interaction for periods on the order of months is possible, that is to say, unlike the situation in the case of a single workshop (i.e., which typically only lasts for roughly a week).

Since the 9th MSJ-IRI 2000 “Integrable Systems in Differential Geometry”, (link http://www.math.tsukuba.ac.jp/~moriya/iri-j2.html), several scientific activities in the area of differential geometry and integrable systems have been carried out extensively and intensively. There has been remarkable progress in differential geometry based on theory of harmonic maps of Riemann surfaces into symmetric spaces and integrable system methods: The DPW(Dorfmeister-Pedit-Wu)method via loop groups and its applications to geometric analysis of surfaces, integrable system approach to the constrained Willmore conjecture; differential geometry of discrete surfaces and discrete geometric analysis; special geometry of minimal submanifolds and their moduli spaces; isoparametric submanifolds of finite and infinite dimensions; Floer homology of Lagragian submanifolds in homogeneous Kähler geometry; special geometry related to infinite integrable systems, Higgs bundles and mirror symmetry; fusion of non-linear PDE methods and integrable system methods based on symmetry of differential equations,and so on.
This research project intends to cultivate new areas of“Mathematics of Symmetry, Stability and Moduli”by enhancing and expanding such research fields of differential geometry and integrable systems and encouraging activities of young researchers. Franz Pedit (UMASS Amherst,USA), Chikako Mese(Johns Hopkins U.,USA),Eric Rains (Caltech,USA),Fernando Codá Marques (Princeton,USA),Jaigyoung Choe(KIAS,Korea)and others will be invited as mid-term to long-term foreign RIMS visiting professors or international leading researchers. Throughout this academic year we conduct activities such as international workshop,special lectures,joint research,satellite seminar and so on from the viewpoints of geometry of submanifolds and integrable systems, geometric PDE and variational problems,mirror symmetry and its applications to differential geometry. A major internatonal conference “Differential Geometry and Integrable Systems”(MSJ-SI) will be held at the end of this project and we intend to greatly output new research results and to educate young researchers widely. The agreement of academic cooperation between RIMS and OCAMI, which was concluded in 2007, will be also used to promote this project.
  • Applications of Harmonic Maps and Higgs Bundles to Differential Geometry (RIMS Satellite Seminars)(Closed seminar)

    Location: Seapal Suma    Period: Canceled
    Organizer: Franz Pedit(The University of Massachusetts Amherst,USA)

  • Generalized Hitchin Systems,Non-commutative Geometry and Special Functions (RIMS Research Project 2021)【RIMS Review Seminar】

    Location: Online via Zoom    Period: 2021-11-16〜2021-11-25
    Organizer: Yoshihiro Ohnita(Osaka City University Advanced Mathematical Institute)

  • Special Geometry, Mirror Symmetry and Integrable Systems (RIMS Research Project 2021)【RIMS Review Seminars】

    Location: Online via Zoom    Period: 2021-11-29〜2021-12-02
    Organizer: Yoshihiro Ohnita(Osaka City University Advanced Mathematical Institute)

  • Symmetry and Stability in Differential Geometry of Surfaces(RIMS Research Project 2021)【RIMS Review Seminar】(partly open)

    Location: Nagoya University    Period: 2022-02-14〜2022-02-16
    Organizer: Yoshihiro Ohnita(Osaka City University Advanced Mathematical Institute)

  • International Workshop on Geometry of Submanifolds and Integrable Systems (RIMS Research Project 2021)【RIMS Symposia】(Open Symposia)

    Location: Hybrid Meeting    Period: 2022-02-20〜2022-02-23
    Organizer: Yoshihiro Ohnita(Osaka City University Advanced Mathematical Institute)

  • Differential Geometry and Integrable Systems (RIMS Research Project 2020)【RIMS Symposia】(Open Symposia)

    Location: Sugimoto Campus, Osaka City University    Period: 2022-03-01〜2022-03-21
    Organizer: Yoshihiro Ohnita(Osaka City University Advanced Mathematical Institute)


As a part of applied mathematical studies, biofluid mechanics has gathered significant attention from various research communities such as physical and material sciences, engineering, biology and medicine. In particular, novel computational and theoretical techniques, mathematical models and methods are all required to understand complex motions in biological phenomena. In this research project, though a series of workshops, tutorial seminars and symposia, we enthusiastically explore newly-born research topics in collaboration with researchers with various research backgrounds to expand the horizons of fluid mechanics and applied mathematics, in addition to deepening the traditional research topics, aiming at cultivating national and international networks of related researchers.

Operator algebra theory is a branch in functional analysis being studied intensively and extensively with strong ties to ergodic theory, topological dynamical systems, analytic group theory,mathematical physics, quantum information, noncommutative geometry, noncommutative probability, etc. The basic idea of operator algebra theory is to study the algebras of operators (duh), which are noncommutative, as opposed to the algebras of functions. Operator algebras come in two basic varieties: von Neumann algebras and C*-algebras. Von Neumann algebras deal with measure theoretic aspects of the operator algebra theory, while C*-algebras do for topological aspects.
The goal of this research project is to promote theory of operator algebras generally and develop the younger generation. For this purpose, we plan to hold three international workshops and conferences, special lecture series by long-term visitors, and a school for younger generation.
  • Workshop on von Neumann algebras and related topics (RIMS Research Project 2021)【RIMS Symposia】

    Location: Room 420    Period: Canceled
    Organizer: Narutaka Ozawa(RIMS, Kyoto University)

  • The Second Australia-China-Japan-Singapore-U.S. Index Theory Conference (RIMS Research Project 2021)【RIMS Symposia】

    Location: Room 420    Period: Canceled
    Organizer: Yasuyuki Kawahigashi(Graduate School of Mathematical Sciences, The University of Tokyo)

  • Workshop on C*-algebras and related topics (RIMS Research Project 2021)【RIMS Symposia】

    Location: Online via Zoom    Period: 2021-09-27〜2021-09-28
    Organizer: Narutaka Ozawa(RIMS, Kyoto University)

  • Workshop on free probability and related topics (RIMS Research Project 2021)【RIMS Symposia】

    Location: Hybrid Meeting    Period: 2022-01-13〜2022-01-14
    Organizer: Benoit Collins(Graduate School of Science, Kyoto University)


Modern variational methods have a wide range of theories; the direct method, the critical point theory represented by the minimax method and the mountain pass theorem, Morse-Conley theory, and so on.
These theories have been extensively studied not only in the field of partial differential equations, such as finding solutions to nonlinear elliptic partial differential equations as the stationary point of some energy functional, but also in connection with various mathematical fields, such as Riemannian geometry, symplectic geometry, mathematical physics, and so on.

We carry out this project based on the following two subjects:
(A) Deepening the insight of the lack of compactness appearing in non-compact variational problems;
(B) Applications of variational methods to evolution equations.

“Non-compact variational problem” in subject (A) is the variational problem such that the approximate sequence (such as minimizing sequence, or Palais-Smale sequence) is not always compact by itself, and it often appears in the application to the fields mentioned in the introduction.
By virtue of the concentration-compactness principle, introduced by Prof. Pierre-Louis Lions in 1980s, the mechanism of the lack of compactness is generally understood and this theory has been widely applied to the study of non-compact variational problems.
Subject (A) aims to refine the understanding of the mechanism and give new applications for non-compact variational problems.

Besides, by the use of the concentration-compactness principle, there are various studies of the asymptotic behavior of global-in-time solutions to nonlinear evolution equations, such as nonlinear parabolic, wave, dissipative equations.
The purpose of subject (B) is to develop the study of evolution equations using variational methods, represented by these applications.

In order to achieve the two subjects, we will invite Prof. Bernhard Ruf (University of Milan) and Prof. Luca Martinazzi (University of Rome ``La Sapienza") for a long-term to give a series of lectures and hold three workshops.
For subject (B), in particular, we will hold a workshop in order to identify current issues and open problems based on the progress since the RIMS workshop “Progress in Variational Problems -Variational Methods in the Study of Evolution Equations-”, which was held under the same theme.
In addition, we will hold a school mainly for young researchers. At these workshops and schools, we will invite young researchers from abroad for a short-term.
This project will be carried out in close collaboration with the Osaka City University Advanced Mathematics Institute (OCAMI).
Singularity theory special months

The singularity theory of smooth maps describes how the singular points of such maps behave. Originating with the pioneering works of Thom, Morse, and Whitney up until about 1960, the basic tools were then developed by Mather in the late 1960s. It continued to develop since then, and continues to be actively researched today. In addition, both inside and outside of mathematics, it has been applied to various areas, particularly in geometric contexts, by considering singular phenomena as singular points of smooth mapsand analyzing them in detail using singularity theory.

A basic tool of singularity theory and its applications is the classification of singularities. For this, the structures of quotient rings of various commutative rings play important roles, but these structures can become very complicated, and investigating them cannot be done by hand, so the use of software becomes necessary. However, well-known general-purpose software is of limited help for this. In recent years, in addition to the significant development of computer performance, a new tool for classification using computer algebra has been developed by Japanese researchers, and the above limits have been greatly exceeded and great progress has been made.
Additionally, there has been remarkable development of criteria for singularities that should be studied hand-in-hand with classification. Furthermore, new kinds of applications have been discovered, such as the analytic Hamiltonian appearing in topological insulators and continuum mechanics, with possibilities for these applications to develop rapidly.

These recently emerging new theories are multidisciplinary and worthy of earnest study. In view of this situation, meetings will be held to present the latest results of singularity theory and to provide lectures by experts in these new theories. Holding such meetings over the course of several months and interacting with young researchers, singularity theory researchers, and researchers who are looking to apply singularity theory as well, we can mutually develop both pure singularity theory and its applications. We can also promote training the next generation of researchers.

Designating October to December 2022 as "Singularity Theory Special Months", and inviting two leading researchers from Brazil and Spain, and representative Japanese researchers as well, we will regularly hold schools and research meetings. In addition, the Mathematical Society of Japan Seasonal Institute (MSJ-SI2022) "Deepening and developing applied singularity theory" will be held during this period.
This project will be linked with this event, with each complementary to the other.
Stochastic Processes and Related Fields

Past RIMS Research Projects
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