Past RIMS Research Projects


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)


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, 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.
  Cluster algebras were original introduced by Fomin and Zelevinsky around 2000 to generalize a class of commutative algebras appearing in Lie theory in view of the Laurent phenomenon. Nowadays they are recognized as a kind of extension of the theory of root systems, and they are actively studied as an underlying algebraic and combinatorial structure ubiquitously appearing in several areas of mathematics.
  In this research project, we will hold an international workshops series "Cluster Algebras 2019" at RIMS in June, 2019, which is the largest comprehensive program on cluster algebras since the semantic program in KIAS, Korea in 2014. We will also hold a mini course on topics in cluster algebras at RIMS in May, 2019 by the visiting professors Bernard Leclerc (Universitè de Caen) and Michael Gekhtman (Notre Dame).
Discrete Optimization and Related Topics
Discrete optimization occurs frequently in our economic and social activities.
The development of discrete optimization in both theory and application has a major impact on our society, since artificial intelligence (AI), machine learning, and big data receive much attention.
In this project research, we aim to promote theoretical research on discrete optimization. The project focus not only on classical research but also on the one related to big data, such as sublinear or constant time optimization algorithm. For example, we plan to have the following three international workshops

1) Hungarian-Japanese Symposium on Discrete Mathematics and Its Applications

2) International Workshop on Innovative Algorithms for Big Data

3) International Workshop on Combinatorial Optimization and Algorithmic Game Theory
Past RIMS Research Projects(2018-)


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