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Past RIMS Research Projects

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Past RIMS Research Projects

2024
Development in Algebraic Geometry related to Integrable Systems and Mathematical Physics

The topological recursion, which is a mathematical formulation of the matrix model of mathematical physics, is expected to be a universal recurrence relation underlying the enumeration of various invariants, such as Gromov-Witten and Donaldson-Thomas invariants of algebraic and symplectic manifolds. Furthermore, the generating functions of these invariants have been shown in various examples to provide τ-functions of integrable systems such as the KdV equation and Painlevé equations. These ideas trace back to Witten-Kontsevich's theorem and continue to be active research areas from various directions. In recent years, there has been deeper understanding, including the construction of Lax pairs associated with integrable systems from the theory of quantum curves.

Moreover, the generating functions of the aforementioned invariants are in the form of formal power series in a perturbation parameter \(h\), and recent research has been actively conducted on their Borel summability and resurgent structures, with a particular focus on their relationship with BPS structures.

In the mathematical understanding of the mirror symmetry of Calabi-Yau manifolds, significant progress has been made in quantum cohomology theory based on Gromov-Witten invariants and the definition of Donaldson-Thomas invariants, Gopakumar-Vafa invariants based on derived category. Recently, there have been constructive theories for computing higher-genus Gromov-Witten invariants, revealing connections with　holomorphic anomaly equations arising from deformation spaces of Calabi-Yau manifolds. Furthermore, non-perturbative solutions of topological string theories have been sought, and progress has been made in their analysis, shedding light on holomorphic anomaly equations in various contexts.

In recent years, there have been advancements in the algebro-geometric construction of moduli spaces of parabolic connections and parabolics Higgs bundles on algebraic curves of arbitrary genus. We can show that generalized Riemann-Hilbert correspondences, which are maps from the moduli spaces of parabolic connections to the moduli spaces of mondoromy and Stokes data, are surjective, proper birational analytic morphisms. This fact shows that the generalized monodromy-preserving deformations give rise to dynamical systems with geometric Painlevé properties on families of moduli spaces of parabolic connections. These moduli spaces are known to admit algebraic symplectic structures, and one has algebraic geometric constructions of Darboux coordinates with respect to these symplectic structures.

These developments have allowed for a detailed treatment of integrable systems and dynamical systems arising from monodromy-preserving deformations in algebraic geometry. Additionally, research has advanced on the relation between expansions of τ-functions of Painlevé equations and those constructed from conformal field theory and WKB analysis. It is a highly intriguing research theme to invesigate the connections between these theories and the theories of topological recursions and mirror symmetry. Furthermore, research on discrete Painlevé systems and quantum Painlevé systems has been progressing, and the study of symmetries associated to these systems is bringing new perspectives in various fields.

In this research project, we will invite researchers from various fields and conduct cutting-edge research presentations on the aforementioned research themes. Our goal is to elucidate the interplay of theories, particularly between generating functions of various invariants, integrable systems, and the underlying geometric frameworks.
2023
Stochastic Processes and Related Fields

Since the founding of the mathematical theory of stochastic differential equations by Kiyosi Itô in 1942, research on stochastic calculus in Japan has had a great influence on the development of probability theory throughout the world. Probability theory itself has grown remarkably over the last century, and now encompasses an extremely wide range of research topics. Within mathematics, it has strong interactions with fields such as partial differential equations, potential theory and geometry. Moreover, the vast expansion of probability research is directly linked to applications across the sciences, in areas such as statistical mechanics, biology, economics and the analysis of big data. As a result of probability theory now having so many different aspects, it is difficult for individual researchers to grasp the subject in its entirety.

With the keywords "stochastic processes and stochastic analysis", this RIMS Research Project aims to further promote international joint research in probability by disseminating results in the area that originate in Japan. Centering on the following three themes, the program will provide a bird's-eye view of modern probability theory. (For each theme, a conference is planned, the title of which is shown in parentheses.)
(i) Stochastic Processes Related to Stochastic Partial Differential Equations (Stochastic Partial Differential Equations and Stochastic Calculus)
(ii) Analysis of Stochastic Models Motivated by Statistical Mechanics (Stochastic Analysis on Large Scale Interacting Systems)
(iii) Random Matrix Theory, Combinatorial Probability and Quantum Information (Random Matrices and Their Applications)
Furthermore, to inspire exchanges between these themes, in September 2023, we will organize a large-scale conference at the Research Institute for Mathematical Sciences, inviting researchers in stochastic processes and stochastic analysis who are active at the front line of probability research internationally.

Mid-career researchers will take responsibility for the practical organization of conferences on each theme and, in doing so, will promote the participation of young researchers, including as speakers, so that the next generation of researchers plays an active role. We encourage the participation of female researchers in all aspects of the program, seeing it as an opportunity to promote gender diversity in research on probability theory in Japan.
2022
Deepening and applications of variational problems

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 Central Advanced Mathematics Institute (OCAMI).
2022
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.
2021
Mathematical Biofluid Mechanics

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.
2021
Theory of operator algebras and its applications

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)

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  • 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)

2020→2021
Expanding Horizons of Inter-universal Teichmüller Theory

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).
2020→2021
Differential Geometry and Integrable Systems
-Mathematics of Symmetry, Stability and Moduli-

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.
2019
Cluster Algebras
  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).
2019
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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