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 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.
Advances in Theoretical Research on Mathematical Optimization

To promote research on theory and algorithms in the field of mathematical optimization and to contribute to the further development of the research community in Japan, we plan to organize workshops and research meetings and invite world-class researchers in related fields.

Mathematical optimization (mathematical programming) problems are those of maximizing or minimizing an objective function under given constraints, and frequently appear in various fields such as natural science, engineering, economics, and information science. For mathematical optimization problems, it is important to elucidate the structure and properties of optimal solutions and to construct algorithms that can compute optimal or good approximate solutions quickly. Especially in recent years, there is an increasing need to solve large-scale optimization problems with more complex constraints such as semidefinite constraints, second-order cone constraints, and mixed integer constraints, and which are represented by large data sets. Therefore, it is necessary to develop a new theoretical framework to understand the mathematical structure behind optimization problems and to design high-performance algorithms based on this theory.

In this project, we plan to organize international workshops and research meetings on mathematical optimization with the aim of further promoting and developing theoretical and algorithmic research in mathematical optimization. In order to further strengthen the reseach areas where research groups in Japan are leading the world, the workshops will focus on lectures on these areas.
・ Discrete optimization: discrete convex analysis, algorithmic graph minor theory
・ Continuous optimization: theory and algorithms of conic optimization
・ Integration of mathematical optimization with machine learning, and practical applications of mathematical optimization

We will also invite world-class researchers in these fields to these workshops to exchange information and promote research exchanges. Two or three of them will be invited to stay at RIMS for a few months as leading researchers to develop mathematical optimization research through joint research with researchers in Japan.
Past RIMS Research Projects
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Research Institute for Mathematical Sciences (RIMS)