About RIMS research project 2024

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 parabolic 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 monodromy 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 for 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 investigate 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 with these systems brings 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.