Research of new developments in moduli spaces and integrable systems

点击翻译按钮获取中文摘要

In the present research, Inaba, Iwasaki and Saito constructed the moduli space of stable parabolic connections of rank 2 over the projective line with at most n-regular singular points and its natural compactification, and showed the smoothness of the moduli space. Moreover, constructing the moduli space of the representations of the fundamental group of the complement of the n-points in projective plane, we showed that Riemann-Hilbert correspondence from the moduli space of connections to the moduli space of the representations is a surjective proper bimeromorphic morphism. This result implies that the non-linear differential equations coming from isomonodromic deformations of connections have Painleve property. Recently Inaba succeeded in generalizing this result to the case of stable parabolic connections with at most regular singularities of any rank over any smooth algebraic curve. We expect further to obtain the same result in the case of connections with irregular singular point … More s. Noumi and Yamada gave a geometric description of elliptic difference Painleve equations with Ohta, Masuda, Kajiwara, and construct the elliptic hypergeometric solutions and its degeneration.By using the algebro-geometric construction of Painleve VI equations and applying the Riemann-Hilbert correspondence to ergotic theory of algebraic surfaces, Iwasaki and Uehara showed that the dynamics of almost loops of non-linear monodromy of Painleve VI equation are chaotic. Hosono and Doran determined the differential equation of the period integrals of certain class of Calabi-Yau hypersurfaces by oscillatory integrals. Moreover solving the Stokes phenomena associated to the integral, they observed a natural understanding from the view point of Mirror symmetry. Yoshioka and Hiraku Nakajima proved the Nekrasov conjecture for the instanton numbers. Moreover with Lotha Gottsche, they obtained a wall-crossing formula for Donaldson invariants for the rational surfaces, and extend the results. Fukaya, Ono, Ohta, Oh, are completing a big book of Floer theory of Lagrangian submanifolds. Less