Fusion of geometry and the theory of integrable systems

我们对几乎所有的等参超曲面进行了分类，并用矩映射对它们进行了刻画，证明了它们与可积系统的关系。建立了奇异曲面的基本理论和利用勒让德映射的新方法。通过黎曼-希尔伯特对应关系，研究了Painleve方程的动力系统，并提出了混沌的观点。讨论了高属Gromov-Witten的模性和镜像对称性。构造了量子上同调中出现的具有势的曲面，这有助于tt*几何。

We classified almost all isoparametric hypersurface, and characterize them in terms of the moment map, which proves the evidence of a relation with integrable systems. A basic theory of surfaces with singularities, and a new method using the Legendre map have been established. Via the Riemann-Hilbert correspondence, the dynamical system of Painleve equations is investigated, and the view point of the chaos has been developed. The modularity of higher genus Gromov-Witten and the mirror symmetry are discussed. A surface with potential appeared in quantum cohomology is constructed, which contributes to the tt* geometry.