The study of vector bundles on an algebraic manifold with the trivial canonical bundle

Yoshioka等人在K3和Abel曲面上找到了稳定层存在的条件，并证明了模空间的连通性。如果模空间是紧凑的，然后他确定了阿尔巴尼亚地图，并表明，纤维是一个hyperkaehler流形。此外，他表明，变形类型的流形是由尺寸。利用这一结果，将模空间拓扑型的研究归结为秩1的情形。为了得到这些结果，他使用了傅立叶-向井变换和底层K3曲面的变形。因此，他还研究了稳定性和傅立叶-向井变换之间的关系。此外，他还研究了椭圆曲面上稳定层的模、K3曲面上稳定层模的曲面分量以及Gromov-Witten不变量，得到了一些结果。他提出了一个数学定义的BPS不变量使用模的纯1维层，并检查了一致性的某些情况下。山田研究D-膜上的理性椭圆曲面。数学上的美结构如单值群、Mordell-Weil群、仿射Weyl群都是从物理学的角度来研究的，Saito和yamada从对称性和几何学的角度来研究Painleve方程。特别是，齐藤表明，贝克兰德变换是触发器中的双有理几何和Painleve方程是来自变形理论的合理表面，其中有一个应用的分类Riccati解决方案。

Yoshioka find the condition for the existence of stable sheaves on K3 and abelian surfaces and showed the connectedness of the moduli spaces. If the moduli space is compact, then he determined the albanese map and showed that the fiber is a hyperkaehler manifold. Moreover he showed that the deformation type of the manifold is determined by the dimension. By this result, the study of the topological type of the moduli space is reduced to the rank 1 case. In order to get these results, he used the Fourier-Mukai transform and the deformation of the underlying K3 surfaces. Hence he also studied the relation between the stability and the Fourier-Mukai transform. Moreover he studied the moduli of stable sheaves on an elliptic surface, surface components of the moduli of stable sheaves on a K3 surface and the Gromov-Witten invariants and got some results.Saito studied Gopakumar-Vafa conjecture on BPS states. He proposed a mathematical definition of the BPS invariants by using the moduli of purely 1-dimensional sheaves and checked the consistency for some cases.Yamada studied D-brane on a rational elliptic surface. Mathematical beautiful structures such as monodromy group, Mordell-Weil group, affine Weyl group are studied from the Physical point of view.Saito and yamada studied the Painleve equation in terms of the symmetry and the geometry. In particular, Saito showed that the Backlund transform is the flop in the birational geometry and the Painleve equation is derived from the deformation theory of a rational surface, which has an application of the classification of the Riccati solution.