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

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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.