Geometry on String Theory and Moduli spaces

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During the period of the project, we have investigated the following subjects and obtained the following results. (i) Gromov-Witten invariants and BPS invariants for Calabi-Yau manifolds-Gopakumar-Vafa conjecture, (ii) Homological Mirror Symmetry and Geometry of derived categories, (iii) Algebraic Geometry of spaces of initial conditions of Painleve equations, (iv) Lie theoretic studies for Painleve equations and its generalizations, (v) Symmetries of the moduli spaces of vector bundles, (vi) New development of invariant theory. As for (i), we proposed a mathematical definition of BPS invariants for Calabi-Yau 3-fold and verified their compatibility for the known calculation of Gromov-Witten invariants assuming Gopakumar-Vafa conjecture. In (iii), we introduce the notion of Okamoto-Painleve pairs and proved that one can derive Painleve equations through deformation theory of Okamoto-Painleve pairs. As for the studies of Painleve equations (iv), our group have been developing Lie theoretic approach and algebro-geometric approach, which clarify the relations among Painleve equations, the symmetry of Affine Weyl groups and the geometry of rational surfaces.