Studies on moduli spaces from the view point of mathematical physics

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Moduli spaces of algebraic varieties and vector bundles play am important role not only in algebraic geometry but alsomathematical physics. We studied moduli spaces from the view point of mathematical physics.Ueno studied mainly conformal field theory which has deep relationship with moduli spaces of pointed Riemann surfaces. He showed that non-abelian conformal field theory, so called the WZW modelis defined overtherational number field and even more it can be defined over a discrete valuation ring. With Y.Shimizu and T.Suzuki he also studied explicit description of protectively flat connection of the sheaf of conformal blocks.Maruyama showed a new method how to construct moduli sapce of stable sheaves on algebraic surfaces, which will play an important role to study boundaries of the moduli spaces. Mukai studied the moduli spaces of vector bundles on K3 surfaces and found interesting relationship with the moduli spaces of algebraic curves. Kawamata showed unobstractedness of deformations of Calabi-Yau manifolds by purely algebraic method. Katsura studied pluricanonical sytems of elliptic surfaces in positive characteristics and showed that they have similar properties as in characteristic O.In this way we found many interesting properties of moduli spaces and also showed deep relationship with mathematicalphysics.