Moduli spaces and symmetries in string theory

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On the theme of Seiberg-Witten theory approached through elliptic surfaces, Shimizu did the joint research with A.Tsuchiya (Nagoya Univ.), A.Kato (Unlv. Tokyo) Y.Saito (Univ. Tokyo), H.Awata (Univ. Tokyo) etc. Among others, we constructed the moduli space of marked rational elliptic surfaces and their period map, studied the inversion problem, the relation with simply elliptic singularities, constructed the elliptic Lie algebra and the elliptic Weyl group, and carried out the realization of a Seiberg-Witten integrable system. In 2002, we introduced the periods of elliptic K3 surfaces, the moduli and the derived category of coherent sheaves in order to enlarge our study and to look for the quantum analog of classical periodsUeno and Shimizu studied the degeneration behavior of the conformal blocks of the CFT with abelian currents as symmetries, and published a book on moduli theory. Ueno constructed a modular functor using CFT. From the viewpoint of deformation theory of complex structure, Saito reconsidered the space of initial values of the Painleve equations, introduced the notion of Okamoto-Painleve pairs, classified them and rewrited the differential equationsSuzuki studied weakly distance-regular digraphs and type II matrices associated with link invariants. Morimoto expressed the analytic functionals on the sphere as initial values of the heat functions. Yamakawa studied codimension one locally free action of solvable Lie groups