Geometry of integrable systems with infinite degrees of freedom and new development of moduli theory.

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From non-abelian conformal field theory (WSWN-model) and abelian conformal field Ueno and J.E. Andersen constructed modular functor and studied properties of the associated topological field theory. They also showed that S-matrices of non-abelian conformal field theory are determined by the genus 0 data. Moreover they showed that the Nielsen-Thurston classification of mapping class groups of 4-pointed spheres is determined by their quantum SU (n) representations. F. Kato and his collaborators have constructed the most general rigid geometry so that it can be applied to study moduli spaces by analytic method. S. Mochizuki has studied categorical aspect of moduli space of algebraic curves from different viewpoints. For example, he constructed theory of Frobenioids and that of etale theta functions, which give new direction of study of moduli spaces. Moreover, many interesting results on geometric and arithmetic properties of moduli spaces were obtained.For theory of integrable systems K. Takasaki and his collaborators found that moduli spaces play important roles in the soliton theory. Also interesting relationship between geometry of special algebraic curves and Painleve equations were found. Moreover several important properties of quantum cohomology and quantum K-theory of flag manifolds have been found.Our studies show that mathematical structure of moduli space is deeper than what we thought at the beginning and we need new mathematical tools for further investigations. Our results give a part of such tools and also show new directions to further investigations.