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

从非交换共形场论（WSWN模型）和交换共形场Ueno和J.E. Andersen构造了模函子并研究了其相关拓扑场论的性质。他们还证明了非交换共形场论的S-矩阵是由亏格0数据决定的。此外，他们还证明了四点球映射类群的Nielsen-Thurston分类是由它们的量子SU（n）表示决定的。F. Kato和他的合作者们构造了最一般的刚性几何，从而可以用解析方法来研究模空间。S. Mochizuki从不同的角度研究了代数曲线的模空间的范畴方面。例如，他建造的理论Frobenioids和的etale θ函数，这给新的方向研究模空间。此外，还得到了模空间的几何和算术性质的许多有趣结果。高崎和他的合作者发现模空间在孤子理论中起着重要的作用。我们发现了特殊代数曲线的几何与Painleve方程之间有趣的关系，并且发现了旗流形的量子上同调和量子K理论的几个重要性质，我们的研究表明模空间的数学结构比我们最初想象的要深刻，我们需要新的数学工具来进行进一步的研究。我们的结果给出了这样的工具的一部分，也显示了新的方向，进一步的调查。

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.