Algebraic Geometry, Differential Geometry and Topology of Manifolds

1. 无穷维李群（规范群等）的同伦理论。Kono和S. Tsukuda利用纤维同伦理论部分解决了有限复上主束伴随束的分类问题。他们确定了在纤维化定位后伴随束的琐碎性的条件。请注意，量规组是伴随管束的部分的空间。不稳定的K-theoryA。Kono和H. Hamanaka确定了2n维有限复到U(n)的映射的同伦类群。另一方面，A. Kono和H. Oshima（Ibaraki university）分类了自同伦类为交换群的紧李群。同调代数同调代数是一种非交换的同调代数。A. Kono和A. Moriwaki考虑了同局部代数在代数几何或算术几何中的应用。在数学物理和弦理论中的应用是由k.f ukaya考虑的。日本京都大学的M. Asaoka定义了二维投影Anosov动力系统的代数不变量，并得到了它们的几个基本性质。

1. Homotopy theory of infinite dimensional Lie groups (gauge groups etc)A. Kono and S. Tsukuda partially solved the classification problem of the adjoint bundles of the principal bundles over finite complexes using the fibrewise homotopy theory. They determined the condition for the triviality of the adojoint bundle after the fibrewize localization. Note that gauge groups are the space of sections of the adojoint bundles.2. Unstable K-theoryA. Kono and H. Hamanaka determined the group of homotopy classes of maps from a 2n dimensional finite complex to U(n). On the other hand A. Kono and H. Oshima(Ibaraki Univ.) classified compact Lie groups whose self homotopy classes are commutative groups.3. Homotopical algebraHomotopical algebra is non -commutative homological algebra. A. Kono and A. Moriwaki considered application of homotopical algebra to alebraic geometry or arithmetic geometry. Applications to mathematical physics and string theory are considered by K. Fukaya.4. Dynamical systemAlgebraic invariants for 2-dimensional projective Anosov dynamical system are defined and several elementary properties of them are obtained by M. Asaoka(Kyoto Univ).