Algebraic Geometry, Differential Geometry and Topology of Manifolds

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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).