Geometric Study of Algebraic Systems

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1.The head investigator, Masayoshi Miyanishi, gave new proofs to the Abhyankar-Moh Theorem and the Lin-Zaidenberg Theorem which are based on the classification theory of open algebraic surfaces. He considered the embeddings of affine curves with one-place points at infinity into the affine plane and classified such embeddings of possibly minimal degree when the genus is low. He, furthermore, simplified a proof in P.Roberts' counterexample to the fourteenth problem of Hilbert.2.Akira Fujiki found a natural partial compactification of hyper kahler manifolds as quaternionic manifolds which are obtained as hyper Kahler quotinet spaces of certain kinds. He also investigated the variation via moment maps of Kahler quotient spaces by making use of equivariant cohomology groups.3.Sanpei Usui proved that one can recover vanishing cycles by using log geometry. As applications of this result, he clarified the Z-structure of the variation of Hodge structures and the description of monodromies.4.Jun Murakami constructed a new invariant for three-dimensional manifolds which is based on the Kontsevich invariant for the knots and studied its properties. He constructed a topological quantum theory by making use of this invariant, and a family of mapping groups of surfaces as its application.5.Kazuhiro Konno defined a new index for surfaces with non-hyperelliptic pencils and found some general method to investigate the lower bound of the slopes of surfaces. He also an index, called the Horikawa index, for degenerate fibers.6.Hiroyuki Yamane obtained a new technique to calculate the index of the BMW algebras which are defined in a paper of Rham.