Various problems related to classifications around the higher dimensional birational geometry

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Mori, together with Kollar, published an book on the birational geometry of algebraic varieties. Topics treated in the book include a simpler alternate definition of dlt singularity, a simpler proof of the rationality of dlt singularities and an alternate proof of the existence of the 3-dimensional stable flips.He also published a review of his work on the existence of rational curves on algebraic varieties, in which he posed problems on the refinement of the existence theorem, the generalization of the cone theorem, etc. Together with Kollar, Miyaoka and Takagi, he finished the proof of the boundedness of the terminal Q-Fano 3-folds, which will be published shortly. (The proof of Reid's conjecture on 3-dimensional flips in the reducible case is in preparation.)Miyaoka is preparing the proof for the assertion that every projective smooth n-fold with an external ray of length at least n+1 is isomorphic to the projective space, which is to be published soon.Nakayama has investigated prob … More lems related to the minimal model theory. He proved that small deformation of terminal singularities are terminal (in preparation). He also proved that, assuming the abundance conjecture, every nonsingular projective manifold whose universal covering is an affine space has an abelian variety as a finite etale covering.Mukai's work are on the algebraic construction of moduli spaces and various geometries on them, including certain duality of polarized K3 surfaces. He is also investigating the Verlinde formula on the moduli spaces of the parabolic vector bundles.Masahiko Saito, together with Hosono and Takahashi, has formulated a generalzation of the holomorphic anomaly equation on the counting of higher genus curves on rational elliptic surfaces and verified that it is consistent with the B-model computation in the case of genus 0,1.Hayakawa has proved that, for every 3-dimensional terminal singularity of index at least two, an arbitrary exceptional divisor with the minimal discrepancy can be obtained by an explicit "weighted blow-up".Dr. Kenji Matsuki at Purdue University was invited for two weeks from the end of June 1999 to present his recent work on the weak factorization of the birational map in a series of lectures (the lecture notes will be published. ) Less