Mathematical problems on manifolds

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Through close contact and discussion among the investigators, comprehensive study was made on various problems on manifolds.1. Toric varieties were studied from the aspects of algebraic geometry, algebraic analysis and differential geometry, and new results were obtained on intersection cohomology, morphisms to toric varieties, classification of toric Fano manifolds and Einstein-Kaehler metrics.2. Arithmetic varieties were studies, and new results were obtained on rational points on Abelian surfaces, the Tate conjecture on the second etale cohomology, crystal fundamental group and rho-adic Hodge theory.3. Rigidity results were obtained for non-Archimedean varieties.4. Various new results were obtained on the global analysis, hyperbolic geometry and fundamental groups of differentiable manifolds, Riemannian manifolds, and conformally flat manifolds.5. Laplacian and Schroedinger operators were studies from the viewpoint of analysis and mathematical physics, and new interesting results were obtained on the discrete analog on graphs.6. Global study was made on reaction-diffusion equations, and new results were obtained on the stability of solutions.7. Various new results were obtained on the stability of vector bundles on Kaehler manifolds and Einstein-Hermitian metics.8. Pseudo-differential operators, maximal operators, bounded linear operators, operator algebras were studies from the viewpoint of real analysis, complex analysis and Fourier analysis.