Mathematical problems on manifolds

通过研究人员之间的密切接触和讨论，对歧管的各种问题进行了全面的研究.从代数几何、代数分析和微分几何等方面对复曲面簇进行了研究，在复曲面簇的交上同调、到复曲面簇的态射、复曲面Fano流形的分类和Einstein-Kaehler度量等方面获得了新的结果.研究了算法的多样性，在Abel曲面上的有理点、关于二阶上同调的Tate猜想、晶体基本群和rho-adic Hodge理论等方面获得了新的结果.获得了非阿基米德品种的刚度结果。在整体分析、双曲几何、可微流形、黎曼流形和共形平坦流形的基本群等方面获得了许多新的结果.从分析和数学物理的角度研究了Laplacian算子和Schroedinger算子，在图的离散模拟上得到了一些新的有趣结果.对反应扩散方程进行了整体研究，获得了解的稳定性的新结果.在Kaehler流形和Einstein-Hermitian度量上得到了向量丛稳定性的各种新结果.从真实的分析、复分析和傅立叶分析的角度研究了伪微分算子、极大算子、有界线性算子、算子代数。

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.