Problems on varieties

1.从代数几何、算术几何和复解析几何的观点出发，得到了环簇、Fano簇、正特征椭圆曲面和对数模式的交上同调的各种重要结果。从数论和算术几何的观点出发，得到了关于代数簇上的有理点，特别是代数曲面，以及p-进解析曲面的新结果。从代数分析、表示论和数学物理的角度，对等变完整系统在代数簇、p进对称空间、非平衡模型、量子群表示和保形场论等方面的研究进展进行了综述。从整体分析、微分几何和复几何的观点出发，我们在调和映射、调和叶层、电磁场的几何和量子遍历、爱因斯坦-厄米特度规、箭图以及自对偶度量等方面取得了新的结果。我们能够获得这些结果，这要归功于通过这项资助可以很容易地访问关于变种的大量结果的数据库。各种不同观点的专家自由地聚集在一起，提供了个人的见解和想法，以建立全面的成果。我们试图将我们的结果传达给世界各地的相关专家。

1. From the viewpoints of algebraic geometry, arithmetic geometry and complex analytic geometry, various important results were obtained on the intersection cohomology of toric varieties, Fano varieties, elliptic surfaces in positive characteristics and logarithmic schemes.2. From the viewpoints of number theory and arithmetic geometry, new results were obtaied on rational points on algebraic varieties, algebraic surfaces in particular, as well as p-adic analytic surfaces.3. From the viewpoints of algebraic analysis, representation theory and mathematical physics, progress was made on equivariant holonomic systems on alegbraic varieties, p-adic symmetric spaces, nonequilibrium models, representations of quantum groups and conformal field theory.4. From the viewpoints of global analysis, differential geometry and complex geometry, new results were obtained on harmonic maps, harmonic foliations, the geometry and quantum ergodicity of eletro-magnetic fields, Einstein-Hermitian metrics, quivers as well as self-dual metrics.We were able to obtain these results thanks to easy access to databases on voluminous results on varieties made possible through this grant. Specialists on various different viewpoints freely got together, provided individual insights and and ideas to build up comprehensive results. We tried to communicate our results to relevant experts throughout the world.