Problems on varieties

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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.