Theory of mixed motives and theory of scissors congruence groups of algebraic polyhedra

在文1中，构造了域D(K)上混合动机的三角范畴。这一理论自20世纪80年代初S以来一直被猜测存在，并有望有更多的应用。在文献2中，在一些“标准猜想”的假设下，研究了D(K)上适当的t-结构的存在性。在文3中，我展示了如何将投射簇的动机--D(K)的一个对象联系起来。我们在文4中探索了适当映射下常数层的直接映象的“分解定理”的基元类比。这包括任意簇上的基元簇的框架。在文5中，Hanamura和M.Yoshida研究了关于扭曲上同调的Hodge理论，并应用它来推导Riemann不等式的类比。

In paper 1 the triangulated category of mixed motives over a field D(k) was constructed. This theory has been conjectured to exist since the early 1980's. The theory is expected have much application. In paper 2, the existence of an appropriate t-structure on D(k) has been studied assuming some "standard conjectures". I showed how to associate to a projective variety its motive, an object of D(k), in paper 3. We explored the motivic analogue of the "decomposition theorem" for the direct image of the constant sheaf under a proper map, paper 4. This includes the framework of motivic sheaves over an arbitrary variety.In paper 5, Hanamura and M.Yoshida studied Hodge theory on twisted cohomology, and applied it to derive the analogue of Riemann's inequality.