Cohomology Theories for Algebraic Varieties

研究者研究了最近由研究者和F. Morel建立的代数共调理论，代数变量和代数方案的广义上同调理论，Morel- voevodsky代数同伦范畴中的理性等价关系，Asakura的算术Hodge上同调的改进。研究者试图沿着Bloch的高周群的思路构建一个高等代数协共论，考察协共动机的范畴，并试图给出一个代数高等椭圆属的理论。此外，研究者还研究了k -理论谱序列的动机上同调对其他代数变体上同调理论的推广，并利用代数同伦范畴中的有理等价概念考虑了该谱序列与Voevodsky的切片谱序列的关系。最后，研究者定义了Asakura的算术Hodge上同调的一个变体，并利用它构造了一个上同调理论的无穷序列，该序列推测地给出了越来越好的动机上同调近似。研究者的研究涉及代数几何和代数拓扑的混合。代数几何是使用代数和几何来研究方程解的学科。例如，一个人可以通过检查一个圆的方程来研究它，或者通过观察它的几何性质。代数拓扑学通过在空间上附加代数不变量来研究空间，这些不变量通常可以显式计算。研究者采用代数拓扑中的方法和结构，然后对它们进行修改和改进，使它们可以用于定义拓扑不变量的代数版本。这些新的代数不变量适用于研究方程解的微妙性质。

The investigator studies the theory of algebraic cobordism, recently constructed by the investigor and F. Morel, as well as generalized cohomology theories for algebraic varieties and schemes, the relation of rational equivalence in the Morel-Voevodsky algebraic-homotopy category, and a refinement of Asakura's arithmetic Hodge cohomology. The investigator attempts to construct a theory of higher algebraic cobordism, along the lines of Bloch's higher Chow groups, examines the category of cobordism motives, and tries to give a theory of algebraic higher elliptic genera. In addition, the investigator examines the generalization of the motivic cohomology to K-theoryspectral sequence to other cohomology theories on algebraic varieties, and considers the relationship of this spectral sequence to the slice spectral sequence of Voevodsky, using the notion of rational equivalence in the algebraic homotopy category. Finally the investigator defines a variation of Asakura's arithmetic Hodge cohomology and uses this to construct an infinitesequence of cohomology theories which conjecturely give a better and better approximation to motivic cohomology.The investigator's research involves a mixture of algebraic geometry and algebraic topology. Algebraic geometry is the study of solutions of equations using both algebra and geometry. For example, one can study a circle by examining its equation, or by looking at its geometric properties. Algebraic topology studies spaces by attaching algebraic invariants to them, invariants which can often be computed explicitly. The investigator takes methods and constructions in algebraic topology, and then modifies and refines them so that they can be used to define algebraic versions of the topological invariants. These new algebraic invariants are then applicable for studying subtle properties of solutions of equations.