Cohomology Theories for Algebraic Varieties

研究者研究了最近由Elderor和F. Morel，以及代数簇和计划的广义上同调理论，Morel-Voevodsky代数同伦范畴中的有理等价关系，以及Asakura算术Hodge上同调的改进。研究者试图构造一个高级代数协边理论，沿着Bloch的高级Chow群的路线，考察协边动机的范畴，并试图给出一个代数高级椭圆型的理论。此外，调查员检查的推广motivic上同调的K-theory谱序列的其他上同调理论代数簇，并考虑这种谱序列的关系，切片谱序列Voevodsky，使用的概念，合理的等价代数同伦范畴。最后，研究者定义了Asakura算术Hodge上同调的一个变体，并利用它构造了一个无穷序列的上同调理论，这些上同调理论逐渐给出了对motivic上同调的一个越来越好的逼近。代数几何学是用代数学和几何学来研究方程的解的学科。例如，人们可以通过检查圆的方程或观察其几何性质来研究圆。代数拓扑学通过将代数不变量附加到空间来研究空间，这些不变量通常可以显式计算。 调查人员采取的方法和结构在代数拓扑，然后修改和完善它们，使它们可以用来定义代数版本的拓扑不变量。这些新的代数不变量，然后适用于研究微妙的性质方程的解决方案。

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.