Geometry and arithmetics through the theory of algebraic cycles

代数几何的基本对象是代数簇。这些被局部定义为多项式方程的零轨迹。代数几何的主要目标是对簇进行分类。一种方法是将不变量附加到变种上。一些不变量是算术性质的，例如X上闭点的度的gcd。有些是拓扑性质的，例如X的基本拓扑空间的奇异上同调。有些是几何性质的，例如X的Chow群。X上的余维n代数圈是余维n的不可约子簇的形式和，而Chow群CH^n（X）是以X中的余维n的不可约子簇模一个称为有理等价的等价关系为基的阿贝尔群。粗略地说，有理等价是代数圈上最精细的等价关系，它可以明确地定义圈上的交积。此外，X的上述不变量被编码（或至少预期被编码）在Chow群X中。因此，在某种意义上，代数圈是代数簇的最好的不变量，代数圈理论是几何，拓扑和数论的核心，我将综合K-理论，伽罗瓦上同调和数论的方法，得出新的结果，在代数圈理论上定义在代数生成域或其他领域的算术兴趣的簇。相反地，我将使用代数圈的理论来推导出新的算术结果。此外，这些结果的结果将揭示新的光的几何形状的品种。因此，就其本质而言，我对代数圈理论的研究建议是数学科学中的学科内的。

The basic objects of algebraic geometry are algebraic varieties. These are defined locally as the zero locus of polynomial equations. The main goal of algebraic geometry is to classify varieties. An approach consists in attaching invariants to varieties. Some invariants are of an arithmetic nature, e.g. the gcd of the degrees of closed points on X. Some are of a topological nature, e.g. the singular cohomology of the underlying topological space of X. Some are of a geometric nature, e.g. the Chow groups of X. A codimension-n algebraic cycle on X is a formal sum of irreducible subvarieties of codimension n and the Chow group CH^n(X) is the abelian group with basis the irreducible subvarieties of codimension n in X modulo a certain equivalence relation called rational equivalence. Roughly, rational equivalence is the finest equivalence relation on algebraic cycles that makes it possible to define unambiguously an intersection product on cycles. Moreover, the aforementioned invariants for X are encoded (or at least expected to be) in the Chow groups X. Therefore, in some sense, algebraic cycles are the finest invariants for algebraic varieties, and the theory of algebraic cycles lies at the very heart of geometry, topology and number theory.I will integrate methods from K-theory, Galois cohomology and number theory to derive new results in the theory of algebraic cycles on varieties defined over finitely generated fields or other fields of arithmetic interest. Conversely, I will use the theory of algebraic cycles to derive new results of arithmeticinterest. In addition, the outcome of such results will shed new light on the geometry of such varieties. Thus, by its very nature, my research proposal on the theory of algebraic cycles is intradisciplinary within the mathematical sciences.