Algebraic groups and cohomological invariants

数学及其应用的每一个领域都得益于几何观点。有时只需使用拓扑学就可以解决问题，拓扑学描述的是物体的粗略形状。但拓扑学并不总是足以告诉我们给定的方程是否可以解。我们被引导建立了更微妙的理论，例如代数K-理论，它包含了关于哪些代数方程可以解的更多信息。但数学家在尝试计算代数K-理论时面临着巨大的挑战。这个问题包括Clay研究所著名的百万美元问题中的两个，Hodge猜想和Birch-Swinnerton-Dyer猜想。拟议的研究旨在计算一类特殊的基于群的例子的代数K-理论和相关理论。群论是关于对称性的数学理论。在这种情况下，我们看到了代数K-理论及相关理论完全计算的前景。我们的计算将有助于解决代数中古老而困难的分类问题，例如二次型的分类。

Every area of mathematics and its applications gains froma geometric point of view. Sometimes one can solve a problemjust using topology, which describes the rough shapeof an object. But topology is not always enough to tell uswhether a given equation can be solved. We are led to set upmore subtle theories such as algebraic K-theory whichinclude more information about which algebraic equationscan be solved. But mathematicians face massive challengesin trying to compute algebraic K-theory. This problem includestwo of the Clay Institute's famousmillion-dollar problems, the Hodge conjectureand the Birch-Swinnerton-Dyer conjecture.The proposed research seeks to compute algebraic K-theoryand related theories for a special class of examplesbased on groups. Group theory is the mathematical theoryof symmetry. In this situation, we see the prospectof complete calculations of algebraic K-theory and relatedtheories. Our calculations willcontribute to ancient and difficult classification problemsin algebra, such as the classification of quadratic forms.