Algebraic groups and cohomological invariants

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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.