Homology of Linear Groups with Applications to Algebraic K-theory

克努森研究了代数K-理论中各种感兴趣的环上的线性群的同调。 在这个项目中，他专注于低维同调群的一般线性群的坐标环的椭圆曲线在一个数字领域的目标是证明第二个K-群这样的曲线有有限的秩。 这扩展了克努森以前的工作同源的秩一组等环。 此外，基于代数群的单分类方案与方案乘积的代数圈，构造了一个新的同调理论，并研究了它的性质。 希望是，这一建设将导致一个证明弗里德兰德-米尔诺猜想有关的同源性代数群离散。 此外，调查员以前的工作的结构，特殊的线性群的积分洛朗多项式环被用于研究的布劳表示的辫子群。 这种表示法对三个字符串是忠实的，对五个或更多字符串是不忠实的;剩下的四个字符串的情况在这个项目中被挑出来研究。 最后，克努森研究了一个正特征域上的约化群中的一个离散群相对于一个Zebraki稠密表示的完备性。 这推广了群的经典幂幺完备化，并推广到正特征R。海恩在特征零点上的工作。 代数K理论将一个方案与一系列编码该方案信息的群相关联。 该项目的一个方面是研究椭圆曲线的K群。 这样的曲线具有非常丰富的结构，并出现在数学的各个分支，如代数几何和编码理论。 这个项目的一个看似无关的部分涉及所谓的布劳表示的辫子群，这是密切相关的结理论。 这些不同的主题是统一的研究结构组的矩阵条目在各种环（坐标环的椭圆曲线在第一种情况下和环的积分洛朗多项式在第二）。希望能解决几个关于这些物体的悬而未决的问题。

Knudson studies the homology of linear groups over various rings of interest in algebraic K-theory. In this project he focuses attention on the low-dimensional homology groups of the general linear group over the coordinate ring of an elliptic curve over a number field with the goal of proving that the second K-group of such a curve has finite rank. This extends Knudson's previous work on the homology of rank one groups over such rings. In addition, a new homology theory based on algebraic cycles in the product of a scheme with the simplicial classifying scheme of an algebraic group is constructed and its properties investigated. The hope is that this construction will lead to a proof of the Friedlander-Milnor conjecture concerning the homology of algebraic groups made discrete. Also, the investigator's previous work on the structure of special linear groups over integral Laurent polynomial rings is used in the study of the Burau representation of the braid groups. This representation is known to be faithful for three strings and unfaithful for five or more strings; the remaining case of four strings is singled out for study in this project. Finally, Knudson studies the completion of a discrete group relative to a Zariski dense representation in a reductive group over a field of positive characteristic. This generalizes the classical unipotent completion of a group and extends to positive characteristic R. Hain's work in characteristic zero.A scheme is a geometric object constructed from solution sets of polynomial equations. Algebraic K-theory associates to a scheme a sequence of groups which encode information about the scheme. One aspect of this project is the study of the K-groups of an elliptic curve. Such curves have remarkably rich structure and appear in various branches of mathematics such as algebraic geometry and coding theory. A seemingly unrelated part of this project concerns the so-called Burau representation of the braid groups which is intimately connected with knot theory. These diverse topics are unified by studying the structure of groups of matrices with entries in various rings (the coordinate ring of the elliptic curve in the first case and the ring of integral Laurent polynomials in the second). The hope is to solve several outstanding conjectures about these objects.