Mathematical Sciences: Homotopy Fixed Point Problems in K-Theory

利用群作用于“有界k群”的同伦不动点集，研究无穷离散群群环的代数k理论。他希望将这些方法应用于具有边界的非正弯曲流形的所有基本群、s -算术群和p-进群的离散子群的群的类别加以推广。此外，他计划研究从流形M的稳定伪同位素空间到其自由环空间S1-同伦不动点集的Dennis迹映射，对称群表示中的奇异轨迹，以及等变稳定同伦理论中的轨道谱。k理论是寻找空间拓扑不变量和确定这些不变量性质的标准技术之一。这涉及到复杂的代数。通常将几何对象与代数对象联系起来，用代数对象进行计算，然后将结果转换回几何。

The investigator will work on the algebraic K-theory of group rings of infinite discrete groups, using the homotopy fixed point set of the action of the group on a "bounded K-group." He hopes to extend the class of groups to which the methods apply to all fundamental groups of non-positively curved manifolds with boundary, to S-arithmetic groups, and to discrete subgroups of p- adic groups. In addition, he plans to study the Dennis trace map from the stable pseudoisotopy space of a manifold M to the S1- homotopy fixed point set of its free loop space, the singular loci in representations of the symmetric group, and orbit spectra in equivariant stable homotopy theory. K-theory is one of the standard techniques for finding topological invariants of spaces and for determining properties of these invariants. Sophisticated algebra is involved. Typically one relates geometric objects to algebraic ones, calculates with the algebraic objects, and then translates the results back into geometry.