Mathematical Sciences: Index Theories for G-Spaces and FixedPoints

G-空间和G-对的数值指标理论， 当G是一个紧李群时， Ljusternik-Schnirelmann方法和Bourgin定理 杨型。 研究人员已经开始了一项理想的研究- 有价值的理论，可以处理的情况下，数字- 有价值的指数失败。 一个引人注目的应用是博苏克-乌拉姆 具有正交群的Stiefel流形上映射的一个定理 在数字索引不充分的情况下采取行动。 这个理想- 价值理论可以结转的情况下，对称性 群G不是紧的，在无穷小的上下文中 deRham G-上同调，并可用于获得Borsuk-Ulam 定理，其中G不是紧的。 对理想的更深入研究- 价值理论，因为它适用于临界点理论， 不变泛函，Borsuk-Ulam定理和 等变J-同态是一个主要目标。 研究人员还计划继续他们的研究， 不动点理论，特别强调Nielsen不动点 点理论和局部Reidemeister迹理论。 他们 已经成功地应用了他们以前的工作，沿着这些 以确定平衡点的存在性 经济模式。

Numerical-valued index theories for G-spaces and G-pairs, when G is a compact Lie group, play an important role in the Ljusternik-Schnirelmann method and in theorems of the Bourgin- Yang type. The investigators have begun a study of an ideal- valued theory which can handle situations where the numerical- valued indices fail. A striking application is a Borsuk-Ulam theorem for maps on a Stiefel manifold with orthogonal group action where the numerical index is inadequate. This ideal- valued theory can be carried over to the case when the symmetry group G is not compact, within the context of infinitesimal deRham G-cohomology, and can be applied to obtain Borsuk-Ulam theorems where G is not compact. A deeper study of the ideal- valued theory as it applies to the critical point theory of invariant functionals, to Borsuk-Ulam theorems and to the equivariant J-homomorphism is a primary objective. The investigators also plan to continue their research in fixed point theory with special emphasis on the Nielsen fixed point theory of G-maps and local Reidemeister trace theory. They have made successful application of their prior work along these lines to establish the existence of equilibria for certain economic models.