Mathematical Sciences: Index Theories for G-Spaces and FixedPoints

当G是紧李群时，G-空间和G-对的数值指标理论在Ljusternik-Schnirelmann方法和Bourgin-Yang类型的定理中起着重要的作用。研究人员已经开始研究一种理想值理论，该理论可以处理数值指数失效的情况。一个引人注目的应用是关于具有正交群作用的Stiefel流形上的映射的Borsuk-Ulam定理，其中数值指标是不适当的。这个理想值理论可以推广到对称群G不紧的情形，在无限小的Derham G-上同调的背景下，并且可以用来得到G不紧的Borsuk-Ulam定理。更深入地研究理想值理论应用于不变泛函的临界点理论、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在不动点理论方面的研究，特别是G-映射的Nielsen不动点理论和局部Reidemister迹理论。他们沿着这些思路成功地应用了他们先前的工作，建立了某些经济模型的均衡的存在性。

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.