Mathematical Sciences: Loop Spaces and Classical Homotopy Theory

首席研究员正在研究几个相关领域。许多问题都与由希尔顿-米尔诺定理得到的经典分配律公式有关。这里的第一个问题是对Kervaire不变量强形式的重新表述，第一步是分析Dickson的Cayley数类比中的零因子。第二个问题是分析一个更简单、稳定的分配律。一个方向是Barratt有限指数猜想的稳定模拟。在一个相关的项目中，他继续了他过去关于构形空间同调的一些工作。这些研究主要是为了理解和完善代数技术。具体地说，这些技术允许计算几何流形的定性性质，例如连通性、结性、不动点性质等等。

The principal investigator is studying several related areas. A number of problems are connected to the classical distributivity formula obtained from the Hilton-Milnor theorem. Problem one here is a reformulation of the strong form of the Kervaire invariant, the first step being the analysis of zero divisors in Dickson's analogues of the Cayley numbers. The second problem is the analysis of a simpler, stable distributivity law. One direction is a certain stable analogue of Barratt's finite exponent conjecture. In a related project, he continues some of his past work on the homology of configuration spaces. These investigations are mainly attempts to understand and perfect algebraic techniques. Specifically, the techniques permit calculations concerning qualitative properties of geometric manifolds, e.g. connectedness, knottedness, the fixed- point property, and so forth.