Mathematical Sciences: Classical Homotopy Theory

9400587 Cohen F.R. Cohen打算考虑经典同伦理论中的问题和其他学科的分支，涉及离散群的上同调，某些模空间，以及Whitehead方的进一步研究。他的主要课题之一是将Barratt有限指数猜想的部分解推广到一般情况。他打算继续分析在这项工作中出现的群，以及它们与涉及经典同伦理论的其他数学部分的联系，以及它们与某些模空间的联系。他将继续研究Whitehead平方，涉及到Cayley-Dickson代数和有理函数空间之间的相互作用。F.R. Cohen和S. Gitler将继续合作研究嵌入空间的特征类，这些特征类在他们早期的工作中是隐含的，并且与Vassiliev定义的结的不变量有关。这个项目研究的是如何将大维度的球体包裹在小维度的物体上，至少达到连续变形的程度。这种类型的几何问题在数学和其他相关领域（如物理）中无处不在。一些基本结构是由在表面上运动的粒子的几何和代数结果给出的，这些粒子是由时间坐标参数化的。在不同的时间，粒子可能回到它们原来的位置。这种几何性质的代数类比称为扭转。各种自然发生的类似曲面的空间的扭转增长边界是本项目的主要课题之一。这项工作的一个目标是攻击一个突出的问题，在这个问题上被称为Barratt的有限指数猜想。第二个出乎意料的分支是，这项工作与其他一些对物理学感兴趣的人的工作重叠。＊＊＊

9400587 Cohen F.R. Cohen intends to consider problems in classical homotopy theory and offshoots to other subjects involving cohomology of discrete groups, certain moduli spaces, and a further study of the Whitehead square. One of the main projects is to extend his recent partial solution of M.G. Barratt's finite exponent conjecture to the general case. He intends to continue an analysis of the groups appearing in this work together with their connections with other parts of mathematics involving classical homotopy theory and their connections to certain moduli spaces. He will continue his study of the Whitehead square involving an interplay between Cayley-Dickson algebras and spaces of rational functions. F.R. Cohen and S. Gitler will continue joint work in a study of characteristic classes for spaces of embeddings which are implicit in their earlier work and which are related to invariants of knots defined by Vassiliev. This project is a study of how large dimensional spheres can be wrapped around smaller dimensional objects at least up to continuous deformations. This type of geometric problem occurs ubiquitously in mathematics and other related areas such as physics. Some of the basic structure is given by the geometric and algebraic consequences of particles which move on a surface and which are parametrized by a time coordinate. At various times, the particles may return to their original position. The algebraic analogue of this geometric property is known as torsion. Bounds on the growth of torsion for various naturally occurring spaces which are analogues of surfaces is one of the main subjects of this project. One goal of this work is an attack on an outstanding problem in the subject known as Barratt's finite exponent conjecture. A second and unexpected offshoot is that this work overlaps with work of some others who are interested in physics. ***