Mathematical Sciences: Calculus of Funtors and Pseudo-Isotopy Theory

我们将通过“函子演算”来研究任意两连通空间映射的相对伪同伦。“函子演算”是主要研究者为此目的而发明的一种一般的同伦论方法。其中一个重点将是完善这一演算。另一个是关于根据某种循环同调或S1-等变同伦计算幂零或连通理想的相对代数K-理论的一般问题。这项研究的进展将应用于有关几何空间变形的微妙几何问题，即流形，这反过来又影响理论物理的研究。

The relative pseudoisotopy of any two-connected map of spaces will be studied by means of the "calculus of functors," a general homotopy-theoretic method which was invented by the principal investigator primarily for this purpose. One emphasis will be on refining this calculus. Another will be on the general problem of computing the relative algebraic K-theory of a nilpotent or connected ideal in terms of some sort of cyclic homology or S1-equivariant homotopy. Advances in this research will have application to subtle geometric questions about deformations of geometric spaces known as manifolds, which in turn impact research in theoretical physics.