Mathematical Sciences: Equivariant Bordism and Formal Group Laws

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9704437 Sadofsky Recall Quillen's theorem that the formal group law over the ring of complex cobordism classes of manifolds induced by the multiplication on infinite complex projective space is isomorphic in a natural way to the Lazard's universal formal group law. Complex cobordism occupies a central place in stable homotopy theory, in part due to the Nilpotence and Periodicity theorems of Hopkins, Devinatz and Smith. Quillen's theorem is of fundamental use, as it allows one to use techniques from the study of formal groups to produce, and calculate with, various cohomology theories based on complex cobordism. A problem of longstanding importance in equivariant theory is to understand complex cobordism classes of G-manifolds (where G is some Lie group). This research will explore this question from the point of view of formal group laws. The goal is to use the methods from Quillen's paper in the Advances to get a handle on equivariant complex cobordism, and to link this with the notion of equivariant formal group laws, preliminaries of which are set forth in Mike Cole's University of Chicago thesis. When is the set of positions of some mechanical system the boundary of the set of positions of some other mechanical system? As a simple example, think of the circle around some point in the plane as being the set of positions of a pendulum. This is the boundary of the set of positions of a pendulum of the same length, but with a joint halfway along the rod joining the bob to the center (said joint able to bend from 0 to 180 degrees). This set of positions describes a disk. This question has a very beautiful and elegant answer in terms of the algebraic topology of the manifold, given by the Fields medal winning work of Rene Thom in the 1950s. In certain cases, the answer to this question turns out to be related to "formal group laws," a class of algebraic rules for adding two quantities, with properties similar to, but not identicial to ordinary add ition. This research is directed at answering our original question when, in addition, it is required that the set of positions of the mechanical system be the boundary of the set of positions of some other mechanical system in a way that respects some symmetries of the original system. Some of this is already known in principle in simple cases, but formal group laws with additional symmetries will provide a framework for the answer and may allow extensions beyond special cases. ***