Mathematical Sciences: Equivariant Bordism and Formal Group Laws

9704437萨多夫斯基回忆了奎伦的定理，即由无限复射影空间上的乘法诱导的流形的复余边类环上的形式群律以自然的方式与Lazard的普适形式群律同构。复余边论在稳定同伦理论中占有中心地位，这部分归因于Hopkins、Devinatz和Smith的幂零定理和周期定理。奎伦的定理是非常有用的，因为它允许人们使用形式群研究中的技巧来产生和计算基于复余边线的各种上同调理论。在等变理论中，一个长期存在的重要问题是理解G-流形(其中G是某个李群)的复余边类。本研究将从形式群法的角度来探讨这一问题。其目的是使用Quillen在《高级进展》中的论文中的方法来处理等变复余边论，并将其与等变形式群律的概念联系起来，其初步内容已在Mike Cole的芝加哥大学论文中提出。什么时候某个机械系统的位置集是另一个机械系统的位置集的边界？举个简单的例子，把平面上某一点周围的圆想象成一组摆动的位置。这是相同长度的摆的一组位置的边界，但在杆的中途有一个关节将杆子连接到中心(所述关节可以从0度到180度弯曲)。这组位置描述了一个圆盘。这个问题在流形的代数拓扑方面有一个非常美丽和优雅的答案，这是20世纪50年代雷内·托姆的菲尔兹奖获奖作品给出的。在某些情况下，这个问题的答案被证明与“形式群律”有关，这是一类用于将两个量相加的代数规则，具有类似于普通加法的性质，但与普通加法不同。这项研究旨在回答我们最初的问题，即当要求机械系统的位置集是另一个机械系统的位置集的边界时，以一种尊重原始系统的某些对称性的方式。其中一些在简单情况下原则上是已知的，但具有额外对称性的正式群律将为答案提供一个框架，并可能允许扩展到特殊情况之外。***

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. ***