Mathematical Sciences: Isogenies and Operations in Complex Oriented Cohomology

数学科学：复向上同调中的同构和运算

• 批准号: 9401550
• 负责人: Haynes Miller
• 金额: $ 4.56万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-08-01 至 1998-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401550&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Isogenies; Operations; Complex

9401550 Miller A number of moduli problems for elliptic curves have been studied, such as that of classifying elliptic curves with a given point of specified exact order. Analogous but more intricate problems can be studied in the context of one-dimensional formal groups. The resulting algebraic objects also arise as generalized cohomology rings in algebraic topology. In particular, the theory of subgroups and isogenies of deformations of formal groups over a finite field is closely connected with the Morava K-theory and completed E(n) cohomology of ring spectra with higher commutativity properties. Neil Strickland, the postdoctoral associate, aims to prove a number of conjectures making this connection more precise. This research is part of algebraic topology, which studies certain features of higher dimensional shapes, such as how many holes they have, how the holes are linked together or twisted around each other and so on. These problems are meaningful even though physical space has only three dimensions. For example, if one quantity depends on seven others, then one can imagine drawing a graph of the functional relationship in an abstract eight-dimensional space. It is possible and useful to study the properties of this graph, even though it cannot be realized physically. Because it is so hard to visualize such a space, it is helpful to convert questions about it into algebraic problems. Broadly speaking, there are two methods of conversion, appropriate to different types of questions. In geometry, the basic algebraic operation is to measure distances and angles. In topology, one might say that the basic operation is to count holes (although it is not so easy to say what precisely is meant by a three-dimensional hole in a six-dimensional object). An interesting feature of the subject is that certain topological questions give rise in an unexpected way to algebraic problems which have previously been addressed in apparently unrelated are as of mathematics, such as number theory. The current project aims to clarify one quite spectacular example of this phenomenon, and to extend the relevant algebraic knowledge so as to cover a wider family of topological situations. ***

9401550米勒 椭圆曲线的模问题已经被研究了很多，例如对具有指定精确阶的给定点的椭圆曲线的分类问题。 类似但更复杂的问题可以在一维形式群的背景下研究。 由此产生的代数对象也出现在代数拓扑中的广义上同调环。 特别地，有限域上的子群和形式群变形的同构理论与Morava K-理论和具有更高交换性的环谱的完全E（n）上同调密切相关。 尼尔·斯特里克兰，博士后助理，旨在证明一些使这种联系更加精确的假设。 这个研究是代数拓扑学的一部分，它研究高维形状的某些特征，例如它们有多少个孔，孔如何连接在一起或相互缠绕等等。这些问题是有意义的，即使物理空间只有三维。 例如，如果一个量依赖于其他七个量，那么我们可以想象在一个抽象的八维空间中画一个函数关系图。 尽管在物理上无法实现，但研究这种图的性质是可能的，也是有用的。 因为很难想象这样一个空间，所以将有关它的问题转化为代数问题是很有帮助的。一般来说，有两种转换方法，适用于不同类型的问题。 在几何中，基本的代数运算是测量距离和角度。 在拓扑学中，人们可能会说，基本的操作是计算孔（尽管在一个六维物体中精确地说一个三维孔是什么意思并不那么容易）。 一个有趣的特点的主题是，某些拓扑问题产生了意想不到的方式代数问题，以前已解决的显然无关的是数学，如数论。 目前的项目旨在澄清这种现象的一个相当壮观的例子，并扩展相关的代数知识，以涵盖更广泛的家庭拓扑的情况。 ***