Mathematical Sciences: Isogenies and Operations in Complex Oriented Cohomology

9401550 Miller研究了椭圆曲线的若干模问题，如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作为代数拓扑学中的广义上同调环出现。特别地，有限域上形式群的变形的子群和同调理论与具有较高交换性的环谱的Morava K-理论和完备E(N)上同调密切相关。博士后助理尼尔·斯特里克兰(Neil Strickland)的目标是证明一些猜想，使这种联系更加准确。这项研究是代数拓扑学的一部分，它研究高维形状的某些特征，例如它们有多少个洞，这些洞是如何连接在一起或相互缠绕的，等等。即使物理空间只有三个维度，这些问题也是有意义的。例如，如果一个量依赖于其他七个量，那么可以想象在抽象的八维空间中绘制函数关系的图形。研究这个图的性质是可能的，也是有用的，尽管它不能在物理上实现。因为这样的空间很难想象，所以把关于它的问题转化成代数问题是很有帮助的。一般说来，有两种转换方法，适用于不同类型的问题。在几何学中，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，基本的操作是计算空洞(尽管要准确地说出六维物体中的三维洞是什么意思并不那么容易)。这门学科的一个有趣的特点是，某些拓扑问题以一种意想不到的方式引起代数问题，这些问题以前在数学中显然是无关的，例如数论。目前的项目旨在阐明这一现象的一个相当壮观的例子，并扩展相关的代数知识，以涵盖更广泛的拓扑情况。***

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. ***