Mathematical Sciences: Isogenies and Operations in Complex Oriented Cohomology

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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. ***