Jordans Theorem in Number Theory, Group Theory, and Quantum Topology

Indiana University, Bloomington IN

Investigators

Abstract

The investigator intends to work on several projects related to variants of Jordan's theorem characterizing finite subgroups of GL(n). These include a joint project with Richard Pink to analyze the adelic image of Galois representations with coefficients in a function field, a joint project with Alex Lubotzky to understand specializations of Zariski-dense representations of discrete groups, and a joint project with Michael Freedman and Zhenghan Wang to understand monodromy of representations of mapping class groups arising from TQFTs. The last project is part of Freedman's project of replacing the qubit model of quantum computation with a new model based on ideas from quantum topology. Symmetry is a unifying theme in many areas of mathematics and physics, including the subjects touched on in this proposal, number theory, algebra, topology, and quantum field theory. Mathematicians have studied groups, that is, abstract symmetry types, for almost two hundred years. One of the earliest major results is Jordan's theorem, which asserts, more or less, that a geometric figure can have a complicated symmetry group only if it lives in a space of many dimensions. This proposal deals with several extensions and applications of Jordan's result. The motivating problem comes from algebraic number theory, the study of number systems. These systems can have intricate groups of symmetries, which can be externalized as symmetries of "spaces" analogous to the usual spaces of geometry. The investigator intends to probe the symmetry of certain number systems by means of a new extension of Jordan's theorem. In a different direction, the investigator intends to use a similar class of methods to analyze the internal symmetry of certain physical systems. Michael Freedman has recently proposed using the systems in question as the basis for a fundamentally new type of quantum computer which should be much less vulnerable to the decoherence problem which has plagued existing designs. For this to work, one needs a large enough symmetry group to allow the new machine to simulate the internal state of a machine of the old type.