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

研究人员打算研究几个与刻画GL(N)的有限子群的Jordan定理的变体有关的项目。其中包括与Richard Pink的联合项目以分析函数域中系数的Galois表示的ADELIC映象，与Alex Lubotzky的联合项目以了解离散群的Zariski稠密表示的专门化，以及与Michael Freedman和郑汉Wang的联合项目以了解由TQFT产生的映射类群的表示的单行。最后一个项目是弗里德曼基于量子拓扑学思想用新的模型取代量子计算的量子比特模型的项目的一部分。对称性是许多数学和物理领域的统一主题，包括本提案中涉及的主题、数论、代数、拓扑学和量子场论。近两百年来，数学家一直在研究群，也就是抽象的对称类型。最早的主要结果之一是乔丹定理，该定理或多或少地断言，一个几何图形只有当它生活在多维空间时才能有一个复杂的对称群。该提案涉及对Jordan结果的几个推广和应用。激励问题来自代数数论，也就是研究数字系统的学科。这些系统可以有错综复杂的对称组，这些对称组可以外化为类似于通常几何空间的“空间”的对称性。研究人员打算通过乔丹定理的一个新的推广来探索某些数系的对称性。在不同的方向上，研究人员打算使用一类类似的方法来分析某些物理系统的内部对称性。迈克尔·弗里德曼最近提议将这些系统作为一种全新类型量子计算机的基础，这种计算机应该不太容易受到困扰现有设计的退相干问题的影响。要做到这一点，人们需要一个足够大的对称群，以允许新机器模拟旧机器的内部状态。

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.