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

调查人员打算工作的几个项目有关的变种约旦定理的特点有限子群的GL（n）。 其中包括与Richard Pink的联合项目，分析函数域中系数的Galois表示的adelic图像，与Alex Lubotzky的联合项目，以了解离散群的Zakirki密集表示的专业化，以及与Michael Freedman和Zhenghan Wang的联合项目，以了解TQFT产生的映射类群表示的单值性。 最后一个项目是弗里德曼的项目的一部分，该项目是用基于量子拓扑思想的新模型取代量子计算的量子比特模型。对称性是数学和物理学许多领域的统一主题，包括本提案涉及的主题，数论，代数，拓扑和量子场论。 数学家研究群，即抽象的对称类型，已经有近两百年的历史了。 最早的主要结果之一是乔丹定理，它或多或少地断言，一个几何图形只有在多维空间中才能有复杂的对称群。 本文讨论了Jordan结果的几个推广和应用。 激励问题来自代数数论，研究数字系统。 这些系统可以具有复杂的对称群，这些对称群可以具体化为类似于通常的几何空间的“空间”的对称。 研究者试图通过对约当定理的一个新的推广来探讨某些数系的对称性。 在另一个方向，研究者打算使用类似的方法来分析某些物理系统的内部对称性。 Michael Freedman最近提出，将这些系统作为一种全新的量子计算机的基础，这种计算机应该不那么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，需要一个足够大的对称群，以允许新机器模拟旧类型机器的内部状态。

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.