Non-Commutative Spaces, Their Symmetries, and Geometric Quantum Group Theory

非交换空间、它们的对称性和几何量子群论

基本信息

批准号：
2001128
负责人：
Alexandru Chirvasitu
金额：
$ 17.5万
依托单位：
SUNY at Buffalo
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001128&HistoricalAwards=false
关键词：
Non Commutative Spaces Their Symmetries

项目摘要

The project fits within the scope of the branch of mathematics known as noncommutative geometry, originating primarily in the discovery early in the 20th century that the then-newly-discovered phenomena of quantum mechanics require a novel mathematical formalism. The main point of departure from classical (or non-quantum) mathematics is the fact that by the very small-scale nature of our ambient world, certain pairs of measurable physical quantities cannot be measured simultaneously (with the momentum and position of a particle serving as the preeminent example). Mathematically, this manifests as the non-commutativity of a pair of transformations on a physical system, justifying the name "noncommutative" for the relevant field of study. The formal objects that model the symmetries (that is, structure-preserving transformations) of a physical system modeled according to this paradigm are known as a "quantum groups", and they are the central theme of the present research proposal. The training of graduate students is an important part of this project.A number of broader themes inform the problems under consideration. As one example, discrete quantum groups, like their classical counterparts, fall into a constellation of taxonomic classes based on the approximation properties enjoyed by their group algebras. The quantum versions of the classical results are typically more technically demanding, only partially settled, and good test beds for the strengths of the geometric-group-theoretic and operator-algebraic techniques that jointly make classical discrete groups such rich geometric and analytical objects. In another direction, much light can be shed on the structure and above-mentioned approximation properties of quantum groups (discrete or more generally, locally compact) by category and representation-theoretic means. For that reason, results to the effect that group-theoretic data (e.g. the center of a locally compact quantum group) can be reconstructed from categories of unitary representations with their underlying structure are of some interest in the field and the project. As a third example, randomness features heavily in the study of groups and other discrete objects (probabilistic methods are very important in graph theory, for instance); the general phenomenon whereby a "generic" object, constructed randomly (with the technical meaning of that phrase depending on the specifics of the problem) tends to be highly asymmetric appears to replicate in the quantum setting, with "most" finite graphs, finite metric spaces, etc. admitting no quantum symmetries. Such generic rigidity results always recover their classical counterparts (given that quantum symmetries always encompass ordinary ones), but usually require more involved and often more enlightening proof techniques. It is hoped the requisite eclectic mix of approaches to the problems (combinatorial, representation-theoretic, probabilistic, operator-algebraic, etc.) will offer insight not only into the nature of the quantum-mathematical objects ostensibly being studied, but also into the classical versions thereof.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目符合数学分支的范围，称为非交通性几何形状，主要起源于20世纪初期的发现，当时量子力学的新发现现象需要一种新颖的数学形式主义。从古典（或非量词）数学的出发点是，由于我们周围世界的非常小的本质，无法同时测量某些可测量的物理数量（以作为杰出示例的粒子的势头和位置）。从数学上讲，这表现为物理系统上一对转换的非交换性，证明了相关研究领域的“非交通”名称。根据该范式建模的物理系统对称对称的形式对称对称的对称对象（即结构保存转换）被称为“量子组”，它们是本研究建议的中心主题。对研究生的培训是该项目的重要组成部分。许多广泛的主题介绍了所考虑的问题。作为一个例子，离散的量子群（例如其经典对应物），基于其组代数所享有的近似属性，属于分类类别的星座。经典结果的量子版本通常在技术要求更高，只有部分定居，并且良好的测试床，用于几何组理论和操作符和操作符 - 算子 - 代码技术的优势，这些技术共同使经典的离散组这样丰富的几何学和分析对象。在另一个方向上，可以通过类别和表示理论手段的量子组（离散或更普遍的局部紧凑）的结构和上述近似特性散发出许多光。因此，可以从具有其基础结构的单一表示类别重建群体理论数据（例如，本地紧凑量量子组的中心）的结果是在该领域和项目中具有一定的兴趣。作为第三个例子，随机性在研究组和其他离散对象的研究中很大程度（例如，概率方法在图理论中非常重要）；一般的现象，即“通用”对象随机构造（根据问题的具体细节，该短语的技术含义）倾向于在量子设置中复制高度不对称，并具有“大多数”有限图，有限的度量空间等。这样的通用刚度结果总是恢复其经典的对应物（鉴于量子对称性总是包含普通的对称性），但通常需要更多涉及的量子，而且通常需要更多启发性的证明技术。希望将问题（组合，代表性理论，概率，概率，操作员 - 地面等）的折衷混合在一起，不仅可以洞悉量子存在对象的本质，从表面上研究了量子存在的对象的性质，而且还可以通过其构建奖项来构建元素。和更广泛的影响审查标准。