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

该项目符合被称为非对易几何的数学分支的范围，主要源于20世纪初的一项发现，即当时新发现的量子力学现象需要一种新的数学形式主义。经典(或非量子)数学的主要出发点是，由于我们周围世界的小尺度性质，某些可测量的物理量对不能同时测量(粒子的动量和位置是突出的例子)。在数学上，这表现为物理系统上的一对变换的非对易，证明了相关研究领域的“非对易”的名称是合理的。根据这一范例对物理系统的对称性(即保持结构的变换)建模的形式对象被称为“量子群”，它们是本研究提案的中心主题。研究生的培养是这一项目的重要组成部分。一些更广泛的主题反映了正在考虑的问题。举个例子，离散量子群，就像它们的经典对应群一样，根据它们的群代数所享有的近似性质，属于分类类别的星座。经典结果的量子版本通常在技术上要求更高，只是部分确定的，并且是几何群论和算符代数技术优势的良好试验台，这些技术共同使经典离散群成为如此丰富的几何和分析对象。在另一个方向上，可以通过范畴和表示理论的方法来揭示量子群(离散的或更一般地说，局部紧的)的结构和上述近似性质。因此，群论数据(例如，局部紧量子群的中心)可以从具有其底层结构的酉表示范畴中重构的结果在该领域和该项目中具有一定的兴趣。作为第三个例子，随机性在群体和其他离散对象的研究中有很大的特点(例如，概率方法在图论中非常重要)；在这种普遍现象中，随机构造的“一般”对象(该短语的技术含义取决于问题的细节)往往是高度不对称的，似乎在量子环境中重复出现，“大多数”有限图形、有限度量空间等不允许存在量子对称性。这种类属刚性结果总是恢复它们的经典对应结果(考虑到量子对称总是包含普通的)，但通常需要更复杂的、往往更具启发性的证明技术。希望有必要的折衷方法来解决这些问题(组合论、表示论、概率论、算子代数等)。将不仅提供对表面上正在研究的量子数学对象的性质的洞察，而且还提供对其经典版本的洞察。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.