CAREER: Invariants and Entropy of Square Integrable Functions

熵是热力学和信息论中产生的一个量。它量化了物理系统的随机性或不确定性状态，在机器学习、数据压缩和量子力学中有许多应用。在这个项目中，熵在两种情况下进行了研究。第一种方法考虑了动力系统的熵。动态系统描述某一物理系统随时间变化的状态（例如，房间内某种气体的量，病毒的传播）。把“时间”抽象起来，用一个叫做“群”的对称的离散系统来代替它，这是有用的，也是自然的。动态熵最初是通过信息论的观点来定义的，最近的发展表明，这一观点允许人们将熵理论扩展到某一类被称为“sofic groups”的群体下的进化。在这个新框架中，熵描述了一个无限系统有多少有限近似；科学探究的一个基本问题。熵的第二种设置是在冯·诺伊曼代数的背景下（特别是在自由概率中），它作为量子力学的设置自然出现，并为量子计算提供了精确的框架。密码学的潜在应用是巨大的。在这种情况下研究熵等于理解量子力学系统的无序性。这些问题与泛函分析、遍历理论、算子代数、随机矩阵和几何有关，其中一些问题将被明确地解决。教育部分的直接目的是扩大对数学和科学的参与。这包括在弗吉尼亚大学启动一个教学和多样性研讨会，在系里创造一个更加多样化的包容性环境，并举办一个暑期学校，把不同领域的研究人员聚集在一起。计划扩大弗吉尼亚大学桥梁项目的作用，以及该部门与数学联盟的参与。这两项努力明确旨在增加数学中代表性不足的群体的代表性。该提案围绕两个主要项目展开。首先是研究代数作用中的熵，代数作用是指离散群通过自同构作用于紧群上的作用。一个目标是完全解决自由群的代数作用的f不变熵与由希尔伯特空间定义的非紧黎曼流形的广义扭转理论（在拓扑意义上）之间的联系。计划是我们目前对哪些代数动作同构于伯努利位移的理解的扩展。在冯·诺伊曼代数中，首席研究员将扩展他最近的联合工作，表明性质(T)冯·诺伊曼代数很少有有限维近似。这些概念将推广到与左正则表示中的值具有消失第一上同调的群，以及内可服从群。将这些结果扩展到扭曲群von Neumann代数和紧量子群，只要这些对象在左正则表示的自然类似物中具有消失的第一上同调。最后，将“平均维数”一词从遍历理论设置改为冯·诺伊曼代数设置，最终目的是解决著名的负冯诺依曼代数生成问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Entropy is a quantity arising in thermodynamics and information theory. It quantifies the state of randomness or uncertainty of a physical system and has numerous applications in machine learning, data compression, and quantum mechanics. In this project entropy is studied in two settings. In the first the entropy of a dynamical system is considered. Dynamical systems describe the state of a certain physical system (e.g., amount of a certain gas inside a room, spread of viruses) as it changes in time. It is useful and natural to make “time” abstract and replace it with a discrete system of symmetries called “groups.” Dynamical entropy was originally defined through an information theory viewpoint, and recent developments show that this perspective allows one to expand entropy theory to evolution under a certain class of groups called “sofic groups.” In this new framework, entropy describes how many finitary approximations an infinitary system has; a fundamental question of scientific inquiry. The second setting for entropy is in the context of von Neumann algebras (specifically in free probability), which arise naturally as the setting for quantum mechanics and provide a precise framework for quantum computing. Potential applications in cryptography are vast. Investigating entropy in this setting amounts to understanding the disorder of a quantum mechanical system. These problems have links to functional analysis, ergodic theory, operator algebras, random matrices, and geometry, some of which will be explicitly addressed. The educational component is directly aimed at broadening participation in mathematics and the sciences. This includes starting a teaching and diversity seminar at the University of Virginia to create a more diverse inclusive environment in the department and running a summer school to bring researchers in different fields together. An expansion is planned for the role of the bridge program at the University of Virginia, as well as the department's involvement with the math alliance. These two endeavors are explicitly aimed at increasing the representation of underrepresented groups in mathematics. This proposal revolves around two main projects. The first is the study of entropy in algebraic actions, which are actions of a discrete group on a compact group by automorphisms. One goal is to completely settle the connections between f-invariant entropy for algebraic actions of free groups and a generalized torsion theory (in the sense of topology) for noncompact Riemannian manifolds defined via Hilbert spaces. Planned is an extension of our current understanding of which algebraic actions are isomorphic to Bernoulli shifts. In von Neumann algebras, the principal investigator will expand on his recent joint work showing that Property (T) von Neumann algebras have few finite-dimensional approximations. These concepts will be generalized to groups with vanishing first cohomology with values in the left regular representation, as well as to inner amenable groups. Expansion of these results is planned to twisted group von Neumann algebras, and compact quantum groups, provided such objects have vanishing first cohomology in natural analogues of the left regular representation. Lastly, the term mean dimension will be adapted from the ergodic theory setting to the von Neumann algebraic setting with the ultimate goal of settling the famous generator problem for von Neumann algebras in the negative.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.