CAREER: Invariants and Entropy of Square Integrable Functions

职业：平方可积函数的不变量和熵

• 批准号: 2144739
• 负责人: Benjamin Hayes
• 金额: $ 40万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2027-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2144739&HistoricalAwards=false
• 关键词: CAREER; Invariants; Entropy; Square; Integrable

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.

熵是热力学和信息论中出现的一个量。它量化了物理系统的随机性或不确定性的状态，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。第一章考虑了动力系统的熵。动态系统描述某个物理系统的状态（例如，房间内某种气体的量，病毒的传播）随着时间的变化而变化。将“时间”抽象化并用称为“群”的离散对称系统取代它是有用且自然的。动力学熵最初是通过信息论的观点来定义的，最近的发展表明，这种观点允许人们将熵理论扩展到称为“sofic群”的一类群下的进化。在这个新的框架中，熵描述了一个无限系统有多少个有限近似;这是科学探索的一个基本问题。熵的第二个设定是在冯·诺依曼代数的背景下（特别是在自由概率中），它作为量子力学的设定自然出现，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相当于理解量子力学系统的无序。这些问题与泛函分析、遍历理论、算子代数、随机矩阵和几何学有关，其中一些将被明确地讨论。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开始一个教学和多样性研讨会，在部门创造一个更加多样化的包容性环境，并运行一个暑期学校，使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。这两项努力的明确目的是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。第一个是研究代数作用中的熵，代数作用是离散群在紧群上通过自同构的作用。一个目标是完全解决自由群的代数作用的f-不变熵和通过希尔伯特空间定义的非紧黎曼流形的广义扭理论（在拓扑意义上）之间的联系。Planned是我们目前对代数作用同构于伯努利移位的理解的延伸。在冯·诺依曼代数中，主要研究者将扩展他最近的联合工作，表明性质（T）冯·诺依曼代数几乎没有有限维近似。这些概念将推广到左正则表示的值为零的第一上同调群，以及内部顺从群。扩展这些结果计划扭曲群冯诺依曼代数，和紧凑的量子群，提供这样的对象有消失的第一上同调在自然类似物的左正规表示。最后，平均维数一词将从遍历理论的设定调整到冯·诺依曼代数的设定，最终目标是解决冯·诺依曼代数的著名生成器问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。