CAREER: Invariants and Entropy of Square Integrable Functions

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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.