Collaborative Research: NSF-BSF - Dynamics Through the Viewpoint of Self-Similar Algebras: Topological Markov Chains and Self-Similar Groups.

Cuny City College, New York NY

Investigators

Abstract

This award supports a project exploring the connections between algebra and dynamics through the study of mathematical structures known as operator algebras. Operator algebras are a fundamental object of study in analysis and mathematical physics, with connections and applications to many other mathematical fields. In this context, they arise from dynamical systems called shifts of finite type, and from groups that exhibit self-repeating patterns, like the patterns found in fractals. These systems appear in a wide range of mathematical areas, including data encryption, statistical physics, neural biology, and fractal geometry. The research supported by this award will lead to new insights into how symmetry and complexity interact in mathematical systems, expanding foundational knowledge in pure mathematics. In addition to advancing theory, the project will support undergraduate and graduate students as well as earlier career researchers. This is a project funded jointly by the National Science Foundation's Division of Mathematical Sciences, in the Directorate for Mathematical and Physical Sciences (NSF-MPS-DMS), and the Israel Binational Science Foundation (BSF) in accordance with the Memorandum of Understanding between the NSF and the BSF. A major goal in the field of operator algebra is to produce new invariants for dynamical systems by studying algebras associated to them. This goal has been continuously advanced over the years, especially in Elliott's classification program for simple C*-algebras. The project aims to strengthen such connections between dynamics, group theory, operator algebra theory, and ring theory. Through (operator) algebras, a bridgehead will be created for studying some of the most subtle anomalies related to subshifts of finite type (SFTs) and self-similar groups. This will be achieved by studying specific interactions between graph algebras, as well as new invariants for SFTs. Inspired by several prominent interactions between graph algebras and symbolic dynamics, the project also aims to characterize structure-preserving isomorphisms between algebras associated to self-similar groups with the goal of uncovering new dynamical phenomena. This will shed light on some of the most important problems in symbolic dynamics, graph algebras, as well as on self-similar groups and their associated algebras. In particular, by working in the more general framework of self-similar groupoids, it will be possible to generalize shift equivalence and strong shift equivalence to self-similar groups in order to understand these structure-preserving isomorphisms and Morita equivalences in the context of self-similar groups. 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.