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 classifica