Operator Algebras and Dynamics

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This project will study relationships between L^2-invariant theory and ergodic theory. The former provides invariants for CW-complexes through the process of applying group von Neumann algebra techniques to the algebraic topology of universal covering spaces, and it has applications to geometry and K-theory. Ergodic theory provides invariants such as entropy and mean dimension. These were originally defined for integer group actions, subsequently for amenable group actions, and most recently for sofic group actions. The Principal Investigator also intends to extend the entropy theory of sofic group actions to actions on operator algebras and actions of hyperlinear groups. Classical topology studies the shape of geometric objects. The invariant theory involved in this project achieves such a goal by using so-called von Neumann algebras, which were first introduced as a mathematical tool to study quantum mechanics in physics. Ergodic theory studies the long-time behavior of systems under evolution (e.g., the "sandpile model" in physics). In the last few years, some surprising connections between these two fields have been discovered, connections that lead to applications in both areas. The Principal Investigator plans to deepen and broaden such connections.