Ergodic Embeddings, Bimodule Decomposition, and the Structure of Type II1 Factors

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Rigidity in mathematics occurs when certain geometric objects, such as a group of transformations, can be recognized by merely knowing some partial information, such as their factors. Introduced by von Neumann in the 1930s to study quantum mechanics, factors are irreducible algebras of infinite matrices, where the product of two elements, A times B, is in general different from the product in reverse order, B times A. This project aims to combine tools including deformation-rigidity theory, ergodic embeddings, approximation and simulation techniques, and reconstruction methods to study rigidity in factors and to tackle several longstanding questions in this area. Rigidity results can be relevant to many areas of mathematics and its applications, including computer science, complexity theory, quantum information theory, the design of computer networks, and the theory of error-correcting codes. This project contributes to workforce development through the training of graduate students in topics related to the project research.A striking feature of the II1 factor framework is its ability to host both rigidity and randomness phenomena. Earlier work exploiting the tension between these opposing paradigms led to striking discoveries and to fruitful interaction between study of II1 factors and other areas, such as C*-algebras, free probability, ergodic theory, group theory (measured, geometric, arithmetic, etc.), quantum groups, random matrices, and descriptive set theory. This work developed several important techniques to study II1 factors: finite dimensional approximation and reconstruction methods in subfactor theory, deformation rigidity theory, intertwining by bimodules, and incremental patching. This project aims to employ the technique of iterative ergodic embeddings with control of bimodule structure in combination with previous techniques to tackle several important questions: (1) the non-isomorphism of the free group factors and their "coarseness," a deep structural property that generalizes many prior results and conjectures about these factors; (2) Connes' embedding conjecture (notably for groups) and the sofic group problem; and (3) Connes' bicentralizer conjecture for III1 factors. The project aims to broaden the scope of the techniques under development and to deepen the interaction of operator algebra with other areas of mathematics.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.