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This talk presents a relatively easy entry point into the world of operator algebras and noncommutative geometry. The two main goals for the talk are to provide some basic definitions from the theory of free group factors and to present a problem which has an operator-theoretic statement and an algebraic flavor, but outlines some of the subtle difficulties in the theory. More specifically, we introduce a notion of transitive family of subspaces relative to a type II_{1} factor, and hence a notion of transitive family of projections in such a factor. We show that whenever \cal{M} is a factor of type II_{1} and \cal{M} is generated by two self-adjoint elements, then \cal{M} \otimes M_{2}(C) contains a transitive family of 5 projections. Finally, we exhibit a free transitive family of 12 projections that generate a factor of type II_{1}.
Refreshments at 4:00 pm in ES 152