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Wednesday, September 28, 2016

**Abstract:** Von Neumann developed a theory of vector spaces over certain algebras where the dimension of the vector space is a non-negative real number (or infinity). The algebras are called type II_1 factors and the vector spaces are Hilbert spaces. Applying the dimension to subrings of the rings in question one gets a notion of index of a subring or degree of a ring extension. This index turns out to NOT be any real number bigger than 1, but is in some sense "quantized". Thanks to theorems of Popa, the subfactors are actually classified by systems of planar diagrams and this knowledge has been exploited to actually enumerate all subfactors of index up to 5.25.