Noncommutative ring theory and its applications

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The study of rings began as a way to gain a deeper, unifying understanding of the laws which govern the integers, matrices, and other mathematical concepts.  In the 1960s, Gelfand and Kirillov noticed that one can associate a number to each ring; today, we call this number the Gelfand-Kirillov dimension of the ring.  It gives one an idea of the growth rate of the ring.  While the work of Gelfand and Kirillov greatly advanced our understanding of rings, at the same time it showed that in many ways our understanding of rings was still lacking.  Specifically, rings with small Gelfand Kirillov dimension are not well-understood once the dimension is larger than one.  This proposal deals with finding the underlying laws which govern algebras of low Gelfand-Kirillov dimension.