Title: When classical algebra meets modern geometry
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Abstract: Modern algebraic geometry is usually understood as a
mechanism linking commutative algebra to geometry. I will discuss how
elementary non-commutative algebra fits into the picture, with a focus
on the modern geometry hidden in the most classical of algebraic
objects: Hamilton's quaternions and their generalizations. The
discovery of this hidden geometry has allowed dramatic progress in our
understanding of the fine structural properties of these algebras,
further strengthening and deepening the link between algebra, number
theory, and geometry. I'll begin at the beginning - with Hamilton in
1843 - and end with a description of the limits of our knowledge in
2011. In spite of our new powerful tools, the distance between 1843
and 2011 is remarkably small.