Families of Geometries, Real Algebras, and Transitions

Abstract

This thesis details the results of four interrelated projects. The first of these presents a new proof of the theorem of Cooper, Danciger and Wienhard classifying the limits under conjugacy of the orthogonal groups in GL(n; R). The second provides a detailed investigation into Heisenberg geometry, which is the maximally degenerate such limit in dimension two. The remaining two projects concern understanding geometric transitions which do not occur naturally as limits under conjugacy in some ambient geometry. The third project describes a new degeneration of complex hyperbolic space, formed by degenerating the complex numbers as a real algebra, into the algebra R+R. Inspired by this example, the final project attempts to build the beginnings of a framework for studying transitions between geometries abstractly. As a first application of this, we generalize the previous result and describe a collection of new geometric transitions, defined by constructing analogs of familiar geometries (projective geometry, hyperbolic geometry, etc) over real algebras, and then allowing this algebra to vary.