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Affine algebra

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  • UserVaughan Pratt (Stanford University)
  • ClockThursday 25 September 2008, 14:15-16:00
  • HouseMR12.

If you have a question about this talk, please contact Helen Innes.

We axiomatize affine algebra equationally in terms of the geometric primitives of reflection and centroid. This reduces the usual three-stage algebraic development of affine geometry (fields then vector spaces then torsors) to a single stage. The expansion of this variety with a lawless constant is equivalent to vector spaces over the rationals (or complex rationals when further expanded with quarter-turn rotation). Completion to the continuum is accomplished entirely within this framework via a notion of convergence of pairs of point sets constituting a partial equivalence relation, with Cauchy sets defined as those sets that converge with themselves. The additive fragment based on reflection, whose models we call groves, is of independent interest as a weak generalization of heaps/herds as an affine counterpart of abelian groups.

This talk is part of the DPMMS Pure Maths Seminar series.

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