Graduate Student Colloquium
William Simmons
UIC
Complete differential varieties, calculations, and model theory
Abstract: A differential ring is a ring with an additive homomorphism $\delta$
satisfying the usual product rule. The algebraic and geometric behavior of
differential rings presents interesting variations on standard commutative
algebra and algebraic geometry. The classical fundamental theorem of
elimination theory provides a good case study. It asserts that projective
varieties over an algebraically closed field $K$ are \emph{complete}. That
is, if $V$ is such a projective variety and $W$ is any algebraic variety
defined over $K$, then the projection $V\times W \rightarrow W$ takes
Zariski closed sets to Zariski closed sets. The differential case is more
complicated and has features not present in the original. We discuss how
insights from computational examples and model theory suggest a path
through the complexity.
Monday February 25, 2013 at 4:15 PM in SEO 636