Geometry Seminar

Date: September 11, 2015

Time: 4:00PM - 5:00PM

Location: BLOC 117

Speaker: Michael DiPasquale

Title: Regularity of Planar Splines

Abstract: The algebra of piecewise polynomial functions (splines) of smoothness r on a subdivision by convex polytopes in an n-dimensional real vector space is of fundamental interest in approximation theory and numerical analysis. In the late 1980s, Billera pioneered the use of tools from commutative and homological algebra in the study of splines. Using this approach, Mcdonald and Schenck provided a formula for the dimension of the vector space of splines of smoothness r and degree at most r over a planar polyhedral subdivision, when d is large enough. In this talk we present an answer to how large d needs to be in order for this formula to hold. The tool we use is the Castelnuovo-Mumford regularity of the algebra of planar splines. No knowledge of spline theory will be assumed.