Place: Chauvenet 201 at 12:00 pm
Time: Friday, April 11Speaker: Henry Schenck
Title: Splines, Homological Algebra, and some interesting ideals
Abstract: For a simplicial subdivison $\Delta$ of a region in ${\bf R}^n$, we analyze the dimension of the vector space $C^r_k(\Delta)$ of $C^r$ piecewise polynomial functions (splines) on $\Delta$ of degree at most $k$. This problem is closely related to the structure of the graded ${\bf R}[x_0 \ldots x_n]$ module $C^r(\hat \Delta)$, where $\hat \Delta$ is the join of $\Delta$ with the origin in ${\bf R}^{n+1}$.
I will define a chain complex, such that $C^r(\hat \Delta)$ appears as the top homology module, and will prove that if $n=2$ (i.e. $\Delta$ is planar), then $C^r(\hat \Delta)$ is free if and only if the first homology module of the complex vanishes. Finally, I will show that there is a short exact sequence which relates the behavior of $C^r(\hat \Delta)$ to the behavior of ideals of the form $<(x+a_1y)^{r+1},\cdots,(x+a_ky)^{r+1}>$, with $a_i \ne a_j$ if $i \ne j$, and will analyze the structure of these ideals.