# COLLOQUIUM at Ayres Hall

## Friday, 14 February, 2014

**SPEAKER: **Prof. Henry K. Schenck, University of Illinois

**TITLE: ***Algebra of generalized barycentric coordinates*

**ABSTRACT: **In 1827, M"obius defined barycentric coordinates for a simplex. Motivated by his work in approximation theory and numerical analysis, in 1975 Wachspress generalized the construction, introducing barycentric coordinates $w_d$ for an arbitrary convex polygon $P_d$ on $d$ vertices. These coordinates are rational functions depending on the vertices, and define a rational map from $mathbb{P}^2$ to $mathbb{P}^{d-1}$. Garcia-Sottile asked if it was possible to determine the image, which is useful for a number of reasons. We answer this question, showing $w_d$ is a regular map on a blowup $X_d$ of $mathbb{P}^2$, given by a very ample divisor on $X_d$, so has a smooth image $W_d$. We determine generators for the ideal of $W_d$, and prove that in graded lex order, the initial ideal of I(W_d) is given by a Stanley-Reisner ideal. A flat degeneration argument then allows us to prove that the surface has a number of wonderful properties, which (time permitting) I'll discuss. Rather than dwelling on the esoteric technical details, I'll focus on the concrete and computational aspects, emphasizing how many of the tools appearing in the talk are useful for attacking other problems. Joint work with Corey Irving, Santa Clara Univ.

Refreshments will be available at 3:10 p.m.

## Event Contact

Phone: 974-2463