Combinatorics & algebra seminar-Polytopes and their Minkowski rings

Speaker: Geir Agnarsson, GMU

Date and time: Tuesday, March 3, 4–5 pm
Place: Rome 206

 

Abstract: From a given collection ${\cal{P}}$ of convex polytopes in a Euclidean space one can form the \emph{Minkowski ring}, which is a commutative ring generated by the indicator functions $\{[P]: P\in {\cal{P}}\}$ where the addition is given naturally as the sum of the functions and where the multiplication is determined by $[P]\cdot[Q] = [P + Q]$; the indicator function of the Minkowski sum of the polytopes $P$ and $Q$. If ${\cal{P}}$ is finite containing $d$ polytopes, then the Minkowski ring can be viewed as an algebra over the complex numbers and can be presented as ${\mathbb{C}}[x_1,\ldots,x_d]/I$ where $I$ is the ideal describing all the algebraic relations implied by the polytopes in ${\cal{P}}$. In this talk we will present algebraic and topological properties of such ideals. -- This is joint work with Jim Lawrence.