Benjamin G. Thompson

Statement

I've always loved mathematics, patterns, and bold colors. While I predominantly do research, I regularly think about how to bring advanced mathematics to wider audiences too. Making art out of mathematical tools and ideas is a part of this practice, as I do below with some recently developed objects from algebraic geometry. Most people in the world have no idea what "bumpless pipe dreams" are (nor do they need to!), but I hope my work is able to provide a wordless introduction to them nonetheless. I had a lot of fun doing the mathematics required to make this artwork, as well as the creative choices that followed in assembling it. Working with bumpless pipe dreams in this way was something new for me, hopefully it will be for viewers too.

Artworks

My work illustrates all 12 bumpless pipe dreams (BPDs) of the permutation (125634) in the symmetric group of six elements. BPDs were created by Lam, Lee and Shimozono in 2018 to compute (double) Schubert polynomials. Schubert polynomials represent cohomology classes of Schubert varieties, which are objects of fundamental importance in intersection theory. BDPs are fillings of a n x n square with a certain set of tiles in which the connectivity of the pipes between a pair of adjacent edges is specified by a permutation (in this case (125634)). Schubert polynomials can be recovered from BPDs with a weighted sum calculation determined by blank tiles, which in my work correspond to areas of the squares through which no pipe traverses.