Algebra for Topology in Biology and Statistics

With "Big Data" comes a surge of geometric data sets and data objects, particularly in biology and medicine, where shapes, images, videos, networks, and small samples in high dimension are common. One way to deal with such data is to summarize topologically. That, in turn, leads to algebraic structures which take their cue from graded polynomial rings and their modules, although the theory is complicated by the passage from integer exponent vectors to real exponent vectors. The path to effective methods requires finitness conditions to replace the usual ones from commutative algebra. Statistical considerations interact with these finiteness conditions in fundamental ways, some of them relying on an understanding of how datasets of this nature relate to moduli of modules.

The talk will start with a general discussion of biological and medical investigations that lend themselves to techniques based on geometry, including concrete datasets and underlying statistical questions. The primary focus will be on a fundamental problem in evolutionary biology that drove the genesis of the new algebraic ideas covered here; it concerns evolution of changes to discrete morphological features, for which we are doing statistics on a dataset comprising images of fruit fly wing veins.

The main material is joint work in progress with David Houle (Biology, Florida State), Ashleigh Thomas (grad student, Duke Math), Justin Curry (postdoc, Duke Math), and Surabhi Beriwal (undergrad, Duke Math).