Algebraic data structures for topological summaries

Abstract:
This talk introduces a combinatorial algebraic framework to encode,
compute, and analyze topological summaries of geometric data.  The
motivating problem from evolutionary biology involves statistics on
a dataset comprising images of fruit fly wing veins.  The algebraic
structures take their cue from graded polynomial rings and their
modules, but the theory is complicated by the passage from integer
exponent vectors to real exponent vectors.  The path to effective
methods is built on appropriate finiteness conditions, to replace the
usual ones from commutative algebra, and on an understanding of how
datasets of this nature interact with moduli of modules.  I will
introduce the biology, algebra, and topology from first principles.
Joint work with David Houle (Biology, Florida State), Ashleigh Thomas
(grad student, Duke Math), and Justin Curry (postdoc, Duke Math).