Ezra Miller

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  Real multiparameter persistent homology
  Ezra Miller

  Persistent homology with multiple continuous parameters presents fundamental challenges different from those arising with one real or multiple discrete parameters: no existing algebraic theory applies (even poorly or inadequately). In part that is because the relevant modules are wildly infinitely generated. This talk explains how and why real multiparameter persistence should nonetheless be practical for data science applications. The key is a finiteness condition that encodes topological tameness -- which occurs in all modules arising from data -- robustly, in equivalent combinatorial and homological algebraic ways. Out of the tameness condition surprisingly falls much of ordinary (that is, noetherian) commutative algebra, crucially including finite minimal primary decomposition and a concept of minimal generator. The geometry and relevance of these algebraic notions will be explained from scratch, assuming no prior experience with commutative algebra, in the context of two genuine motivating applications: summarizing probability distributions and topology of fruit fly wing veins.

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  Sticky central limit theorems at singularities
  Ezra Miller
  Applications to areas such as biology, medicine, and image analysis require understanding the asymptotics of distributions on stratified spaces, such as tree spaces. In the surprisingly common circumstance when Frechet (intrinsic) means of distributions on stratified spaces lie on strata of low dimension, central limit theorems can exhibit non-classical "sticky" behavior: positive mass can be supported on thin subsets of the ambient space. This talk reports on investigations initiated by a Working Group at the Statistical and Applied Mathematical Sciences Institute (SAMSI) program on Analysis of Object Data, and continued jointly with Stephan Huckemann, Jonathan Mattingly, and Jim Nolen.
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  Data structures for real multiparameter persistence
  Ezra Miller
  Biological data, such as the images of fruit fly wing veins that drive the ongoing investigations reported in this talk, generate persistent homology with multiple parameters each of which varies continuously. Statistical analysis of persistence in this context presents fundamental challenges, such as how to encode persistence summaries for automatic computation and how to carry out statistical analyses with the summaries---theoretically and algorithmically---particularly in view of nontrivial moduli for multiparameter persistence diagrams. This talk presents an algebraic and geometric framework that renders these challenges surmountable while also clarifying the topological interpretation of each multiparameter persistence summary. The framework is new and useful already for two discrete parameters but works equally well for continuous parameters, or even for filtrations by arbitrary partially ordered sets. Joint work with David Houle (Biology, Florida State), Ashleigh Thomas (grad student, Duke Math), and Justin Curry (postdoc, Duke Math).