We construct an operad Phyl whose operations are the edge-labelled trees used in phylogenetics. This operad is the coproduct of Com, the operad for nonunital commutative algebras, and [0, ∞), the operad with unary ope... more abstractWe construct an operad Phyl whose operations are the edge-labelled trees used in phylogenetics. This operad is the coproduct of Com, the operad for nonunital commutative algebras, and [0, ∞), the operad with unary operations corresponding to nonnegative real numbers, where composition is addition. We show that the Markov models used to reconstruct phylogenetic trees from genome data give coalgebras of Phyl. These always extend to coalgebras of the larger operad Com + [0, ∞], since Markov processes on finite sets converge to an equilibrium as time approaches infinity. We show that for any operad O, its coproduct with [0,∞] contains the Boardman-Vogt operad W(O). To prove these results, we need to describe the coproduct of operads using labelled trees.

Phylogenetics, Markov Processes, Homotopy theory of operads, and Operads

Persistent homology is a method used in topological data analysis to study qualitative features of data. It is robust to perturbations, independent of dimensions and scale, and provides a compact representation of the... more abstractPersistent homology is a method used in topological data analysis to study qualitative features of data. It is robust to perturbations, independent of dimensions and scale, and provides a compact representation of the outputs. The computation of persistent homology, despite recent progress, remains a wide open area with numerous important and fascinating challenges. We investigate the challenges of computing persistent homology, and we navigate the various algorithms that can be used for it. Specifically, we evaluate the (currently available) open-source implementations of persistent-homology computations on a range of synthetic and real-world data sets, and we indicate which algorithms and implementations are best suited to these data. We provide guidelines for the computation of persistent homology, make our own implementations used in this study available to the public, and suggest measures to quantify the challenges of the computation of persistent homology.

Topological Data Analysis and Computational Algebraic Topology

Homotopy Theory and Tropical Mathematics

Elliptic curves and Tropical Geometry