I am a CAM Adjunct Assistant Professor in the Mathematics Department at UCLA. I obtained a DPhil in Mathematics from the University of Oxford in June 2018. My research focuses on using methods from pure mathematics (such as algebraic topology, algebraic geometry and category theory) to study data.
A fundamental tool in topological data analysis is persistent homology, which allows extraction o... more A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter family of spaces obtained from the data. In applications data often depend on several parameters, and in this case one is interested in studying the persistent homology of a multiparameter family of spaces associated to the data. While the theory of persistent homology for one-parameter families is well-understood, the situation for multiparameter families is more delicate. Following Carlsson and Zomorodian we recast the problem in the setting of multigraded algebra, and we propose multigraded Hilbert series, multigraded associated primes and local cohomology as invariants for studying multiparameter persistent homology. Multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case.
We construct an operad Phyl whose operations are the edge-labelled trees used in phylogenetics. T... more 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 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.
Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative... more Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input. Despite recent progress, the computation of PH remains a wide open area with numerous important and fascinating challenges. The field of PH computation is evolving rapidly, and new algorithms and software implementations are being updated and released at a rapid pace. The purposes of our article are to (1) introduce theory and computational methods for PH to a broad range of applied mathematicians and computational scientists and (2) provide benchmarks of state-of-the-art implementations for the computation of PH. We give a friendly introduction to PH, navigate the pipeline for the computation of PH with an eye towards applications, and use a range of synthetic and real-world data sets to evaluate currently available open-source implementations for the computation of PH. Based on our benchmarking, we indicate which algorithms and implementations are best suited to different types of data sets. In an accompanying tutorial, we provide guidelines for the computation of PH. We make publicly available all scripts that we wrote for the tutorial, and we make available the processed version of the data sets used in the benchmarking.
A fundamental tool in topological data analysis is persistent homology, which allows extraction o... more A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter family of spaces obtained from the data. In applications data often depend on several parameters, and in this case one is interested in studying the persistent homology of a multiparameter family of spaces associated to the data. While the theory of persistent homology for one-parameter families is well-understood, the situation for multiparameter families is more delicate. Following Carlsson and Zomorodian we recast the problem in the setting of multigraded algebra, and we propose multigraded Hilbert series, multigraded associated primes and local cohomology as invariants for studying multiparameter persistent homology. Multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case.
We construct an operad Phyl whose operations are the edge-labelled trees used in phylogenetics. T... more 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 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.
Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative... more Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input. Despite recent progress, the computation of PH remains a wide open area with numerous important and fascinating challenges. The field of PH computation is evolving rapidly, and new algorithms and software implementations are being updated and released at a rapid pace. The purposes of our article are to (1) introduce theory and computational methods for PH to a broad range of applied mathematicians and computational scientists and (2) provide benchmarks of state-of-the-art implementations for the computation of PH. We give a friendly introduction to PH, navigate the pipeline for the computation of PH with an eye towards applications, and use a range of synthetic and real-world data sets to evaluate currently available open-source implementations for the computation of PH. Based on our benchmarking, we indicate which algorithms and implementations are best suited to different types of data sets. In an accompanying tutorial, we provide guidelines for the computation of PH. We make publicly available all scripts that we wrote for the tutorial, and we make available the processed version of the data sets used in the benchmarking.
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