Operads and Phylogenetic Trees

We construct an operad $\mathrm{Phyl}$ whose operations are the edge-labelled trees used in phylogenetics. This operad is the coproduct of $\mathrm{Com}$, the operad for commutative semigroups, and $[0,\infty)$, the operad with unary operations corresponding to nonnegative real numbers, where composition is addition. We show that there is a homeomorphism between the space of $n$-ary operations of $\mathrm{Phyl}$ and $\mathcal{T}_n\times [0,\infty)^{n+1}$, where $\mathcal{T}_n$ is the space of metric $n$-trees introduced by Billera, Holmes and Vogtmann. Furthermore, we show that the Markov models used to reconstruct phylogenetic trees from genome data give coalgebras of $\mathrm{Phyl}$. These always extend to coalgebras of the larger operad $\mathrm{Com} + [0,\infty]$, since Markov processes on finite sets converge to an equilibrium as time approaches infinity. We show that for any operad $O$, its coproduct with $[0,\infty]$ contains the operad $W(O)$ constucted by Boardman and Vogt. To prove these results, we explicitly describe the coproduct of operads in terms of labelled trees.Comment: 48 pages, 3 figure

Magnitude meets persistence. Homology theories for filtered simplicial sets

The Euler characteristic is an invariant of a topological space that in a precise sense captures its canonical notion of size, akin to the cardinality of a set. The Euler characteristic is closely related to the homology of a space, as it can be expressed as the alternating sum of its Betti numbers, whenever the sum is well-defined. Thus, one says that homology categorifies the Euler characteristic. In his work on the generalisation of cardinality-like invariants, Leinster introduced the magnitude of a metric space, a real number that gives the "effective number of points" of the space. Recently, Leinster and Shulman introduced a homology theory for metric spaces, called magnitude homology, which categorifies the magnitude of a space. When studying a metric space, one is often only interested in the metric space up to a rescaling of the distance of the points by a non-negative real number. The magnitude function describes how the effective number of points changes as one scales the distance, and it is completely encoded in the Euler characteristic of magnitude homology. When studying a finite metric space in topological data analysis using persistent homology, one approximates the space through a nested sequence of simplicial complexes so as to recover topological information about the space by studying the homology of this sequence. Here we relate magnitude homology and persistent homology as two different ways of computing homology of filtered simplicial sets.Comment: 21 pages. Part of PhD thesis chapte

Stratifying multiparameter persistent homology

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.Comment: Minor improvements throughout. In particular: we extended the introduction, added Table 1, which gives a dictionary between terms used in PH and commutative algebra; we streamlined Section 3; we added Proposition 4.49 about the information captured by the cp-rank; we moved the code from the appendix to github. Final version, to appear in SIAG

Amplitudes on abelian categories

The use of persistent homology in applications is justified by the validity of certain stability results. At the core of such results is a notion of distance between the invariants that one associates to data sets. While such distances are well-understood in the one-parameter case, the situation for multiparameter persistence modules is more challenging, since there exists no generalisation of the barcode. Here we introduce a general framework to study stability questions in multiparameter persistence. We introduce amplitudes -- invariants that arise from assigning a non-negative real number to each persistence module, and which are monotone and subadditive in an appropriate sense -- and then study different ways to associate distances to such invariants. Our framework is very comprehensive, as many different invariants that have been introduced in the Topological Data Analysis literature are examples of amplitudes, and furthermore many known distances for multiparameter persistence can be shown to be distances from amplitudes. Finally, we show how our framework can be used to prove new stability results.Comment: 49 pages, major revisions throughout; added section on preliminaries, reorganised/improved main sections in the paper, removed discussion about general finitely encoded module

Topology-Aware Surface Reconstruction for Point Clouds

We present an approach to inform the reconstruction of a surface from a point scan through topological priors. The reconstruction is based on basis functions which are optimized to provide a good fit to the point scan while satisfying predefined topological constraints. We optimize the parameters of a model to obtain likelihood function over the reconstruction domain. The topological constraints are captured by persistence diagrams which are incorporated in the optimization algorithm promote the correct topology. The result is a novel topology-aware technique which can: 1.) weed out topological noise from point scans, and 2.) capture certain nuanced properties of the underlying shape which could otherwise be lost while performing surface reconstruction. We showcase results reconstructing shapes with multiple potential topologies, compare to other classical surface construction techniques, and show the completion of real scan data

Changing Patterns of Bloodstream Infections in the Community and Acute Care Across 2 Coronavirus Disease 2019 Epidemic Waves: A Retrospective Analysis Using Data Linkage

BackgroundWe examined community- and hospital-acquired bloodstream infections (BSIs) in coronavirus disease 2019 (COVID-19) and non-COVID-19 patients across 2 epidemic waves.MethodsWe analyzed blood cultures of patients presenting to a London hospital group between January 2020 and February 2021. We reported BSI incidence, changes in sampling, case mix, healthcare capacity, and COVID-19 variants.ResultsWe identified 1047 BSIs from 34 044 blood cultures, including 653 (62.4%) community-acquired and 394 (37.6%) hospital-acquired. Important pattern changes were seen. Community-acquired Escherichia coli BSIs remained below prepandemic level during COVID-19 waves, but peaked following lockdown easing in May 2020, deviating from the historical trend of peaking in August. The hospital-acquired BSI rate was 100.4 per 100 000 patient-days across the pandemic, increasing to 132.3 during the first wave and 190.9 during the second, with significant increase in elective inpatients. Patients with a hospital-acquired BSI, including those without COVID-19, experienced 20.2 excess days of hospital stay and 26.7% higher mortality, higher than reported in prepandemic literature. In intensive care, the BSI rate was 421.0 per 100 000 intensive care unit patient-days during the second wave, compared to 101.3 pre-COVID-19. The BSI incidence in those infected with the severe acute respiratory syndrome coronavirus 2 Alpha variant was similar to that seen with earlier variants.ConclusionsThe pandemic have impacted the patterns of community- and hospital-acquired BSIs, in COVID-19 and non-COVID-19 patients. Factors driving the patterns are complex. Infection surveillance needs to consider key aspects of pandemic response and changes in healthcare practice

Differential Interactions of Sex Pheromone and Plant Odour in the Olfactory Pathway of a Male Moth

Most animals rely on olfaction to find sexual partners, food or a habitat. The olfactory system faces the challenge of extracting meaningful information from a noisy odorous environment. In most moth species, males respond to sex pheromone emitted by females in an environment with abundant plant volatiles. Plant odours could either facilitate the localization of females (females calling on host plants), mask the female pheromone or they could be neutral without any effect on the pheromone. Here we studied how mixtures of a behaviourally-attractive floral odour, heptanal, and the sex pheromone are encoded at different levels of the olfactory pathway in males of the noctuid moth Agrotis ipsilon. In addition, we asked how interactions between the two odorants change as a function of the males' mating status. We investigated mixture detection in both the pheromone-specific and in the general odorant pathway. We used a) recordings from individual sensilla to study responses of olfactory receptor neurons, b) in vivo calcium imaging with a bath-applied dye to characterize the global input response in the primary olfactory centre, the antennal lobe and c) intracellular recordings of antennal lobe output neurons, projection neurons, in virgin and newly-mated males. Our results show that heptanal reduces pheromone sensitivity at the peripheral and central olfactory level independently of the mating status. Contrarily, heptanal-responding olfactory receptor neurons are not influenced by pheromone in a mixture, although some post-mating modulation occurs at the input of the sexually isomorphic ordinary glomeruli, where general odours are processed within the antennal lobe. The results are discussed in the context of mate localization

Stratifying multi-parameter persistent homology

In their paper "The theory of multidimensional persistence", Carlsson and Zomorodian write "Our study of multigraded objects shows that no complete discrete invariant exists for multidimensional persistence. We still desire a discriminating invariant that captures persistent information, that is, homology classes with large persistence." In this talk I will discuss how tools from commutative algebra give computable invariants able to capture homology classes with large persistence. Specifically, 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.Non UBCUnreviewedAuthor affiliation: University of OxfordGraduat