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A ciliate is a phylum of protozoa that has two types of nuclei, macronuclei and micronuclei. There may be more than one of each type of nucleus in the organism [1]. The macronucleus is the structure where protein synthesis and cell metabolism occur [1]. The micronucleus stores genetic information and is mobilized during a sexual reproduction process called conjugation [1]. The somatic macronucleus (MAC) is developed from the germ-line micronucleus (MIC) through genome rearrangement during a sexual reproduction process called conjugation [6, 8]. Segments of the MIC that form the MAC during conjugation are called macronuclear destined sequences (MDSs) [8]. During sequencing each MDS is given coordinates where the MDS sequences begin and end in the MIC. The orientation of a MDS in the MIC can be taken to be positive or negative. If the direction of the MDS in the MIC agrees with the direction in the MAC then the orientation is positive otherwise it is a negative orientation. In this thesis we analyze various aspects of the gene assembly during the rearrangment process of the ciliate Oxytricha trifallax that were recently sequenced [15]. Some of the properties analyzed include overlapping MDSs, orientation, MDSs starting and ending position in the MIC and the gaps of overlapping MDS pairs. A gap of an overlapping MDS pair is the order difference of two MDSs for a particular MAC contig that overlap in the MIC contig. We use 120 MAC contigs from [15] that have overlaps among their own MDSs. These 120 MAC contigs make up the data set we call D4. We explore the patterns of overlapping MDSs in the MIC in D4. To quantify such patterns, we associate a vector V (An) to each MAC contig An, where V (An) = (v1(An), v2(An), v3(An)) is a vector in R3. The first entry is the number of overlapping MDS pairs divided by the number of MDSs. The second entry is the sum of gaps of overlapping MDS pairs divided by the sum of all possible gaps. The final entry is the total number of overlapping base pairs divided by the total length of the MAC contig. We computed the distance matrixM = (dij) where dij is the Euclidean distance between V (Ai) and V (Aj). The MAC contig vectors and M were computed using Python. To analyze D4 we applied Topological Data Analysis (TDA). TDA uses topological constructs to assess shapes in data [3, 12]. From the data entries of the distance matrix M = (dij) we applied a Vietoris-Rips filtration to generate the barcodes of the persistent homology in dimensions 0, 1 and 2. The persistence barcode of 0-dimensional homology illustrates clusters of the data while the 1-dimensional homology represents non-trivial loops in the simplicial complex [3, 13]. The application of TDA on the ciliate Oxytricha trifallax identified ten MAC contig clusters at epsilon= 0.1 in D4 and several loops that were persistent for two or three epsilon values. Other TDA methods can be applied to the Vietoris-Rips filtration to further identify which MAC contigs appear in each cluster.

Studying lymphocyte development using mass cytometry Abstract Development of mature lymphocytes, a white blood cell subtype, is crucial for the correct function of the human immune system. Currently, developmental pathways of lymphocytes can be studied using high-throughput single-cell measurements. In particular, mass cytometry enables the study of immunologically relevant pheno- typic and functional markers on a vast scale. In this work I present my individual contribution to tviblindi, a powerful software tool for analysis of cytometric data aimed at uncovering developmental trajectories. tviblindi is a package written in R, Python and C++. It provides a means to integrate prior knowledge with data analyses grounded in graph theory and algebraic topology. tviblindi is accessible to biological researchers without background in computer science or mathematics. It is an addition to the expanding field of trajectory inference in single-cell data. Furthermore, I review current knowledge of T-cell development and conduct a tviblindi analysis thereof using human thymus and peripheral blood datasets and evaluate the results. 1

<p>Applied topology is a rapidly growing discipline aiming at using ideas coming from algebraic topology to solve problems in the real world, including analyzing point cloud data, shape analysis, etc. Semi-algebraic geometry deals with studying properties of semi-algebraic sets that are subsets of $\mathbb{R}^n$ and defined in terms of polynomial inequalities. Semi-algebraic sets are ubiquitous in applications in areas such as modeling, motion planning, etc. Developing efficient algorithms for computing topological invariants of semi-algebraic sets is a rich and well-developed field.</p> <p>However, applied topology has thrown up new invariants---such as persistent homology and barcodes---which give us new ways of looking at the topology of semi-algebraic sets. In this thesis, we investigate the interplay between these two areas. We aim to develop new efficient algorithms for computing topological invariants of semi-algebraic sets, such as persistent homology, and to develop new mathematical tools to make such algorithms possible.</p>

Cette thèse propose une collection des nouveaux outils pour la représentation musicale. Ces modèles ont deux caractéristiques principales. D'un côté, ils sont inspirés par la géométrie et la topologie. De l'autre côté, ils ont une basse dimensionnalité, afin de garantir une visualisation intuitive des caractéristiques musicales qu'ils représentent. On s'est attaqué au problème de l'analyse musicale à partir de trois points de vue. On a représenté le contrepoint en utilisant des séries temporelles multivariées de matrices de permutations partielles. On a visualisé la conduite des voix en utilisant une classe particulière des tresses partielles et singulières. On donne ensuite une interpretation du Tonnetz comme complex simplicial et on utilise l'homologie persistante, afin de classifier des formes obtenues en déformant les sommets du Tonnetz. Ces déformations sont induites soit par des fonctions qui prennent en compte la nature symbolique de la musique, soit l'interaction symbol/signal. Les modèles basés sur la persistence topologique ont été testés sur une collection hétérogène de bases de données. Ces deux approches sont finalement combinées pour donner un troisième point de vue, qui a donné deux applications. Premièrement, on utilise l'alignement multiple des sequences, pour comparer plusieurs structures harmoniques et sémantiques déduites du signal audio, afin de visualiser et quantifier la propagation d’idée musicales entre artistes, genres et différentes époques. Ensuite on développe la théorie nécessaire pour comparer deux systèmes qui varient dans le temps, en représentant leurs caractéristiques géométriques comme des séries temporelles de diagrammes de persistence. / In this work, we suggest a collection of novel models for the representation of music. These models are endowed with two main features. First, they originate from a topological and geometrical inspiration; second, their low dimensionality allows to build simple and informative visualisations. We tackle the problem of music representation following three non-orthogonal directions. First, we propose an interpretation of counterpoint as a multivariate time series of partial permutation matrices, whose observations are characterised by a degree of complexity. After providing both a static and a dynamic representation of counterpoint, voice leadings are reinterpreted as a special class of partial singular braids, and their main features are visualised. Thereafter, we give a topological interpretation of the Tonnetz (a graph commonly used in computational musicology), whose vertices are deformed by both a harmonic and a consonance-oriented function. The shapes derived from these deformations are classified using the formalism of persistent homology. Thus, this novel representation of music is evaluated on a collection of heterogenous musical datasets. Finally, a combination of the two approaches is proposed. A model at the crossroad between the signal and symbolic analysis of music uses multiple sequences alignment to provide an encompassing, novel viewpoint on the musical inspiration transfer among compositions belonging to different artists, genres and time. Then, music is represented as a time series of topological fingerprints, allowing the comparison of pairs of time-varying shapes in both topological and musical terms.

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