This thesis is about how to interpret metric data with topological tools, such as homology. We show how to go from a metric space to a topological space via Vietoris-Rips complexes. We use the usual approach to Topological Data Analysis (TDA), and transform our metric space into tame parametrised vector spaces. It is then shown how to simplify tame parametrised vector spaces. We also present another approach to TDA, where we transform our metric space into a filtrated tame parametrised chain complex. We then show how to simplify chain complexes over fields in order to simplify tame parametrised filtrated chain complexes. / Denna uppsats handlar om att tolka metrisk data med hjälp utav topologiska verktyg, som exempelvis homologi. Vi visar hur man går från ett metriskt rum till ett topologiskt rum via Vieteris-Rips komplex. Vi använder den vanliga metoden till Topologisk Data Analys (TDA), och transformerar vårat metriska rum till tama parametriserade vektorrum. Det visas sedan hur vi kan förenkla tama parametriserade vektorrum. Vi presenterar även en annan metod för TDA, där vi går från ett metriskt rum till ett filtrerat tamt parametriserat kedjekomplex. Sedan visar vi hur man förenklar kedjekomplex över kroppar för att kunna förenkla filtrerade tama parametriserade kedjekomplex.

We present a rigorous yet accessible introduction to structures on finite sets foundational for a formal study of complex networks. This includes a thorough treatment of binary relations, distance spaces, their properties and similarities. Correspondences between relations and graphs are given and a brief introduction to graph theory is followed by a more detailed study of cohesiveness and centrality. We show how graph degeneracy is equivalent to the concept of k-cores, which give a measure of the cohesiveness or interconnectedness of a subgraph. We then further extend this to d-cores of directed graphs. After a brief introduction to topology, focusing on topological spaces from distances, we present a historical discussion on the early developments of algebraic topology. This is followed by a more formal introduction to simplicial homology where we define the homology groups. In the context of algebraic topology, the d-cores of a digraph give rise to a partially ordered set of subgraphs, leading to a set of filtrations that is two-dimensional in nature. Directed clique complexes of digraphs are defined in order to encode the directionality of complete subdigraphs. Finally, we apply these methods to the neuronal network of C.elegans. Persistent homology with respect to directed core filtrations as well as robustness of homology to targeted edge percolations in different directed cores is analyzed. Much importance is placed on intuition and on unifying methods of such dispersed disciplines as sociology and network neuroscience, by rooting them in pure mathematics. / Vi presenterar en rigorös men lättillgänglig introduktion till de abstrakta strukturer på ändliga mängder som är grundläggande för en formell studie av komplexa nätverk. Detta inkluderar en grundlig redogörelse av binära relationer och distansrum, deras egenskaper samt likheter. Korrespondenser mellan olika typer av relationer och grafer förklaras och en kort introduktion till grafteori följs av en mer detaljerad studie av sammanhållning och centralitet. Vi visar hur begreppet 'degeneracy' är ekvivalent med begreppet k-kärnor (eng: k-cores), vilket ger ett mått på sammanhållningen hos en delgraf. Vi utökar sedan detta till konceptet d-kärnor (eng: d-cores) för riktade grafer. Efter en kort introduktion till topologi med fokus på topologiska rum från distansrum, så presenterar vi en historisk diskussion kring den tidiga utvecklingen av algebraisk topologi. Detta följs av en mer formell introduktion till homologi, där vi bl.a. definierar homologigrupperna. Vi definierar sedan så kallade riktade klick-komplex som simplistiska komplex (eng: simplicial complexes) från riktade grafer, där d-kärnorna av en riktad graf då ger upphov till filtrerade komplex i två parametrar. Persistent homologi med avseende på dessa riktade kärnfiltreringar såväl som robusthet mot kantpercolationer i olika kärnor analyseras sedan för det neurala nätverket hos C.Elegans. Stor vikt läggs vid intuition och förståelse, samt vid att förena metodiker för så spridda discipliner som sociologi och neurovetenskap.

Data mining, and in particular topological data analysis (TDA), had proven to be successful inabstracting insights from big arrays of data. This thesis utilizes the TDA software AyasdiTM inorder to improve the accuracy of the energy model of an Electric Arc Furnace (EAF), pinpointingthe causes of a wrong calculation of the steel temperature. Almost 50% of the charges analyzedpresented an underestimation of temperature, while under 30% an overestimation.First a dataset was created by filtering the data obtained by the company. After an initialscreening, around 700 charges built the dataset, each one characterized by 104 parameters. Thedataset was subsequently used to create a topological network through the TDA software. Bycomparing the distribution of each parameter with the distribution of the wrong temperatureestimation, it was possible to identify which parameters provided a biased trend. In particular, itwas found that an overestimation of temperature was caused by an underestimation of themelting energy of materials not having through a melting test. It was also found a possible biasedtrend in some distribution of parameters like %O in steel and slag weight, which it is believedare all connected together. Despite not finding a global solution for the reasons behind theunderestimation of temperature, it is believed that a different settings more focused around thematerials used as scrap mix can highlight more on that subject. In conclusion TDA proved itselfefficient as a problem solving technique in the steel industry.

Coupled oscillators, such as groups of fireflies or clusters of neurons, are found throughout nature and are frequently modeled in the applied mathematics literature. Earlier work by Kuramoto, Strogatz, and others has led to a deep understanding of the emergent behavior of systems of such oscillators using traditional dynamical systems methods. In this project we outline the application of techniques from topological data analysis to understanding the dynamics of systems of coupled oscillators. This includes the examination of partitions, partial synchronization, and attractors. By looking for clustering in a data space consisting of the phase change of oscillators over a set of time delays we hope to reconstruct attractors and identify members of these clusters.

abstract: Robotic swarms can potentially perform complicated tasks such as exploration and mapping at large space and time scales in a parallel and robust fashion. This thesis presents strategies for mapping environmental features of interest – specifically obstacles, collision-free paths, generating a metric map and estimating scalar density fields– in an unknown domain using data obtained by a swarm of resource-constrained robots. First, an approach was developed for mapping a single obstacle using a swarm of point-mass robots with both directed and random motion. The swarm population dynamics are modeled by a set of advection-diffusion-reaction partial differential equations (PDEs) in which a spatially-dependent indicator function marks the presence or absence of the obstacle in the domain. The indicator function is estimated by solving an optimization problem with PDEs as constraints. Second, a methodology for constructing a topological map of an unknown environment was proposed, which indicates collision-free paths for navigation, from data collected by a swarm of finite-sized robots. As an initial step, the number of topological features in the domain was quantified by applying tools from algebraic topology, to a probability function over the explored region that indicates the presence of obstacles. A topological map of the domain is then generated using a graph-based wave propagation algorithm. This approach is further extended, enabling the technique to construct a metric map of an unknown domain with obstacles using uncertain position data collected by a swarm of resource-constrained robots, filtered using intensity measurements of an external signal. Next, a distributed method was developed to construct the occupancy grid map of an unknown environment using a swarm of inexpensive robots or mobile sensors with limited communication. In addition to this, an exploration strategy which combines information theoretic ideas with Levy walks was also proposed. Finally, the problem of reconstructing a two-dimensional scalar field using observations from a subset of a sensor network in which each node communicates its local measurements to its neighboring nodes was addressed. This problem reduces to estimating the initial condition of a large interconnected system with first-order linear dynamics, which can be solved as an optimization problem. / Dissertation/Thesis / Doctoral Dissertation Mechanical Engineering 2018

Indiana University-Purdue University Indianapolis (IUPUI) / Background: Cancer is the leading cause of death worldwide and as a result is one of the most studied topics in public health. Breast cancer and prostate cancer are the most common cancers among women and men respectively. Gene expression and image features are independently prognostic of patient survival. However, it is sometimes difficult to discern how the molecular profile, e.g., gene expression, of given cells relate to their spatial layout, i.e., topology, in the tumor microenvironment (TME). However, with the advent of spatial transcriptomics (ST) and integrative bioinformatics analysis techniques, we are now able to better understand the TME of common cancers. Method: In this paper, we aim to determine the genes that are correlated with image topology features (ITFs) in common cancers which we denote topology associated genes (TAGs). To achieve this objective, we generate the correlation coefficient between genes and image features after identifying the optimal number of clusters for each of them. Applying this correlation matrix to heatmap using R package pheatmap to visualize the correlation between the two sets. The objective of this study is to identify common themes for the genes correlated with ITFs and we can pursue this using functional enrichment analysis. Moreover, we also find the similarity between gene clusters and some image features clusters using the ranking of correlation coefficient in order to identify, compare and contrast the TAGs across breast and prostate cancer ST slides. Result: The analysis shows that there are groups of gene ontology terms that are common within breast cancer, prostate cancer, and across both cancers. Notably, extracellular matrix (ECM) related terms appeared regularly in all ST slides. Conclusion: We identified TAGs in every ST slide regardless of cancer type. These TAGs were enriched for ontology terms that add context to the ITFs generated from ST cancer slides.

<p>Harnessing the power of data has been a driving force for computing in recently years. However, the non-vectorized or even non-Euclidean nature of certain data with complex structures also poses new challenges to the data science community. Topological data analysis (TDA) has proven effective in several scenarios for alleviating the challenges, by providing techniques that can reveal hidden structures and high-order connectivity for data. A central technique in TDA is called persistent homology, which provides intervals tracking the birth and death of topological features in a growing sequence of topological spaces. In this dissertation, we study the representative problem for persistent homology, motivated by the observation that persistent homology does not pinpoint a specific homology class or cycle born and dying with the persistence intervals. Furthermore, studying the representatives also leads us to new findings for related problems such as persistence computation.<br> </p> <p>First, we look into the representative problem for (standard) persistence homology and term the representatives as persistent cycles. We define persistent cycles as cycles born and dying with given persistence intervals and connect the definition to interval decomposition for persistence modules. We also look into the computation of optimal (minimum) persistent cycles which have guaranteed quality. We prove that it is NP-hard to compute minimum persistent p-cycles for the two types of intervals in persistent homology in general dimensions (p>1). In view of the NP-hardness results, we then identify a special but important class of inputs called weak (p+1)-pseudomanifolds whose minimum persistent p-cycles can be computed in polynomial time. The algorithms are based on a reduction to minimum (s,t)-cuts on dual graphs.<br> </p> <p>Second, we propose alternative persistent cycles capturing the dynamic changes of homological features born and dying with persistence intervals, which the previous persistent cycles do not reveal. We focus on persistent homology generated by piecewise linear (PL) functions and base our definition on an extension of persistence called the levelset zigzag persistence. We define a sequence of cycles called levelset persistent cycles containing a cycle between each consecutive critical points within the persistence interval. Due to the NP-harness results proven previously, we propose polynomial-time algorithms computing optimal sequences of levelset persistent p-cycles for weak (p+1)-pseudomanifolds. Our algorithms draw upon the idea of relating optimal cycles to min-cuts in a graph that we exploited earlier for standard persistent cycles. Note that levelset zigzag poses non-trivial challenges for the approach because a sequence of optimal cycles instead of a single one needs to be computed in this case.<br> </p> <p>Third, we investigate the computation of zigzag persistence on graph inputs, motivated by the fact that graphs model real-world circumstances in many applications where they may constantly change to capture dynamic behaviors of phenomena. Zigzag persistence, an extension of the standard persistence incorporating both insertions and deletions of simplices, is one appropriate instrument for analyzing such changing graph data. However, unlike standard persistence which admits nearly linear-time algorithms for graphs, such results for the zigzag version improving the general $O(m^\omega)$ time complexity are not known, where $\omega< 2.37286$ is the matrix multiplication exponent. We propose algorithms for zigzag persistence on graphs which run in near-linear time. Specifically, given a filtration of length m on a graph of size n, the algorithm for 0-dimension runs in $O(m\log^2 n+m\log m)$ time and the algorithm for 1-dimension runs in $O(m\log^4 n)$ time. The algorithm for 0-dimension draws upon another algorithm designed originally for pairing critical points of Morse functions on 2-manifolds. The correctness proof of the algorithm, which is a major contribution, is achieved with the help of representatives. The algorithm for 1-dimension pairs a negative edge with the earliest positive edge so that a representative 1-cycle containing both edges resides in all intermediate graphs.</p>

<p>Recent innovations in combinatorial dynamical systems permit them to be studied with algorithmic methods. One such method from topological data analysis, called persistent homology, allows one to summarize the changing homology of a sequence of simplicial complexes. This dissertation explicates three methods for capturing and summarizing changes in combinatorial dynamical systems through the lens of persistent homology. The first places the Conley index in the persistent homology setting. This permits one to capture the persistence of salient features of a combinatorial dynamical system. The second shows how to capture changes in combinatorial dynamical systems at different resolutions by computing the persistence of the Conley-Morse graph. Finally, the third places Conley's notion of continuation in the combinatorial setting and permits the tracking of isolated invariant sets across a sequence of combinatorial dynamical systems. </p>

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