The exploration of complex systems is a fundamental pursuit in various scientific disciplines, includingphysics, biology, finance and engineering. The inherent complexity and dynamics within these systemspose significant challenges for traditional analytical methods. In recent years, the emergence of Topological Data Analysis (TDA) has provided a promising framework for uncovering hidden structures andpatterns in dynamic data sets. This thesis investigates the application of Topological Data Analysis to analyze system dynamics,aiming to enhance our understanding of their behavior. Through a detailed review of existing literature,we examine the theoretical foundations of TDA and its relevance to discrete and continuous processes.We discuss conceptual underpinnings of persistent homology, a key technique in TDA, and its potentialfor capturing essential features of system dynamics. By applying TDA to two distinct models, thestochastic ODE and the discrete logistic equation, we demonstrate its effectiveness in revealing underlyingstructures that traditional methods might overlook, thereby offering new insights into the analysis ofstochastic and discrete dynamical systems.

A major impediment to scientific progress in many fields is the inability to make sense of the huge amounts of data that have been collected via experiment or computer simulation. This dissertation provides tools to visualize, represent, and analyze the collection of sensors and data all at once in a single combinatorial geometric object. Encoding and translating heterogeneous data into common language are modeled by supporting objects. In this methodology, the behavior of the system based on the detection of noise in the system, possible failure in data exchange and recognition of the redundant or complimentary sensors are studied via some related geometric objects. Applications of the constructed methodology are described by two case studies: one from wildfire threat monitoring and the other from air traffic monitoring. Both cases are distributed (spatial and temporal) information systems. The systems deal with temporal and spatial fusion of heterogeneous data obtained from multiple sources, where the schema, availability and quality vary. The behavior of both systems is explained thoroughly in terms of the detection of the failure in the systems and the recognition of the redundant and complimentary sensors. A comparison between the methodology in this dissertation and the alternative methods is described to further verify the validity of the sheaf theory method. It is seen that the method has less computational complexity in both space and time.

Homologous DNA recombination and rearrangement has been modeled with a class of four-regular rigid vertex graphs called assembly graphs which can also be represented by double occurrence words. Various invariants have been suggested for these graphs, some based on the structure of the graphs, and some biologically motivated. In this thesis we use a novel method of data analysis based on a technique known as partial-clustering analysis and an algorithm known as Mapper to examine the relationships between these invariants. We introduce some of the basic machinery of topological data analysis, including the construction of simplicial complexes on a data set, clustering analysis, and the workings of the Mapper algorithm. We define assembly graphs and three specific invariants of these graphs: assembly number, nesting index, and genus range. We apply Mapper to the set of all assembly graphs up to 6 vertices and compare relationships between these three properties. We make several observations based upon the results of the analysis we obtained. We conclude with some suggestions for further research based upon our findings.

archives@tulane.edu / The current system for evaluating prostate cancer architecture is the Gleason Grade system, which divides the morphology of cancer into five distinct architectural patterns, labeled numerically in increasing levels of cancer aggressiveness and generates a score by summing the labels of the two most dominant patterns. The Gleason score is currently the most powerful prognostic predictor of patient outcomes; however, it suffers from problems in reproducibility and consistency due to the high intra-observer and inter-observer variability among pathologists. In addition, the Gleason system lacks the granularity to address potentially prognostic architectural features beyond Gleason patterns. We look towards persistent homology, a tool from topological data analysis, to provide a means of evaluating prostate cancer glandular architecture. The objective of this work is to demonstrate the capacity of persistent homology to capture architectural features independently of Gleason patterns in a representation suitable for unsupervised and supervised machine learning. Specifically, using persistent homology, we compute topological representations of purely graded prostate cancer histopathology images of Gleason patterns and show that persistent homology is capable of clustering prostate cancer histology into architectural groups through discrete representations of persistent homology in both two-dimensional and three-dimensional histopathology. We then demonstrate the performance of persistent homology based features in common machine learning classifiers, indicating that persistent homology can both separate unique architectures in prostate cancer, but is also predictive of prostate cancer aggressiveness. Our results indicate the ability of persistent homology to cluster into unique groups with dominant architectural patterns consistent with the continuum of Gleason patterns. In addition, of particular interest, is the sensitivity of persistent homology to identify specific sub-architectural groups within single Gleason patterns, suggesting that persistent homology could represent a robust quantification method for prostate cancer architecture with higher granularity than the existing semi-quantitative measures. This work develops a framework for segregating prostate cancer aggressiveness by architectural subtype using topological representations, in a supervised machine learning setting, and lays the groundwork for augmenting traditional approaches with topological features for improved diagnosis and prognosis. / 1 / Peter Lawson

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The scale of the scholarly community complicates searches within scholarly databases, necessitating keywords to index the topics of any given work. As a result, an author’s choice in keywords affects the visibility of each publication; making the sum of these choices a key representation of the author’s academic profile. As such the underlying network of investigators are often viewed through the lens of their keyword networks. Current keyword networks connect publications only if they use the exact same keyword, meaning uncontrolled keyword choice prevents connections despite semantic similarity. Computational understanding of semantic similarity has already been achieved through the process of word embedding, which transforms words to numerical vectors with context-correlated values. The resulting vectors preserve semantic relations and can be analyzed mathematically. Here we develop a model that uses embedded keywords to construct a network which circumvents the limitations caused by uncontrolled vocabulary. The model pipeline begins with a set of faculty, the publications and keywords of which are retrieved by SCOPUS API. These keywords are processed and then embedded. This work develops a novel method of network construction that leverages the interdisciplinarity of each publication, resulting in a unique network construction for any given set of publications. Postconstruction the network is visualized and analyzed with topological data analysis (TDA). TDA is used to calculate the connectivity and the holes within the network, referred to as the zero and first homology. These homologies inform how each author connects and where publication data is sparse. This platform has successfully modelled collaborations within the biomedical department at Purdue University and provides insight into potential future collaborations.