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 Title
 A topological study of toroidal dynamics
 Creator
 Gakhar, Hitesh
 Date
 2020
 Collection
 Electronic Theses & Dissertations
 Description

This dissertation focuses on developing theoretical tools in the field of Topological Data Analysis and more specifically, in the study of toroidal dynamical systems. We make contributions to the development of persistent homology by proving Kunnethtype theorems, to topological time series analysis by further developing the theory of sliding window embeddings, and to multiscale data coordinatization in topological spaces by proving stability theorems. First, in classical algebraic topology,...
Show moreThis dissertation focuses on developing theoretical tools in the field of Topological Data Analysis and more specifically, in the study of toroidal dynamical systems. We make contributions to the development of persistent homology by proving Kunnethtype theorems, to topological time series analysis by further developing the theory of sliding window embeddings, and to multiscale data coordinatization in topological spaces by proving stability theorems. First, in classical algebraic topology, the Kunneth theorem relates the homology of two topological spaces with that of their product. We prove Kunneth theorems for the persistent homology of the categorical and tensor product of filtered spaces. That is, we describe the persistent homology of these product filtrations in terms of that of the filtered components. Using these theorems, we also develop novel methods for algorithmic and abstract computations of persistent homology. One of the direct applications of these results is the abstract computation of Rips persistent homology of the Ndimensional torus.Next, we develop the general theory of sliding window embeddings of quasiperiodic functions and their persistent homology. We show that the sliding window embeddings of quasiperiodic functions, under appropriate choices of the embedding dimension and time delay, are dense in higher dimensional tori. We also explicitly provide methods to choose these parameters. Furthermore, we prove lower bounds on Rips persistent homology of these embeddings. Using one of the persistent Kunneth formulae, we provide an alternate algorithm to compute the Rips persistent homology of the sliding window embedding, which outperforms the traditional methods of landmark sampling in both accuracy and time. We also apply our theory to music, where using sliding windows and persistent homology, we characterize dissonant sounds as quasiperiodic in nature.Finally, we prove stability results for sparse multiscale circular coordinates. These coordinates on a data set were first created to aid nonlinear dimensionality reduction analysis. The algorithm identifies a significant integer persistent cohomology class in the Rips filtration on a landmark set and solves a linear least squares optimization problem to construct a circled valued function on the data set. However, these coordinates depend on the choice of the landmarks. We show that these coordinates are stable under Wasserstein noise on the landmark set.
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 Title
 Algebraic topology and machine learning for biomolecular modeling
 Creator
 Cang, Zixuan
 Date
 2018
 Collection
 Electronic Theses & Dissertations
 Description

Data is expanding in an unprecedented speed in both quantity and size. Topological data analysis provides excellent tools for analyzing high dimensional and highly complex data. Inspired by the topological data analysis's ability of robust and multiscale characterization of data and motivated by the demand of practical predictive tools in computational biology and biomedical researches, this dissertation extends the capability of persistent homology toward quantitative and predictive data...
Show moreData is expanding in an unprecedented speed in both quantity and size. Topological data analysis provides excellent tools for analyzing high dimensional and highly complex data. Inspired by the topological data analysis's ability of robust and multiscale characterization of data and motivated by the demand of practical predictive tools in computational biology and biomedical researches, this dissertation extends the capability of persistent homology toward quantitative and predictive data analysis tools with an emphasis in biomolecular systems. Although persistent homology is almost parameter free, careful treatment is still needed toward practically useful prediction models for realistic systems. This dissertation carefully assesses the representability of persistent homology for biomolecular systems and introduces a collection of characterization tools for both macromolecules and small molecules focusing on intra and intermolecular interactions, chemical complexities, electrostatics, and geometry. The representations are then coupled with deep learning and machine learning methods for several problems in drug design and biophysical research. In realworld applications, data often come with heterogeneous dimensions and components. For example, in addition to location, atoms of biomolecules can also be labeled with chemical types, partial charges, and atomic radii. While persistent homology is powerful in analyzing geometry of data, it lacks the ability of handling the nongeometric information. Based on cohomology, we introduce a method that attaches the nongeometric information to the topological invariants in persistent homology analysis. This method is not only useful to handle biomolecules but also can be applied to general situations where the data carries both geometric and nongeometric information. In addition to describing biomolecular systems as a static frame, we are often interested in the dynamics of the systems. An efficient way is to assign an oscillator to each atom and study the coupled dynamical system induced by atomic interactions. To this end, we propose a persistent homology based method for the analysis of the resulting trajectories from the coupled dynamical system. The methods developed in this dissertation have been applied to several problems, namely, prediction of protein stability change upon mutations, proteinligand binding affinity prediction, virtual screening, and protein flexibility analysis. The tools have shown top performance in both commonly used validation benchmarks and communitywide blind prediction challenges in drug design.
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 Title
 Dynamic augmentation of dissipative algebraic loops
 Creator
 Graf, Peter Leo
 Date
 1981
 Collection
 Electronic Theses & Dissertations
 Title
 Equivariant algebraic cobordism and double point relations
 Creator
 Liu, Chun Lung
 Date
 2012
 Collection
 Electronic Theses & Dissertations
 Description

For a reductive connected group or a finite group over a field of characteristic zero, we define an equivariant algebraic cobordism theory by a generalized version of the double point relation of LevinePandharipande. We prove basic properties and the welldefinedness of a canonical fixed point map. We also find explicit generators of the algebraic cobordism ring of the point when the group is finite abelian.
 Title
 Graphs and their associated linegraphs
 Creator
 Chartrand, Gary
 Date
 1964
 Collection
 Electronic Theses & Dissertations
 Title
 Images of certain manifolds under mappings of degree one
 Creator
 Spence, Lawrence Edward, 1946
 Date
 1970
 Collection
 Electronic Theses & Dissertations
 Title
 Integration of topological data analysis and machine learning for small molecule property predictions
 Creator
 Wu, Kedi
 Date
 2018
 Collection
 Electronic Theses & Dissertations
 Description

Accurate prediction of small molecule properties is of paramount importance to drug design and discovery. A variety of quantitative properties of small molecules has been studied in this thesis. These properties include solvation free energy, partition coefficient, aqueous solubility, and toxicity endpoints. The highlight of this thesis is to introduce an algebraic topology based method, called element specific persistent homology (ESPH), to predict small molecule properties. Essentially ESPH...
Show moreAccurate prediction of small molecule properties is of paramount importance to drug design and discovery. A variety of quantitative properties of small molecules has been studied in this thesis. These properties include solvation free energy, partition coefficient, aqueous solubility, and toxicity endpoints. The highlight of this thesis is to introduce an algebraic topology based method, called element specific persistent homology (ESPH), to predict small molecule properties. Essentially ESPH describes molecular properties in terms of multiscale and multicomponent topological invariants and is different from conventional chemical and physical representations. Based on ESPH and its modified version, elementspecific topological descriptors (ESTDs) are constructed. The advantage of ESTDs is that they are systematical, comprehensive, and scalable with respect to molecular size and composition variations, and are readily suitable for machine learning methods, rendering topological learning algorithms. Due to the inherent correlation between different small molecule properties, multitask frameworks are further employed to simultaneously predict related properties. Deep neural networks, along with ensemble methods such as random forest and gradient boosting trees, are used to develop quantitative predictive models. Physical based molecular descriptors and auxiliary descriptors are also used in addition to ESTDs. As a result, we obtain stateoftheart results for various benchmark data sets of small molecule properties. We have also developed two online servers for predicting properties of small molecules, TopPS and TopTox. TopPS is a software for topological learning predictions of partition coefficient and aqueous solubility, and TopTox is a software for computing elementspecific tological descriptors (ESTDs) for toxicity endpoint predictions. They are available at http://weilab.math.msu.edu/TopPS/ and http://weilab.math.msu.edu/TopTox/, respectively.
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 Title
 Manifolds which are homology doubles
 Creator
 Downing, John Scott
 Date
 1969
 Collection
 Electronic Theses & Dissertations
 Title
 Relative bounded cohomology and relative ℓ₁ homology
 Creator
 Park, HeeSook
 Date
 2001
 Collection
 Electronic Theses & Dissertations