(346aq) Topological Data Analysis: Applications to Soft Matter and Molecular Simulations

Smith, A., University of Wisconsin-Madison
Chew, A. K., University of Wisconsin
Van Lehn, R. C., University of Wisconsin-Madison
Abbott, N. L., Cornell University
Zavala, V. M., University of Wisconsin-Madison
Data generated by experiments and complex simulations (e.g., molecular dynamics simulations) is often summarized using descriptive statistics (e.g., averages, moments, and correlation functions) in order to reduce complexity and to facilitate analysis [1]. Unfortunately, descriptive statistics might fail to capture important aspects of complex datasets. Specifically, statistical techniques might fail to capture key geometrical structures (e.g., complex heterogeneous domains). Interesting examples that illustrate these limitations are the anscombe quartet and the datasaurus dozen datasets [2,3]. These datasets are visually distinct (define different geometrical spaces) but have the exact same descriptive statistics (mean, standard deviation, and correlation).

Recent advances in applied topology and geometry have led to the development of a field known as Topological Data Analysis (TDA) [4,5,6]. TDA is a framework that views complex data through the lenses of geometry and topology. A particularly method of TDA, known as persistence homology, represents datasets (e.g., point clouds and images) as geometric objects and performs dimensionality reduction by projecting the data onto a low-dimensional space composed of elementary geometric objects (topological features) that persist at different scales [7,8]. The features are quantifiable and stable to basic deformations (e.g., stretching, rotation, bending) and can be used to perform different tasks (e.g., classification, regression) [9,10]. TDA methods have been applied successfully in materials science [11,12], time series and signal analysis [13,14], and bio-sciences [15,16].

This talk focuses on the application of TDA to complex datasets arising in soft matter and molecular dynamics (MD) simulations. We use TDA to characterize topological features that develop in liquid crystal films when exposed to air contaminants [17]. We also show how TDA can be used to characterize topological features of scatter fields for flow cytometry of emulsions containing liquid crystal droplets [18]. Finally, that TDA can be used to characterize the geometry of three-dimensional liquid-phase environments generated by MD [19]. For all of these studies, we show that the topological features are strongly correlated to emerging properties of interest (e.g., concentration of air contaminant or reactivity of a molecule in a solvent environment).

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[18] Shengli Jiang, JungHyun Noh, Alexander D Smith, Chulsoon Park, Nicholas L Abbott, Victor M Zavala. Identification of Endotoxins from Bacterial Species using Liquid Crystal Droplets and Machine Learning. Under Review, 2020.

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