<p>This dissertation extends the theory of persistent homology to time varying systems. Most of the previous work has been dedicated to using this powerful tool in topological data analysis to study static point clouds. ...

<p>While alternans in a single cardiac cell appears through a simple</p><p>period-doubling bifurcation, in extended tissue the exact nature</p><p>of the bifurcation is unclear. In particular, the phase of</p><p>alternans ...

Observed in many physical systems, coarsening is an orderly decrease in the number of localized structures, such as particles, drops, shear bands, solitons, or point defects. Coarsening is a type of pattern formation in ...

<p>Machine learning is a powerful branch of mathematics and statistics that allows the automation of tasks that would otherwise require humans a long time to perform. Two particular fields of machine learning that have ...

<p>We calculate electromagnetic transmission through periodic gratings using a mode-matching method for solving Maxwell's equations. We record the derivation of the equations involved for several variations of the problem, ...

<p>Noise-induced stabilization occurs when an unstable deterministic system is stabilized by the addition of white noise. Proving that this phenomenon occurs for a particular system is often manifested through the ...

<p>I study differential geometry of 6-manifolds endowed with various $SU(3)$ structures from three perspectives. The first is special Lagrangian geometry; The second is pseudo-Hermitian-Yang-Mills connections or more ...

<p>We consider complete asymptotically flat Riemannian manifolds that are the graphs of smooth functions over $\mathbb R^n$. By recognizing the scalar curvature of such manifolds as a divergence, we express the ADM mass ...

<p>In this work, we apply $L^2$-index theory to compute the index of a non-Fredholm elliptic operator. The operator arises in Type I' string theory, and the index is found to be non-zero, thus implying existence of bound ...

<p>In general relativity, the Riemannian Penrose inequality (RPI) provides a lower bound for the ADM mass of an asymptotically flat manifold of nonnegative scalar curvature in terms of the area of the outermost minimal ...