Topological data analysis (TDA) is a rising field in the intersection of mathematics, statistics, and computer science/data science. The cornerstone of TDA is persistent homology, which produces a summary of topological information called a persistence diagram. To utilize machine and deep learning methods on persistence diagrams, These diagrams are further summarized by transforming them into functions. In this paper we investigate the stability and injectivity of a class of smooth, one-dimensional functional summaries called Gaussian persistence curves.

We prove an equivalence between open questions about the embeddability of the space of persistence diagrams and the space of probability distributions (i.e. Wasserstein space). It is known that for many natural metrics, no coarse embedding of either of these two spaces into Hilbert space exists. Some cases remain open, however. In particular, whether coarse embeddings exist with respect to the p-Wasserstein distance for 1≤ p≤ 2 remains an open question for the space of persistence diagrams and for Wasserstein space on the plane. In this paper, we show that embeddability for persistence diagrams is equivalent to embeddability for Wasserstein space on ℝ^2 . When p > 1 , Wasserstein space on ℝ^2 is snowflake universal (an obstruction to embeddability into any Banach space of non-trivial type) if and only if the space of persistence diagrams is snowflake universal.

In our technology-dependent modern world, it is imperative to monitor the Sun for space weather threats to critical infrastructure. Topological data analysis (TDA) is a new set of mathematical techniques used in data analysis and machine learning. We demonstrate that TDA can robustly detect and classify solar surface and coronal activity. This technique is a promising step toward future application in predictive space weather modeling.

We propose and demonstrate a topological test for spatial dependence based on the framework of persistent homology. We compare our method to Moran's I, a classical measure of spatial auto-correlation, on synthetic datasets as well as on election and COVID data. We find about 65-75% agreement between the main variant of our method and Moran's I on real datasets. While the Moran's I test is more sensitive overall on these datasets, there are instructive instances (synthetic and real) where our method detects a spatial pattern that the Moran's I test does not.

We use Nathanson's $g$-adic representation of integers to relate metric properties of Cayley graphs of the integers with respect to various infinite generating sets $S$ to problems in additive number theory. If $S$ consists of all powers of a fixed integer $g$, we find explicit formulas for the smallest positive integer of a given length. This is related to finding the smallest positive integer expressible as a fixed number of sums and differences of powers of $g$. We also consider $S$ to be the set of all powers of all primes and bound the diameter of Cayley graph by relating it to Goldbach's conjecture.

We define a simple obstruction to Yu's property A that we call $k$-prisms. This structure allows for a straightforward proof that the space of persistence diagrams fails to have property A in a Wasserstein metric.

Cholera is an acute gastro-intestinal infection that affects millions of people throughout the world each year, primarily but not exclusively in developing countries. Because of its public health ramifications, considerable mathematical attention has been paid to the disease. Here we consider one neglected aspect of combating cholera: personal participation in anti-cholera interventions. We construct a game-theoretic model of cholera in which individuals choose whether to participate in either vaccination or clean water consumption programs under assumed costs. We find that relying upon individual compliance significantly lowers the incidence of the disease as long as the cost of intervention is sufficiently low, but does not eliminate it. The relative costs of the measures determined whether a population preferentially adopts a single preventative measure or employs the measure with the strongest early adoption.

We compute metric properties of Cayley graphs of the integers with respect to various infinite generating sets. When the generating set $S$ is the set of all powers of a prime, we find explicit formulas for the smallest positive integer of a given length. We also prove that such graphs are infinite dimensional in a strong sense by showing that they fail to have Yu's property A. Finally, we consider more general generating sets and relate geometric properties of these Cayley graphs to deep unsolved problems in number theory.