Speaker: Joshua Mike, The University of Tennessee, Knoxville

In this talk, we will explore two applications: The first considers simplicial cohomology as a tool to investigate and eliminate inequity in kidney paired donation (KPD). The KPD pool is modeled as a graph wherein cocycles represent fair organ exchanges. Helmholtz decomposition is used to split donation utilities into gradient and harmonic portions. The gradient portion yields a preference score for cocycle allocation. The harmonic portion is isomorphic to the 1-cohomology and is used to guide a new algorithmic search for exchange cocycles. We examine correlation between a patient’s chance to obtain a kidney and their score under various allocation methods and conclude by showing that traditional methods are biased, while our new algorithm is not.
The second considers the persistent homology of a smoothed noisy dynamic. The machinery of persistent homology yields topological structure for discrete data within a metric space. Homology in dynamical systems can capture important features such as periodicity, multistability, and chaos. We consider a hidden Markov dynamic and compare particle filter to optimal smoothing a posteriori. We conclude with a stability theorem for the convergence of the persistent homology of the particle filtered path to that of the optimal smoothed path.