Georgia Institute of Technology College of Sciences

We consider a class of dynamical systems with random switching with the following specifics: Given a finite collection of smooth vector fields on a finite-dimensional smooth manifold, we fix an initial vector field and a starting point on the manifold. We follow the solution trajectory to the corresponding initial-value problem for a random, exponentially distributed time until we switch to a new vector field chosen at random from the given collection. Again, we follow the trajectory induced by the new vector field for an exponential time until we make another switch. This procedure is iterated. The resulting two-component process whose first component records the position on the manifold, and whose second component records the driving vector field at any given time, is a Markov process. We identify sufficient conditions for its invariant measure to be unique and absolutely continuous. In the one-dimensional case, we show that the invariant densities are smooth away from critical points of the vector fields and derive asymptotics for the invariant densities at critical points.

Given a Riemannian manifold $(M,g)$, does there exist a metric $g'$ on $M$ conformal to $g$ such that $g'$ has constant scalar curvature? This question is known as the Yamabe problem. Aim of this talk is to give an overview of the problem and discuss and develop methods that go into solving a few of intermediate results in the solution to the problem in full generality.

A new and exciting breakthrough due to Maynard establishes that there exist infinitely many pairs of distinct primes $p_1,p_2$ with $|p_1-p_2|\leq 600$ as a consequence of the Bombieri-Vinogradov Theorem. We apply his general method to the setting of Chebotarev sets of primes. We study applications of these bounded gaps with an emphasis on ranks of prime quadratic twists of elliptic curves over $\mathbb{Q}$, congruence properties of the Fourier coefficients of normalized Hecke eigenforms, and representations of primes by binary quadratic forms.

Pulmonary embolism (PE) is a potentially-fatal disease in which blood clots (i.e., emboli) break free from the deep veins in the body and migrate to the lungs. In order to prevent PE, anticoagulants are often prescribed; however, for some patients, anticoagulants cannot be used. For such patients, a mechanical filter, namely an inferior vena cava (IVC) filter, is inserted into the IVC to trap the blood clots and prevent them from reaching the lungs. There are numerous IVC filter designs, and it is not well understood which particular IVC filter geometry will result in the best treatment for a given patient. Patient-specific computational fluid dynamic (CFD) simulations may be used to predict the performance of IVC filters and hence can aid physicians in IVC filter selection and placement. In this talk, I will first describe our computational pipeline for prediction of IVC filter performance. Our pipeline involves several steps including image processing, geometric model construction, in vivo stress state estimation, surface and volume mesh generation based on virtual IVC filter placement, and CFD simulation of IVC hemodynamics. I will then present the results of our IVC hemodynamics simulations obtained for two patient IVCs. This talk represents joint work with several researchers at The Pennsylvania State University, Penn State Hershey Medical Center, the Penn State Applied Research Lab, and the University of Utah.