Geodesic nets are types of graphs in Riemannian manifolds where each edge is a geodesic segment. One important object used in the construction of geodesic nets is a balanced vertex, where the sum of unit tangent vectors along adjacent edges is zero. We prove the existence of a balanced vertex of a triangle (with three unbalanced vertices) on a general two-dimensional Riemannian surface when all angles measure less than $2\pi/3$, if the length of the sides of the triangle is not too large. This property is a generalization for the existence of the Fermat point of a planar triangle.

The Cheeger’s constant, also known as the isoperimetric number, is a constant that helps describe the bottleneck present in a graph, if any. Some fields, such as computer networks, have an interest in this constant due to the application of the constant in their field. We examined randomly generated connected graphs and their isoperimetric numbers by developing algorithms to calculate it.

Several viruses can cause cells to fuse into large multinucleated cells called syncytia. Syncytia formation allows the virus to spread without entering the extracellular space, where it might be exposed to immune responses. However, there is evidence that antibodies can also hinder the fusion process. This project uses mathematical analysis to find different possible infection outcomes. A stability analysis of the coinfection model is used to find the fixed points of the model and their stability. This gives us parameter space regions that lead to different possible infection outcomes. Simulations were made to verify the mathematical analysis and see how different syncytia formation properties affect the resulting dynamics. These findings could help develop strategies for controlling viral spread.

Graphene Quantum Dots (GQDs) are nanoscale carbon based graphene sheets that exhibit unique fluorescent properties throughout a wide range of wavelengths. Given their uniquely small size, low toxicity, biocompatibility, and fluorescent capabilities, GQDs have many unique and important roles. To name a few, GQDs are used in drug delivery, fluorescent imaging, and biosensing thanks to their unique ability to fluoresce under different wavelengths of light. Furthermore, there are different types of GQDs with their own unique properties. Knowing this, five amphipathic molecules, called surfactants, were added to two different types of GQDs to test if they would impact the resulting fluorescence. Furthermore, concentrations of these added surfactants were varied to test how different concentrations of a given surfactant might affect the fluorescence for a given GQD. We observed that some of these surfactants provided a beneficial boost to GQDs fluorescence, while others slightly inhibited the fluorescence. Moreover, we saw that the increase in fluorescence varied based on the concentration of surfactant added yielding lower fluorescence for extremely low and high concentrations, while increasing the fluorescence at a more moderate concentration.

With cancer rates increasing at an alarming rate, many traditional methods for cancer treatment begin to feel outdated. This is where engineering nanomaterials, such as Graphene Quantum Dots (GQDs), offer a promising approach to making chemotherapy a more targeted treatment and therefore minimizing the side effects. This study focuses on optimizing drug delivery mechanisms using GQDs, specifically Reduced Graphene Quantum Dots (RGQDs) synthesized via a top-down approach from reduced graphene oxide, and Hyaluronic Acid Graphene Quantum Dots (HAGQDs) synthesized bottom-up from hyaluronic acid. The process is done by loading chemotherapeutics Gemcitabine, Paclitaxel, and Doxorubicin (DOX) HCl onto GQDs through sonication, this is followed by a centrifugal purification which isolates properly drug-loaded GQDs. To evaluate their controlled release, photothermal properties of GQDs are utilized. Samples are excited with an 808 nm laser at 1, 5, and 10 minutes, and they are later compared to a control group. This analysis provides insights into how laser stimulation affects drug release efficiency, paving the way for advancements in GQD based cancer treatments.