In this project we examine 2-dimensional cell-complexes and group actions on those cell complexes. We determine topological invariants of the group actions on these complexes using homology, cohomology, and the Euler characteristic.

In the field of Riemannian geometry, the condition on the Riemannian metric so that a manifold has positive scalar curvature (PSC) is important for a number of reasons. Many famous researchers have contributed gradually to this area of geometry, and in this project, we study more about PSC metrics on such manifolds. Specifically, we refine and provide some details to the proof of Gromov and Lawson that the connected sum of 2 n-dimensional manifolds will admit a PSC metric, provided each of the manifolds has a metric with the same condition. We then derive some useful formulas related to the Riemann curvature tensor, the Ricci tensor, and the scalar curvature in many different scenarios. We compute the quantities for a manifold equipped with an orthonormal frame and its dual coframe, namely the connection one-form and the curvature two-form. Then, we observe the change in the structure functions, defined as a function that determines the Lie derivative of the orthonormal frame, under a nearly conformal change of the said frame. The aim of these calculations is that, by expressing the scalar curvature of a manifold M entirely in terms of the structure functions, we can determine a condition on the conformal factor so that when dividing the tangent bundle of M into two sub-bundles, then the scalar curvature restricted to one sub-bundle will “dominate” that of the other one so that if we know the scalar curvature of the former sub-bundle is positive, we can be assured that the scalar curvature of M as a whole is also positive.

This research project focuses on the spreading of random curves in the differential geometry field which arises in statistical mechanics . It is known from the work of Einstein that random walks are connected to Brownian motion and diffusion. We will examine random curves that are not merely continuous but that are smooth and have prescribed bounds on curvature. We examine the distribution of a finite number of endpoints of such random curves. Using Python, we obtain 2-D histograms, graphs, and charts to research the spreading of random curves. A central goal in statistical mechanics is to describe the large-scale behavior of systems with the distribution of randomly generated data; we compare the distributions of curve endpoints to the Gaussian (normal) distribution.

Omega-3 supplementation in Division I track & field and cross-country athletes: Physiological markers of Omega-3 status, compliance, and likeability

Katie Couris1, Daphne Thomas1, Tatum Johnston1, Austin J Graybeal, PhD, CSCS2, Brooke Helms, MA, RDN, CSSD, LD3, and Jada L. Willis, PhD, RDN, LD, FAND1

1Department of Nutritional Sciences, College of Science & Engineering, Texas Christian University; Fort Worth, TX
2School of Kinesiology and Nutrition, University of Southern Mississippi; Hattiesburg, MS
3TCU Sports Nutrition, Department of Athletics, Texas Christian University, Fort Worth, TX

ABSTRACT
Omega-3 fatty acid (FA) intake is suboptimal in student-athletes. Given this, and the newfound access to supplementation in collegiate athletes, the purpose of this study was to determine if Enhanced Recovery™ (ER) would improve FA profiles, compliance, and likeability versus a control in Division I track & field and cross-country athletes. In this randomized crossover study, 17 athletes were randomly assigned to either ER or a matched, standard control (fish-oil pills) for ~42d each with a 33-35d washout period. FA profiles were measured at baseline and every two-weeks. For omega-3 index (N3I), there were significant effects of time (p<0.001) and interaction (p=0.004). Significant increases were observed up to four-weeks and were higher for the control versus ER at weeks four (ER=7.25%±1.02; CON=7.76%±1.16) and six (ER=7.33%±1.14; CON=8.03%±1.33). There were also significant effects of time for omega-3:6 and arachidonic:eicosapentaenoic acid (p<0.001). However, after adjusting for compliance and consumption of omega-3 food sources, there were no longer significant effects of time, but an interaction effect remained for N3I and was observed for omega-3:6 (p=0.022; p=0.024, respectively) where both measures were better from four-to-six weeks during the control. Consumption of omega-3 food sources was a significant covariate for N3I and omega-3:6 (p=0.037; p=0.017, respectively). Lastly, 57.9% reported liking/being more likely to take ER and felt it was easier to consume (68.4%). As expected, both the ER and control led to improved FA levels. However, supplementation with ER led to improved likability among division I athletes which may enhance long-term omega-3 status.

Dark Matter (DM) is hypothesized to be an exotic particle that is invisible to human observation. But thankfully, its existence is proven through its gravitational interaction with luminous matter (such as stars and galaxies), and it is responsible for the formation of the humongous structures across our universe. The leading interpretation of DM is what we call Cold Dark Matter (CDM), where the DM particles have relatively low velocities and low energies. This causes structures to form quite quickly and easily in the early universe. While CDM can explain many observed properties of the universe, it is not without its flaws (specifically on the scale of low-mass dwarf galaxies). The hypothesis of Warm Dark Matter (WDM) poses a viable solution to the shortcomings of CDM. In WDM, the DM particles are of higher energy and have higher velocities. This would cause the formation of the first gravitationally bound structures in the Universe to be delayed when compared to CDM. Using a model to approximate varying temperatures of DM, we compare the rates and characteristics of early structure formation for the current CDM hypothesis, and that of many other types/temperatures of WDM. We expect that the differences between CDM and WDM will be most apparent during the first billion years after the Big Bang, just as the first stars in the Universe ignite. These results may be indicative of the true nature of dark matter, and finally bring our understanding into the light.