Let $E/\QQ$ be an elliptic curve; for all but finitely many primes $p$, reduction modulo $p$ yields an elliptic curve over the finite field $\mathbb{F}_p$, and it is natural to ask about the properties of these reductions for varying primes. The purpose of this dissertation is to study one such question, namely, how frequently the reductions result in an elliptic curve with cyclic group structure. To be precise, we let $\pi_E^{cyc}(x) $ denote the number of primes less than $x$ for which the reduction of $E$ modulo $p$ is cyclic. The asymptotic behavior of this function has been established by Serre conditional on Generalized Riemann Hypothesis. Furthermore, Banks and Shparlinski showed that this asymptotic holds unconditionally on average over the family of elliptic curves given by short Weierstrass equations with coefficients taken in a `box.' Inspired by the work of Battista, Bayless, Ivanov and James on the Lang-Trotter conjecture, we study the average asymptotic behavior of the functions $\pi_E^{cyc}$ where the average is taken over certain thin families of elliptic curves: elliptic curves with a rational point of order $m$ defined over $\mathbb{Q}$. The results we obtain are again in agreement with the conditional asymptotic. We also extend the study of cyclicity from elliptic curves defined over the rational numbers to elliptic curves defined over a quadratic extension of $\QQ$ and obtain partial results in that case.As a key tool, we prove an analogue of a result of Vl\u{a}du\c{t} that estimates the number of elliptic curves over a finite field which have some specified torsion and which have group structure that is as cyclic as possible.

The Human gastrointestinal tract provides a broad range of environmental niches taht can be colonized by a variety of microbes. The microbes found within the mucosal niches depend on both dietary and endogenous energy for survival (Peled et al., 2021). In return, these microbes are capable of providing numerous beneficial effects for the host including digestion of complex nutrients and the development of the immune systems (Peled et al., 2021, Zheng, 2020). The ability of microbes to metabolize indigestible fibers, especially in low fat, high fiber diets, have been linked to positive health outcomes in both mice and humans (Killinger et al., 2022, Makki et al., 2018). The Bacteroides genus, which is characterized as a Gram-negative, anaerobic bacteria, colonizes the lower gastrointestinal tract and includes Bacteroides fragilis (Wexler et al., 2007). In addition to its prominent role as a commensal bacterium of the gut, B. fragilis is also often attributed as the most virulent among the many species within Bacteroides due to its prevalence in extra-intestinal infections and high mortality rate(Wexler et al., 2007). The capacity of B. fragilis in using these diverse nutrients in the environment, within the gastrointestinal tract and systematically highlights its ability as a microbe (Nakajima et al., 2020, Huang et al., 2011, Sears, 2001). This ability to thrive in different nutritional conditions demonstrates its versatility within the complex gut environment. Here, we demonstrate that strains of B. fragilis isolated from a variety of sources exhibit different growth magnitudes and patterns through in vitro assays with complex media. We performed shotgun whole genome sequencing, and in combination with publicly available sequences, constructed a B. fragilis pan-genome model. We show that genetic differences between isolates was in part due to genes that coded molecular machinery. The growth kinetics for defined energy sources was evaluated, showing that isolate level genetic differences could impact nutrient utilization in B. fragilis. In conclusion, we show that genetic variations within nutrient acquisition genes affect the fitness of B. fragilis strains in specific conditions and therefore, strain-level variation impacts the capability of B. fragilis to survive in various biological contexts.