A common class of problems is that of understanding the large-scale properties of a system defined by small-scale relationships. These problems arise in fields such as statistical mechanics. We have been considering such problems as they apply to graphs, mathematical structures that can be used to represent networks, by investigating the limiting behavior of finite approximations to infinite graphs. In particular, we have been studying the growth rate of the number of spanning trees on these finite graphs. This allows us to quantify the complexity of various infinite graphs. The Matrix-Tree Theorem gives a formula for the number of spanning trees on a finite graph using techniques from linear algebra. This allows us to numerically approximate growth rates. These calculations can help us see patterns and develop theoretical approaches for obtaining exact values, as well as test theoretical results. In the case of graphs with translational symmetry, we have used techniques from group theory, a powerful area of mathematics, to derive an elegant expression for the exact growth rate in terms of Mahler measure. This is an integral formula connected to dynamical systems and number theory, among other research areas.While these considerations on spanning trees are facilitated by the Matrix-Tree Theorem, there is no known formula for counting spanning forests, a structure consisting of one or multiple trees which is more natural in some ways. Our current research generalizes the questions on trees to various types of spanning forests. We have also been exploring different ways to consider how a graph grows. This work can be used to build on this active research area of determining properties of large-scale structures.

The migrant community of Lucerna is located in the last pristine frontier forest of the Madre de Dios region of Peru, and has a fluctuating population of approximately 65 residents. To date, their waste management practices include burning, burial, and dumping waste in the river, which is causing both environmental and human health risks. The aim of this study was to understand the amount of waste and its composition, the levels of water contamination, and the mentalities towards waste management within this remote river town. Data collected during both the dry and wet field seasons and seven waste audits suggest that each family produces 3.5 kilos of waste per week on average. The composition of this waste includes plastics (bottles and bags), cans, glass, and miscellaneous waste. Organic waste is primarily reused as food for livestock and was therefore excluded from this study. The amount of waste acquired per family remained constant between seasons. Thorough water sampling was completed that will later supplement waste audits to correlate water contamination and the current burning, burial, and dumping of waste near water sources. Mentalities towards waste management were measured through informal interviews and observation. The option to transport waste to the closest city of Puerto Maldonado was impractical due to the cost and needed organization within the community. Instead, garbage pits were constructed in January of 2017 to ensure proper waste management and the mitigation of contamination until the necessary infrastructure is in place at the municipal level. This case study will serve as a replicable waste management framework for other remote developing communities within the Las Piedras region, and Amazonia as a whole.

Cacao (Theobroma cacao) is a perennial crop grown in the tropics to produce chocolate worldwide. In the home country of cacao- Peru- wild varieties have been domesticated, cropped, and crossbred to create the optimal plant for specific climates. A study in the Valle Rio Apurímac-Ene (VRAE) region of Peru investigated 120 different varieties of cacao with diverse aromas and tastes, several of which have won awards for producing the best chocolate in the country. When members of these communities fled civil conflict five years ago and came to the Amazon, they brought several varieties not bred for this climate. My study measured the growth of one-year-old cacao seedlings in the experiment plots of Hoja Nueva, a nonprofit working in the lowland Peruvian Amazon. Although VRAE varieties are sun-grown, Hoja Nueva is growing these culturally significant varieties sustainably by keeping native shade trees, using biochar and biopesticides, and implementing reforestation programs that offset deforestation. This study is one of the first to be carried out in the field, since most growth studies have taken place in greenhouses. Environmental and abiotic factors play a pivotal role in seedling development. For 6 weeks abroad, I measured the temperature, light, overstory canopy cover, soil, and micro-climate within Hoja Nueva’s cacao farms, along with the growth of individual parent cacao seedlings. Preliminary results show significantly more seedling growth in areas of higher light intensity and lower overstory canopy cover. This study will serve as the foundation for a replicated study next January, looking at one-year-old grafted cacao plants. Once grafted, it is hypothesized that seedlings will display equal growth in the shade and sun plots. The goal of this study long-term is to produce the same quality and yield of cacao per year as the community is currently, to limit deforestation, and eventually introduce shade-grown cacao varieties to the region.