If you're tiling a circular room with square tiles, how many do you need? What if your tiles are parallelograms? As the circle gets larger, is there a pattern? Mathematically, this can be framed as counting integer points in a large circle. We're interested in how this count changes as you change the shape of the tiles, and to understand this better, we started by counting the number of integer vector pairs within a ball of radius R so that the parallelogram they make has a fixed area (determinant). We created a Python program that would generate all of the primitive points out to radius R, which we then extended to count the number of vector pairs with a certain determinant k. We were able to compute the limiting density of this count, extending known results for the case of determinant 1. We are now studying other discrete sets, such as ones generated by objects from hyperbolic geometry, known as Hecke triangle groups. In the future, we plan to use our research for counting pairs of vectors in different spaces and generalize it for counts of k-tuple vectors. Come for floor tilings, stay for beautiful pictures!

Few communities within the Madre de Dios region of Peru are recognized by local governments, and less are allocated the necessary resources to meet basic needs such as waste management. The aim of this study is to develop a toolkit to understand the amount of waste, the composition, the levels of water contamination, and the mentalities towards waste management, specifically within the Las Piedras district, in the Madre de Dios region, of Peru. The damaging effects of absent waste management and unsustainable landfills affect the environmental, social, and economic sectors of the surrounding environment. To raise awareness of waste management and find tangible solutions, the Waste Management Toolkit includes waste auditing, water sampling, and mentality survey strategies which were conducted in three remote communities, as well as within the Las Piedras Muncipality. Results have shown evidence of both human health and environmental health impacts from improper waste management due to burning, burial, and dumping practices. Mentality surveys have shown distrust in the local municipalities, a lack of education when it comes to waste management practices and impacts, as well a district wide absence of infrastructure to support waste systems. The Waste Management Toolkit has successfully led to the implementation of a communal waste pit in the community of Lucerna in the field season of 2016 and 2017, as well as in the community of Puerto Nuevo in Spring of 2018. The community of Boca Pariamanu has been studied and results have been reported to the local municipality in Spring of 2018 to support waste management systems since they are accessible from the closest large city, Puerto Maldonado to export waste. These three communities have become the pioneers within the region. Future endeavors will include the replication of the Waste Management Toolkit in surrounding communities and municipalities.

Amazonian soils are notoriously nutrient-poor, making it difficult to grow crops without the use of slash and burn practices or synthetic fertilizers. Communities outside of urban areas do not have access to sufficient nutritional variation in their diets. This results in dietary deficiencies, especially in children, as well as more time and money spent importing food items from nearby cities. The objective of my research was to evaluate an approach for developing a community-centered permaculture planting plan. From January through March 2018, I interned for Hoja Nueva, which works with remote native and migrant communities that are generally neglected by the government due to their inaccessibility. The case study focused on a community located on the Las Piedras River in the Peruvian Amazon called Puerto Nuevo, which is known as the ‘forgotten village’ because it lacks an active health post, water treatment, and waste management system. In February, I conducted individual interviews about current diet and desired fruits and vegetables with villagers and held community meetings about potential garden sites. Additionally, two potential sites identified by the community were tested for macronutrient concentrations (nitrogen, phosphorus, and potassium), pH, texture, moisture levels, and environmental variables like light exposure, temperature, relative humidity, and other microclimate variables. These data were critical for determining which site will best support certain plants requested by the community. The complete permaculture plan was delivered in March and included designs for a fruiting forest and vegetable garden with native plants and a compost system to help replenish the soils. Once completed, the mature permaculture site will greatly enhance the diet of the Puerto Nuevo community. The opportunities and challenges inherent in this community-centered approach are discussed.

Having access to clean drinking water is a crucial component of enjoying positive health outcomes, especially for at-risk populations such as children undergoing developmental stages and pregnant mothers. Unfortunately, pristine drinking water is not a universal norm. Peru is a prominent example of a nation with wide diversity in ecology as well as human activity and demography, where the range of access to clean drinking water varies greatly across the country. Contaminants that contribute to polluted drinking water have many probable sources, caused by environmental factors and/or anthropogenic activity. Implications being able to test where contaminants enter the system and their relative levels include relating the outcomes to various socioeconomic and environmental factors which influence community development. Through conducting water quality assessments and interviews with six communities, I understand which economic, cultural, political, and environmental conditions are most related to securing positive opportunities for the respective community’s residents. Three of the communities are remotely located within the Amazonian environment, and three are located along the interoceanic highway with more rapid rates of development. The parameters that have been chosen for this assessment include pH, turbidity, nitrates, chlorine, and total and fecal coliforms. Depending on individual results, communities have been left with economically and technologically feasible recommendations for improving contamination sources as necessary. The water quality assessment was also shared with local government authorities to draw attention to communities most in need as well as to encourage government assistance in resource distribution. Hoja Nueva, a non-profit working in the remote Las Piedras region of Peru, will continue to monitor water quality in the future with respect to the changing environmental and socioeconomic conditions of the communities using the water quality assessment methodology I create as a baseline for comparison.