Papers & Projects

Covariance Uncertainty: Estimation & Visualization, August 2024. [Overleaf draft in progress]

Slides used for intern final presentation.

APL Summer Intern Project (Summer 2024)

Abstract: We consider I.I.D. samples drawn from a bivariate normal distribution. The aim is to quantify and visualize the error about the Maximum Likelihood covariance estimate. Two approaches are considered: using the Fisher Information and using the Wishart Distribution. The former involves calculating the asymptotic covariance matrix using the Fisher Information, then visualizing this using a Cholesky Decomposition. The latter involves estimating the 90% Highest Probability Density (HPD) Region of the Wishart Distribution: proposed methods include using Markov Chain Monte Carlo (MCMC) methods to numerically approximate this.

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The Development and Impact of Gödel's Incompleteness Theorems, May 2024.

Abstract: Kurt Gödel's incompleteness theorems demonstrated that in any sufficiently powerful recursive axiomatic system, there exist true statements that cannot be proven within the system. The first theorem challenged the notion of completeness; the second established that such a system cannot prove its own consistency. In this paper, I explore the historical context of Gödel’s theorems, particularly in relation to Hilbert's program and Russell and Whitehead's Principia Mathematica. I further examine the theorems' impacts on the mathematical community, i.e., shifting the focus from seeking completeness to understanding the limitations of formal systems. The influence of these theorems on 20th-century philosophers, such as Wittgenstein and Quine, is also analyzed. 

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Decomposition of Quiver Representations, April 2024.

Beamer talk given at DRP symposium (Spring 2024).

Abstract: For a non-cyclic quiver Q, the Krull-Schmidt theorem gives us that the decomposition of its representations are unique up to reordering of indecomposables. Furthermore, Gabriel’s theorem tells us that if Q is connected, then it has finite representation type if and only if it is of type ADE, and moreover that its indecomposables are in bijection with its positive roots. In this talk, I will delve into these theorems to develop an algorithmic construction of the indecomposable representations of the quiver A_3 using reflection functors and its positive roots.

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The Impact of Education Expenditure on Economic Growth, December 2023.

Abstract: This paper investigates the impact of government education expenditure on the GDP growth rate in 2016. Despite the widespread belief in the positive effects of education on economic prosperity, I find that education expenditure is not a statistically significant predictor of GDP growth rates within a five-year period. While investments in pre-primary, primary, and secondary education appear to have a negative correlation with GDP growth, tertiary education expenditure shows a positive association. I also find that past economic health, as indicated by the GDP growth rate in 2011, is a significant predictor of future growth. Policy implications include a focus on tertiary education funding due to its immediate impact on economic growth and a reconsideration of the distribution of education expenditure to ensure long-term sustainable development. Additionally, the negative impact of military spending on GDP growth suggests a balanced fiscal strategy that prioritizes educational investments. Further research may include examining longer time lags to fully capture the effects of education on economic growth.

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The Impact of the Presence of Fast Food Restaurants on Obesity, October 2023.

Abstract: This study examines the correlation between fast-food restaurant density and obesity rates across U.S. states. Utilizing a Multiple Linear Regression model, I find that restaurant density does not have a statistically significant impact on obesity rates. However, factors such as education and regional differences, particularly in the South and Midwest, exhibit stronger associations with obesity. This suggests that addressing obesity requires also studying socioeconomic and regional disparities. Limitations include potential biases from self-reported data and the exclusion of confounding factors.

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Gradient Boosting for Fantasy Premier League, December 2021.

Abstract: Fantasy Premier League is a fantasy sport in which points are earned on the basis of real-life statistical performances. A common strategy to maximize performance is effective captain selection (the captain earns twice as many points as other players). To choose the captain, I first applied a gradient boosted trees algorithm (XGBoost classifier) to a dataset of players’ statistical performances to create a ‘candidate pool’. Next, I used Monte Carlo simulations to determine the best captain selection strategy, finding a tradeoff between high, inconsistent scoring (Fixture; mean = 513 points, stdev = 98) and relatively low, stable scoring (Form; mean = 420 points, stdev = 53).

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