Abstract. In this Independent Study, we survey the mathematics of tiling 2-dimensional regions with polyomino shapes of varying sizes. We investigate tile invariants to prove tileability and examine specific tile invariants, such as the Conway/Lagarias invariant. Using “Tile Invariants for Tackling Tiling Questions” by Dr. Michael Hitchman as a guide for exploration, we survey different techniques for finding tile invariants, such as coloring, boundary words, height, and group theoretic techniques. After this background is established, we answer an open problem posed by Hitchman in the affirmative - we prove the requirements for a modified rectangle to be tileable by area 5 ribbon tiles. In the final part of this project, we consider L-pentominoes and conjecture the requirements for a rectangle to be tileable by this tile set. We prove the conjecture in certain cases.

Abstract. Generative machine learning models have achieved unprecedented feats in recent years and look primed to reach even more impressive heights. By learning data distributions through unsupervised training and by leveraging the power of neural networks, these models are responsible for breakthroughs in various domains. The aim of this paper is to cover some of the prominent generative model architectures through the bottom-up construction of an illustrated storybook generating interface that uses transfer learning on a transformer-based text generator, and the Vector Quantized Generative Adversarial Network (VQGAN) coupled with Contrastive Language-Image Pre-training (CLIP) for prompt-driven image generation.

Abstract. Our Great Lakes have been suffering from unstable weather since the 1950’s. The suffering is from climate change, which also affects the people who live in the Midwest. The Midwest heavily relies on the Great Lakes for their source of water and energy. In this study, we used Ordinary Differential Equations and numerical analysis to show the drastic changes in the Great Lakes. We observed Lake Superior’s water temperatures and air temperatures over the years 1995-2010, by month. We dove into some hydrology ideas and all of the different variables and coefficients used. We expanded many hydrology equations using a method of the Taylor series. Then we used the Fourth Order and Fehlberg Runge-Kutta method to numerically analyze the primary ODE we obtained from the Net Heat Equation. We applied the method to our datasets using Python coding. We described the effects of climate change on the Great Lakes.

Abstract. This project is concerned with articulating the necessary background in order to understand the famous result of the undecidability of the continuum hypothesis. The first chapter of this independent study discusses the foundations of set theory, stating fundamental definitions and theorems that will be used throughout the remainder of the project. The second chapter focuses on ordinal and cardinal numbers which will directly relate to the final chapter. First, there is a clear explanation of the notion of order and what it means for a set to be well-ordered. Then ordinal numbers are defined and some properties are listed and proved. The second half of this chapter discusses cardinal numbers. Similarly, they are defined and some of their properties are stated. Some arithmetic rules surrounding cardinal numbers are discussed as an extension of those properties. The next chapter is concerned with the Zermelo-Fraenkel set theory and the axiom of choice (ZFC) which introduces the idea of set-theoretic systems and models. All nine axioms are listed and expanded upon. Additional focus is put on the axiom of choice and its equivalent statements. The final chapter states the continuum hypothesis, as well as the weak continuum hypothesis and the generalized continuum hypothesis. Some additional background of inner models is discussed for subsequent proof. Kurt Godel proved that the continuum hypothesis could not be proven false within ZFC. The outline for this proof is discussed to reflect its main points. Paul Cohen proved that the continuum hypothesis could not be proven true within ZFC, although this is not discussed as extensively. With this last chapter, the end result becomes clear that the continuum hypothesis is independent of ZFC.