Titles are hyperlinked to pdf copies of the final project write-up. Course coordinator: Pat Keef

Abstract:  In 1874, Georg Cantor published an article in which he proved that the set of algebraic numbers are countable, and the set of real numbers are uncountable. This, at the time controversial article, marked the beginning of modern set theory, and it finally gave mathematicians the notation to explore the infinite. In the intervening century and a half, set theory has blossomed into a central part of mathematics, often acting as a language with which to formalize other branches of mathematics.

In this paper, we explore some basic concepts in set theory, and then consider a result that was proven in the Introduction to Higher Mathematics notes: every countable, totally ordered set is embeddable into the rational numbers. We generalize this idea to larger cardinalities and introduce the notion of $\aleph_\alpha$-universal sets. We show that $\eta_\alpha$-sets are, in fact, $\aleph_\alpha$-universal, and conclude by providing a construction of the minimal $\eta_\alpha$-set whenever $\aleph_\alpha$ is a regular ordinal, and the generalized continuum hypothesis holds.

• Author: Yarden Blausapp

Abstract: This paper presents an exploration of the Fibonacci sequence, as well as "multi-nacci sequences" and the Lucas sequence. We compare and contrast various characteristics of these sequences, in particular the existence and repetition of prime factors. We show that for the Fibonacci sequence, and for multi-nacci sequences with the same initial conditions, it follows that every prime divides an infinite number of terms of the sequence. By contrast, we show that this is not the case for the Lucas numbers. We provide conditions for when a prime does divide a Lucas number and give some examples of primes that do not divide any Lucas number.

• Author: Justin Chin

Title: Bayesian Analysis

Abstract: We often think of the field of Statistics simply as data collection and analysis. While the essence of Statistics lies in numeric analysis of observed frequencies and proportions, recent technological advancements facilitate complex computational algorithms. These advancements combined with a way to update prior models and take a person’s belief or beliefs into consideration have encouraged a surge in popularity of Bayesian Analysis. In this paper we will explore the Bayesian style of analysis and compare it to the more commonly known frequentist style.

• Author: Maya McNichol

Title: Classification Trees

Abstract: A classification (or decision) tree is a predictive model used to analyze data. This method allows us to assign a classification to a data point, using a tree that was grown on data from the same measurement space. In this paper, we discuss the methodology for growing these trees. We review background material in probability and information theory, consider the accuracy of a tree, and discuss the advantages of random forests. Finally, we grow a tree using data about mushrooms and use this to classify mushrooms as either edible or poisonous.

• Author: Brooke A. Taylor

Title: Applying Machine Learning Techniques to Whitman Student Admissions Data (copies available upon request)

Abstract: Every year colleges all over the nation are faced with choosing which students to admit to their campuses, and in return, these students must decide where to attend. This admissions process is a source of endless struggle and debate as colleges do their best to predict how many students they think will accept the invitation, and are faced with significant consequences of under-enrollment as well as over-enrollment if their predictions are off. In this paper we investigate how we might help improve these predictions by using three techniques from the field of machine learning: k-Nearest Neighbors, Principal Component Analysis, and decision trees. We apply these techniques to Whitman College’s most recent school year of 2017, and then examine if the 2017 trained models are effective on 2018 data.