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

• TITLE: FROM POSETS TO DERANGEMENTS: AN EXPLORATION OF THE MÖBIUS INVERSION FORMULA

AUTHOR: Allison Beemer
ABSTRACT: This paper works through an independently developed proof of the Möbius inversion formula. Beginning with a simple seed of an idea, that of a partially ordered set, we proceed by considering properties of compatible matrices, walking through a proof of the existence and uniqueness of what is known as the Möbius function, $\mu$, and subsequently offering a proof of the Möbius inversion formula. The development of the Möbius function and the proof of the Möbius inversion formula lead to results in several branches in mathematics, including combinatorics and number theory. In particular, we will prove the principle of inclusion and exclusion and several number theoretic results.

• TITLE: MODEL SELECTION AND SHRINKAGE: A PROJECT IN LINEAR MODELING

AUTHOR: David DeVine
ABSTRACT: Linear modeling is a statistical tool which fleshes predictive information from large and unwieldy data sets. The most familiar linear modeling technique is the Ordinary Least Squares (OLS) technique which minimizes the sum of the squares of residual errors from a predictor curve. As this paper discusses, OLS has significant shortcomings in regression scenarios with many predictors and with correlation between predictors. This paper provides an introduction to modeling techniques that attempt to improve upon these pitfalls. It studies the mathematical machinery behind Ridge Regression, Subset Selection, Forward Stepwise Selection, Lasso, and Adaptive Lasso including a look at the asymptotic behavior of the Lasso estimates and their limiting distribution. It then moves on to apply the theory in a variety of regression scenarios and investigates the relative performances of each technique using the software package R. Under the criteria of Mean Square Error it concludes that Lasso is the most widely applicable and reliable modeling technique of the ones studied.

• TITLE: AN INTRODUCTION TO SURREAL NUMBERS

AUTHOR: Gretchen Grimm
ABSTRACT: In this paper, we investigate John H. Conway's surreal numbers. Surreal numbers are defined by two sets of numbers, which differentiates them from the real numbers. Based on two axioms, we map out all of the surreal numbers, finding infinities greater than any real number and infinitesimal numbers smaller in absolute value than any real number. We investigate addition and multiplication of surreal numbers, and show that they form a totally ordered field, $S$, which contains the real numbers. We also give an introduction to an application of surreal numbers in game theory.

• TITLE: PATHOLOGICAL: APPLICATIONS OF LEBESGUE MEASURE TO THE CANTOR SET AND NON-INTEGRABLE DERIVATIVES

AUTHOR: Price Hardman
ABSTRACT: The Lebesgue integral is a hallmark of modern analysis, and the theoretical foundation of the Lebesgue integral is measure theory. Here, we develop the basics of Lebesgue measure theory on the real line, a theory which is concerned with generalizing the notion of length to sets that are not intervals. We then investigate several fascinating "pathological" examples -- particularly counterintuitive and insightful results -- and use the tools of measure theory to explore both their bizarre features as well as their historical importance. These examples include the Cantor set, an uncountable set of measure zero; the Smith-Volterra-Cantor set, a set of positive measure yet one that contains no intervals; and Volterra's function, a function with a bounded derivative that is not Riemann integrable.

• TITLE: PÓLYA'S COUNTING THEORY

AUTHOR: Mollee Huisinga
ABSTRACT: P$\acute{\text{o}}$lya's Counting Theory is a spectacular tool that allows us to count the number of distinct items given a certain number of colors or other characteristics. We will count two objects as 'the same' if they can be permuted to produce the same configuration. We can use Burnside's Lemma to enumerate the number of distinct objects. However, sometimes we will also want to know more information about the characteristics of these distinct objects. P$\acute{\text{o}}$lya's Counting Theory is uniquely useful because it will act as a picture function - actually producing a polynomial that demonstrates what the different configurations are, and how many of each exist. As such, it has numerous applications. Some that will be explored here include chemical isomer enumeration, graph theory and music theory.

• TITLE: A STUDY OF CONVEX FUNCTIONS WITH APPLICATIONS

AUTHOR: Matthew Liedtke
ABSTRACT: This paper is a rigorous analysis of convex functions and their applications to selected topics in real analysis and economics. In our theoretical survey we explore the continuity and differentiability of convex functions. In addition we state a simplified criteria of convexity for continuous functions. Our topics from real analysis include right-endpoint approximation for integrals and pointwise convergence. In our discussion of economics, we will use the theory of consumer behavior under uncertainty and Jensen's Inequality to show the relation between convex functions and the modeling of risk loving behavior.