Project: Toroidal Polyominoes Mentor: Dr. Aaron Montgomery Prerequisite: None

Overview: Polyominoes (not polynomials) are shapes made up of equally sized squares adjoined edge to edge (for example, the pieces in Tetris are polyominoes). Polyominoes have applications to the study of polymers and of percolation clusters, and are often used in recreational board games and puzzles. While polyominoes are easy to define, it turns out that they are difficult to count. As of 2004, researchers have only been able to count the number of polyominoes made up of 56 or fewer squares, no one knows how many polyominoes are made up of 57 squares. So what are toroidal polyominoes? A torus is a topological shape resembling the surface of a donut. Unlike the plane (which is unbounded), a torus has a finite size, which introduces a new variation to the counting process. Related to the torus is the Klein bottle, a torus with a twist, adding another variation to the problem. We will discuss toruses and Klein bottles (introducing the concept of a surface) and we will examine what it means for two polyominoes to be considered equivalent (introducing the concept of a group). With this information, we will determine which polyominoes are equivalent on these surfaces and develop algorithms to count the number of different polyominoes on these and similar surfaces. This project will introduce all of the necessary background, so you do not need to know anything about polyominoes, surfaces, or groups to join. Of course, there will be time to engage in some applications of polyominoes (the board games, not the polymers or percolation clusters) during the project as well.

Project: Utility Theory Mentor: Dr. Sooie-Hoe Loke Prerequisite: None

Overview: In models of individual consumption, more is better but there is too much of a good thing. For example, that first bite of a potato chip is delicious, as is the second and tenth. But the hundredth? Maybe not so much. How do we capture such feelings of satisfaction when modeling human behavior as it applies to investing one’s capital? Of course, there are many models that attempt to understand this phenomenon, but one of the most widely applied is that of utility theory. This project begins by formally defining what a utility function is, why people are interested in it, and how it is used in practice. We will then study a popular utility framework in economics and behavioral sciences called the expected utility hypothesis. As an application, we will apply these theoretical tools to a popular game-show called Deal or No Deal and address questions such as (i) If there is an opponent playing against you, such as in this game-show, how does that affect your strategy? (ii) In a multi-period game like a televised game-show, does the contestant’s utility remain constant, or does it evolve in measure with the audience cheering? (iii) Is there an optimal wealth level where a player should cash out?

Project: Mathematical modeling and data analysis of abnormal ovulation Mentor: Dr. Brandy Wiegers Prerequisite: Calculus (Recommended but not required: Differential equation

Overview: Mathematical modeling provides us an opportunity to learn more about the world around us by using mathematical functions to approximate real world systems. Our goal in using mathematical modeling is to learn more about the world around us by putting to use our knowledge of calculus and functions to study the properties of the mathematical expressions in our model. This CC-REU project will introduce mathematical modeling and build on a research idea posed by a previous CC-REU student, Vanessa Montano. By the end of the summer we will expand the existing mathematical modeling of female ovulation to create a model of abnormal ovulation, specifically looking at polycystic ovarian syndrome (PCOS). As a syndrome, PCOS has a shared set of symptoms that do not always have a definite biological cause. That said, a majority of women who suffer from PCOS have a shared treatment of hormone intervention (birth control). Mathematically modeling the broader hormone system will provide opportunities for undergraduate researchers to make novel discoveries and recommendations of treatments specific to individual women of different ages, race, and health conditions.