# Current Research Opportunities

If you are interested in discussing undergraduate research, come to my office (CSB 348) and we can chat about possibilities. I currently have undergraduate research projects to consider in the areas of:

#### Combinatorics and Complex Analysis

The level sets of the modulus of a complex analytic function can form complicated “figure-eights”, with many lobes. However the typical level set forms a single figure-eight. In earlier work, I counted the number of ways in which these figure- eights could be arranged with respect to each other (and this allowed me to compute the number of polynomials with a given list of critical values). In this project, you will repeat this work where the analytic function is now allowed to have one pole (so that the figure- eights may now have one lobe inside the other!). NOTE: No complex analysis knowledge is required for this project, only a willingness to draw many pictures, and work with some finite sums.

#### Geometry

A star-shaped room is a room in which there is some single point where one may stand and see all points in the room (any rectangular room is one such example certainly, but also a room in the shape of a star!). In this project we will explore the geometry of the “heart of the star”, that is, the collection of all points in the room where one may stand and see the entire room. Some interesting questions: Must the heart of the star be convex? if not convex, then star-shaped? a polygon? If a polygon, how many edges can it have? What questions can you come up with?

# Past Research

Ashley Maurer, Shannon Wilbourne, Summer 2018 (SOfIA Program)

**“Supporting actuarial science at Monmouth College”**

Poster presented at 2018 SOfIA Presentation.

In this project we explored the ways in which Monmouth College, and especially the Department of Mathematics and Computer Science, can support students interested in pursuing actuarial science as a career. This involved researching the work of actuarial scientists, as well as interviews with many people, both educators and practitioners, involved in the field of actuarial science. We made site visits to the Center for Actuarial Excellence at Illinois State University, and State Farm Insurance headquarters in Bloomington, IL. As part of this project, my students prepared materials for the Department of Admission and Financial Aid at Monmouth College to use in advertising to prospective students.

Molly Schoon, Nathaniel Smolcyk, Allie Warfield Summer 2017 (SOfIA Program)

**“Games of Skill and Games of Chance”**

Poster presented at 2017 SOfIA Presentation.

In this project we analyzed two games of skill (Tic-Tac-Toe or Chess) as though they were games of chance.

1) **Tic-Tac-Toe:** We imagined that we were playing against a computer which played randomly, subject to the rules that it must avoid a one-move loss if possible, and choose a one-move win if possible. We found that subject to these assumptions, the best starting move for X is to play in the corner, in which case the probability of winning with the best playing strategy is approximately .92. Starting in the side gave a winning probability of .85, and starting in the center had a winning probability of only .67.

2) **Chess:** Using a chess engine, we analyzed 22 games of neophyte chess players, and found that the probability that any given move was good or excellent was .63. Treating chess moves as independent “coin-flips”, with .63 probability of making a good or excellent move on any given flip, we found that in a game of 75 moves, the probability of all good or excellent moves is approximately 2.67×10^{-15}. Thus in order for a neophyte to have a .5 probability of playing one perfect game, they would have to play approximately 250 trillion games.

Pengrui Wang Summer 2016 (Summer Scholars Program)

**“Characterizing the Direct Product of Directed Semi-Cycles”**

Poster presented at April 2017 MAA MD-DC-VA sectional meeting.

For two directed graphs G and H, the direct product (or tensor product) GxH is a new directed graph such that:

1) The vertices of GxH are pairs of the form (u,v) such that u is a vertex from G and v is a vertex from H.

2) For two vertices (s,t) and (u,v) of GxH, there is an edge connecting these vertices if and only if there is an edge in G from s to u and an edge in G from t to v.

In this project we explored the direct products of directed graphs G and H where both G and H are directed semi-cycles (that is, a directed cycles with some edges “reversed”). We were able to find formulas for the components of GxH in the special case where one of the following occurs:

i) G is a directed cycle.

ii) At most one edge of each of G and H has been reversed.

Jimmy Yau Fall 2015 (Fall/Winter term independent research project)

**“Recognizing a Difference Quotient”**

An article version of this project has been accepted (with revisions and currently resubmitted) for publication in Pi Mu Epsilon Journal.

For a function of two variables H(a,b), we ask the question of how to determine whether H is the difference quotient of some real function f(x). We developed three different tests to apply to a given H which could determine whether there exists such a function f. The first test detects whether H is the difference quotient of an analytic function. The second test detects whether H is the difference quotient of a differentiable function. The final test detects whether H is the difference quotient of an arbitrary real function.