Dr. Sarah K Mason
 Associate Professor A. J. Sterge Faculty Fellow Department of Mathematics Wake Forest University

# Potential Research Topics

Below are several general topics within combinatorics. If you are interested in doing research in combinatorics, I can tell you about some problems in one of these areas or we can look for a problem in any area of combinatorics that is interesting to you. You can even decide to do research as your summer job since Wake has a great program to fund student research over the summer!

## Symmetric polynomials

A symmetric polynomial is a polynomial in multiple variables that remains the same even when the exponents of the variables are permuted. One very nice basis for symmetric functions is called the Schur function basis and can be generated using diagrams that kind of look like tetris shapes. Many of the problems in this area involve exploring properties of these diagrams but they can also have a very algebraic side to them if you are interested in abstract algebra.

## Weighted voting

Each stockholder in a company chooses how many shares of stocks he/she owns. If it takes a certain number of stocks (a quota) to pass a motion, then the stockholders will usually need to form coalitions to get their motions passed. Situations that can be modeled using stockholders and quotas are called weighted voting systems. Understanding how much power each stockholder actually has is a very interesting task, and one way to do this is to use tools from combinatorics such as partially ordered sets and integer compositions.

## Puzzles and games

Almost any puzzle or game you can think of has some sort of underlying mathematical structure. Often this structure can be modeled by creating a graph to represent the situation. Pick your favorite game and we can explore the mathematics behind it, or pick up where other students have left off in finding graphical solutions to puzzles like "Scramble Squares". This topic has lots of connections to computer science and economics.

# Research students

## Master's Thesis Students at Wake Forest

• Andres Guzman-Guerrero, 2020
• David Nichols, 2020
• Katie Greene, 2017
• Jeremiah Southwick, 2016
• Joel Barnett, 2014
• Kara Stasikelis (co-advised with Jason Parsley), 2013
• Melissa Bechard, 2012

## Senior Thesis Students

• Flora Yang, 2017
• JiaJie Zheng, 2017
• George Story (co-advised with Jason Parsley), 2012
• Katie Daves (Davidson College), 2008

## Summer Research Students

• Will Hu and Lawton Manning, 2017
• Flora Yang and Jiajie Zheng, 2016
• Eagle Jin, 2013
• Mali Zhang (Davidson Research Initiative), 2009
Last updated July 2020
by S K Mason.