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Community of Mathematicians and Statisticians Exploring Research

What is the Community of Mathematicians and Statisticians Exploring Research?

The Community of Mathematicians and Statisticians Exploring Research (Co-MaStER) is a new research program established in 2019 for students interested in conducting research in mathematics and/or statistics (broadly defined) at Villanova University. The program is open to all Villanova students and will be most beneficial for those seeking careers in mathematics - in academia, industry, and government. Below, you will find more information about the program and the application form.

"I am not sure what career I would like to pursue. Should I still apply to a mathematics research program?"

Employers of mathematicians are looking for people with problem solving skills, ability to abstract, attention to detail, ability to learn new ideas and concepts on their own, methodical approach to solving a problem (i.e., breaking a big problem down into smaller pieces that can be solved separately), and the different perspective they bring to problem-solving (as compared to people in other fields and disciplines). Conducting research in mathematics is an excellent way of enhancing these highly desirable skills. Conducting research also exposes you to areas of math and statistics you wouldn’t see in class, which can help you decide a career path.
We encourage everyone to give themselves the opportunity to be part of a research program in mathematics.

"What projects can I work on?"

Each semester, we will post available projects for students to work on below. Read through the current projects before filling out the Application Form.

When completing the Application Form, please indicate all projects that interest you. Each faculty will then reach out to you to let you know if you have been selected to the project. If you have multiple invitations to join a project, we ask that you only accept the one that interests you most.

Slides of the presentation Drs. Muller and Diaz-Lopez gave on September 6, 2019.

Fall 2021 Projects

Spring 2021 Projects are underway. 

Fall 2021 projects and applications will open closer to the Fall semester.

Application Form

Applications for Spring 2021 projects have closed.

If you have questions, please contact Dr. Peter Muller (peter.muller@villanova.edu) or Dr. Alexander Diaz-Lopez (alexander.diaz-lopez@villanova.edu).

Past Projects

1. Lucky colorings of graphs; Led by Dr. Michael Tait

Label the vertices of a graph with positive integers, and for each vertex v let S(v) be the sum of the labels of all of its neighbors. A labelling is called lucky if for every edge uv, S(u) is not equal to S(v). The lucky number of a graph is the minimum number of labels necessary to produce a lucky labelling. In this project we will investigate this graph parameter.
Pre-requisites: MAT 2600 and any previous experience with graph theory or combinatorics.

2. Extremal graph theory; Led by Dr. Michael Tait

Extremal graph theory seeks to quantify the statement "when a graph gets large, it contains structure somewhere" or "complete disorder is impossible". In this project we will consider Ramsey and Turan problems for graphs and hypergraphs.
Pre-requisite: A course in graph theory.

3. Measuring distance between permutations; Led by Dr. Katie Haymaker and Dr. Alexander Diaz-Lopez

In mathematics, it is common to measure the distance between objects. Some common examples are to measure the distance between two points in a cartesian plane and the (Hamming) distance between vectors. In this project, we will explore combinatorial objects called permutations and different ways to measure the distance between permutations that share similar properties. The project is exploratory and in its initial phase. In short, this means we will explore many examples and use these examples to spot patterns, make and (hopefully) prove conjectures, and look for connections to other interesting problems.
Pre-requisites: Foundations of Mathematics (MAT 2600) is required and Modern Algebra (MAT 3600) is preferred but not required.

4. Multi-Sample Rank Tests; Led by Dr. Jesse Frey

Rank tests are statistical tests where only the ordering of the observations matters.  Such tests require fewer distributional assumptions for validity than do tests such as the two-sample t test and the F test.  Two well-known examples are the Wilcoxon rank sum test and the Kruskal-Wallis test.  This project will involve becoming familiar with existing rank tests and learning to compare rank tests by doing power studies in R.  The eventual goal is to create a new multi-sample rank test based on a new nonparametric likelihood.  This is intended for one undergraduate.
Pre-requisites:  STAT 4310.  Additional statistics courses and R experience are also helpful. 

5. Fair Funding and Educational Outcomes in Pennsylvania; led by Dr. Bruce Pollack-Johnson and Dr. Michael Posner.

Pennsylvania passed a law implementing an excellent Fair Funding Formula to ensure all public school districts get fairly funded based on economic circumstances, geography, taxing capacity, etc.  But this is being phased in very slowly (only 10% of funds currently).  In this project, you will be trying to understand better why it is that districts whose funding is closest to the formula get the best educational outcomes, as measured by enrollment in post-secondary education programs, even compared to those that are over-funded according to the formula.  This project will involve manipulating a big data set, looking for models of the relationships between the variables, and doing statistical tests. 
This would be possible for both an undergrad or a grad student.
Pre-requisites:  STAT 4310 (or STAT 7404) and STAT 4380 (or STAT 7500).  Experience with statistical packages such as R or SAS is essential.

6. Assignment Pebbling Graphs; led by Dr. Andy Woldar

The standard pebbling game is an impartial two-player game played on a graph G.  Initially, pebbles are randomly placed on the vertices of G.  A pebbling move consists of removing two pebbles from a vertex v, while simultaneously adding one pebble to a neighbor of v.  The loser of the game is the player who is unable to make a legal pebbling move. 

Related to the initial pebbled graph G is a rooted graph [S_G], which we call the pebble assignment graph.  The root node r of [S_G] is the initial state of the pebbling game, and nodes x and y of [S_G] are adjacent provided y arises from x (or vice versa) via a single pebbling move.  It turns out that [S_G] is always a connected, bipartite graph.  In particular, the girth of [S_G] must be even (or infinite).

Curiously, for every even integer k not equal to 8, we are able to construct an assignment graph of girth k.  We conjecture that girth 8 is not possible.

7. The Kelly Criterion in the presence of maximum payouts; led by Dr. Klaus Volpert

Suppose you get to play a game repeatedly where your chance of winning is 60% each time. Your fund starts with $100. What percentage of your fund should you bet each time to optimize your expected outcome after, say, 100 games? The Kelly Criterion, established in 1955, says that you should bet 20% of your fund each time.
But what if there is a maximum payout in place, say $1000? And what if one is at $900 after only 50 games? Surely one would not bet 20% in the next game! Not even 10%. It is easy to demonstrate that in this situation there is a better strategy than the Kelly criterion. But what is the best strategy possible?
Pre-requisites: basic calculus and statistics. Willingness to learn programming in R or Maple or Python.

1. Measuring Competitive Balance in Sports; Led by Dr. Jesse Frey

Which professional sports league has the greatest amount of in-season competitive balance, and which has the least?  Since some leagues have longer schedules than others, and since some schedules are more unbalanced than others, these can be difficult questions to answer.  In this project, we'll read the existing literature on measures of competitive balance, critique the existing measures, and search for better measures that hopefully have easy interpretations.
Pre-requisites: R programming experience, MAT 5700 (preferred)

2. Extremal spectral graph theory; Led by Dr. Michael Tait

The adjacency matrix of a graph is a square matrix whose rows and columns are indexed by the vertices of the graph, and the ij'th entry of the matrix is 1 if vertex i is adjacent to vertex j and 0 if not. Surprisingly, the eigenvalues and eigenvectors of this matrix can tell us a lot of information about the graph. In this project we will consider the questions like: how large can an eigenvalue of the adjacency matrix be if the graph is triangle-free? If the graph is planar?
Pre-requisite: Some experience with linear algebra

1. Counting colorful tilings of rectangular arrays; Led by Dr. Katie Haymaker

How many ways are there to tile a rectangular board with painted squares and dominoes, when there are "a" available colors for the squares and "b" available colors for the dominoes? What if we include a third type of tile, called a tromino? There is no self-contained formula for the number of tilings of an m×n board with dominoes and squares, but there are ways to find this number for special cases. In this project, we will use a recursive approach to find the number of tilings for certain board sizes.
Pre-requisites: Foundations of Mathematics (MAT 2600) and Diff Eq/Linear Algebra (MAT 2705)

2. Fair Funding and Educational Outcomes in Pennsylvania; Led by Dr. Bruce Pollack-Johnson and Dr. Michael Posner

Pennsylvania passed a law implementing an excellent Fair Funding Formula to ensure all public school districts get fairly funded based on economic circumstances, geography, taxing capacity, etc.  But this is being phased in very slowly (only 10% of funds currently).  In this project, you will be trying to understand better why it is that districts whose funding is closest to the formula get the best educational outcomes, as measured by enrollment in post-secondary education programs, even compared to those that are over-funded according to the formula.  This project will involve manipulating a big data set, looking for models of the relationships between the variables, and doing statistical tests.  Experience with Excel and statistical packages such as R, SAS, SPSS, etc. would be especially helpful.
Pre-requisite: Stat Methods (MAT 4310)

3. Coin-operated black jack machine project; Lead by Prof. John Santomas and Prof. Dominic Canzanese

The aim of this project is to investigate a Reel "21" coin-operated mechanical trade stimulator.  In particular, students will model/study this stimulator through the lens of probability.  For more information, contact Prof. Santomas.
Pre-requisites: In your application, please state all relevant probability coursework you have taken that you feel would be of assistance to you.

4. Any topic students want to research; Supervised by Dr. Osvaldo Marrero

Dr. Marrero is willing to work with students who wish to research whatever mathematical topic they wish. The topic is not limited to one student.
Pre-requisite: In your application state all relevant coursework you have taken that relates to your topic.

5. Student-lead research project in Data Science; Supervised by Dr. Michael Posner. 

If you have your own interesting data science related research question you would like to pursue, Dr. Posner will be happy to offer light supervision for the project.
Pre-requisites: Student must have taken Intro to Data Science (MAT 4380)

1. Extremal Graph Theory; led by Dr. Michael Tait.

This research project has two options depending on student interest and background. First, Turan-type problems ask to maximize the number of edges in a graph on n vertices subject to the constraint that it does not contain some fixed graph F as a subgraph. Students who enjoy counting and abstract algebra might enjoy this type of problem. Second, if students are interested in spectral graph theory, we can consider extremal graph theory problems where one maximizes the largest eigenvalue of the adjacency matrix over the family of F-free graphs. Students who enjoy linear algebra and structural graph theory might enjoy this type of problem.
Pre-requisites: Foundations of Mathematics (Math 2600)


2. Extremal Combinatorics using Automated Conjecturing; led by Dr. Vikram Kamat.

The project is in the area of extremal set theory; broadly, the classic problem in the field is to find the maximum size of a system of subsets of a finite set subject to certain structural constraints, and also characterize the structures of the optimal systems, called the extremal families. The approach in this project will be two-pronged. We will make use of an automated conjecturing program that proposes conjectures involving certain standard invariants, then try to prove or disprove these conjectures. Each counterexample used to disprove a conjecture is then fed to the system allowing it to make better conjectures. The goal is to discover new results in extremal set theory using this computer-aided method.
This project will interest students who are interested in discrete mathematics, but more generally in discovering new mathematics by proving theorems (and disproving false conjectures). Some experience with programming in a language such as Python is helpful, but not necessary, as the computing knowledge needed to learn the conjecturing system can be learned and mastered on the fly.
Prerequisites: Foundations of Mathematics (Math 2600)


3. The volatility of portfolios; led by Dr. Klaus Volpert.

When one puts two or more stocks of different volatilities together, then for the short-term the volatility of the portfolio is an easy calculation based on correlations. But long-term one stock can start to dominate the portfolio and therefore the volatility of the whole. I do not yet understand the probability distribution of this long-term volatility. This distribution affects the pricing of options on such portfolios.
Pre-requisites: The student would run experiments via Monte Carlo simulations and would need to have some programming skills, either in R or Python or C+.


4. Mathematical Modeling Projects; led by Dr. Peter Muller.

This project involves using differential equations to model real-world phenomena.  There are multiple topics I have in mind that can be chosen based on student interest.  One topic is to build off an existing model of how different viewpoints spread through a population. Another broader topic is disease modeling, which can be modeling a disease inside a patient or how a contagion spreads through a population. I am also happy to work with students on project ideas they may have.
Pre-requisites: Calculus I, willingness to learn how to use Matlab/Octave software, Differential Equations (MAT 2705) would be beneficial, but is not required.


5. Combinatorics related to permutations; led by Dr. Alexander Diaz-Lopez.

Permutations are different ways to arrange objects. They are one of the most widely studied objects in combinatorics, yet there is still a lot unknown about them. In this project, students will explore graphs of permutations and model/describe the behavior of peaks in the graphs of permutations. Students that enjoy Foundations, Combinatorics, or Abstract Algebra might enjoy this type of problems.
Pre-requisites: Foundations of Mathematics (Math 2600).


6. Distributional and probabilistic properties of prime numbers and related primality questions; led by Al Marrero.

Throughout history, human beings have been fascinated by prime numbers. We know a lot about the distribution of prime numbers, but often these results are for “large” numbers. We are interested in more specific, concrete results. The project is likely to be largely computational. Students should be familiar with R or Maple. Python is a third option, assuming that the students know that software; however, for now, I won’t be able to provide much specific help with Python. Students should know some probability and  basic analysis--limits, etc. The topic is not limited to one student.


Analysis of publicly available data; led by Al Marrero.

This topic is meant to give students an opportunity to investigate an issue—health, social, etc.—that they are interested in. The students propose the particular topic that they are concerned about, and then we look for publicly available data that may be used for the analyses. Students should know statistics at the minimal level of MAT 4310, and they should be minimally familiar with the software R. The topic is not limited to one student.


Any topic that students want to research; led by Al Marrero.

I am willing to work with students who wish to research whatever mathematical topic they wish. The topic is not limited to one student.

Contact Information

Department of Mathematics & Statistics
SAC Room 305
Villanova University
800 Lancaster Avenue
Villanova, PA 19085 
Tel: 610.519.4850
Fax: 610.519.6928
Email: math@villanova.edu

Chair:
Dr. Jesse Frey

Staff:
Christine Gadonas 610.519.4809
Maria W. Barrett  610.519.4850