MEGL will be running a seven project program with an estimated 35 participants between faculty, graduate research assistants, and undergraduate research interns. For project names and descriptions, please click the boxes below. Apply here by 11:59pm on Monday, December 9: https://forms.office.com/r/4vLid8uvWY
In addition to the research projects, MEGL is running an Outreach program, which exposes K-12 students to important mathematical concepts through visualizations and hands-on activities. There are currently no open Outreach positions for the Spring semester.
One useful description of the stock market is that it is a casino with a long-term uptrend. That is, a casino where the player has the edge.
Philosophically, we have two camps:
1) The stock market is a Markov chain: history does not affect the probabilities of what might happen next
2) Past price trends can be analyzed and used to more accurately assess future price movements
If the latter is correct, then our investigations should not suggest that any particular strategy is more successful than any other.
As an example, consider the well-known strategy of “dollar-cost averaging”. With it, you would invest a constant amount $x, regularly spaced through time. When the market is low, you are automatically buying MORE shares than when the market is high.
Other strategies exist and will be investigated.
The team can be creative in concocting strategies to test.
Faculty Mentor: Dr. Douglas Eckley
Prerequisites: Ability to program in one of R, VBA, or Python
Though quantum computing has exploded in popularity, fundamental questions about its applications and efficacy remain unanswered: when are quantum tools or procedures the right tools for the job? What kinds of experiments benefit from their use? Recently, researchers have used these techniques to tackle logic synthesis, minimizing the complexity and cost of computer circuitry. This project will compare classical and quantum randomized procedures within the logic synthesis problem, connecting ideas in computer science, mathematics, and statistical physics to identify new classes of algorithms.
Faculty Mentor: Dr. Michael Jarret
Graduate Mentor: Anthony Pizzimenti
Prerequisites: Math 203
Returning project; accepting new members
The goal of this project is to build foundational work to study stability in problems involving tipping points in reaction-diffusion equations (RDEs) in one spatial dimension, where the reaction term decays in space (asymptotically homogeneous) and varies linearly with time (nonautonomous) due to an external input. Using a compactification argument and some visualization techniques needed for studying the geometry behind the dynamics of these complex systems, we compute heteroclinic orbits along intersections of stable and unstable invariant manifolds. This will allow us to obtain multiple coexisting pulse and front solutions for the RDEs. Our goal is to apply this framework in a model of a habitat patch that features an Allee effect in population growth and is geographically shrinking or shifting due to human activity or climate change. If time allows, we plan to extend this project by linearizing the system at the standing waves and deriving the eigenvalue problem to investigate their stability.
Faculty Mentors: Dr. Emmanuel Fleurantin, Dr. Matt Holzer
Graduate Mentor: Julia Seay
Prerequisites: Math 214 and familiarity with or desire to learn MatLab
Returning project; not accepting new members
It is important to be able to predict regions that are at risk of flooding when a dam breaks. Topological data analysis (TDA) techniques are new types of mathematical methods for analysis of large data sets that are well suited to finding downstream paths. The goal of this project is to apply TDA methods to topographic data in order to efficiently identify risk regions in the case of a dam breaking. Students should have seen multivariable calculus.
Faculty Mentor: Dr. Thomas Wanner, Dr. Evelyn Sander
Graduate Mentor: Frank Pryor
Prerequisites: Math 213
In this project we will explore hyperbolic polyforms. These are constructed by joining together, edge-to-edge, tiles of a regular hyperbolic {p,q}-tessellation. A hole is a bounded component of the complement of the polyform. We are interested in the question: Given a positive integer h, what is the minimum number of tiles required for a polyform to have h holes. Let’s call this number g(h). The main goals of the project are:
1. Create a table showing g(h) for h=1,2,3,4,5 and for different hyperbolic {p,q}-tessellations.
2. Show that there exist constants c and C (depending on p,q) such that c h < g(h) < C h.
3. Create examples of holey hyperbolic polyforms for large h.
Faculty Mentors: Dr. Ros Toala, Dr. Erika Roldan
Graduate Mentor: Summer Eldridge
Returning project; not accepting new members
Percolation is a model of fluid flow through a random medium. Take an N x N square grid and randomly declare squares open, one at a time, until there is an open path from one side of the square to the other. It turns out that this path behaves like a fractal! The fractal properties of random paths in percolation have been of great interest in both mathematics and physics. Recently, researchers have introduced higher-dimensional versions of these models for which the objects of interest are random surfaces rather than random paths. In this project, we will perform computational experiments to investigate the geometry of these random surfaces, and to see whether they have fractal properties.
Faculty Mentor: Dr. Ben Schweinhart
Graduate Mentors: Anthony Pizzimenti and Morgan Schuman
Prerequisites: Math 213 and some programming experience