MEGL will run nine projects with an estimated 50 members, including undergraduate researchers, graduate mentors and faculty mentors. For project names and descriptions, please click the boxes below. To apply, click here.
A geometric manifold is a manifold whose geometry (metric) locally looks like a familiar space, i.e. Euclidean, hyperbolic, or real projective space. One can endow a given surface with uncountably many different local geometric structures. A natural question to ask is how one can distinguish between two different geometric structures on the same surface. Length spectrum rigidity asks “can we answer this question only knowing the length of certain curves on the surface?” There are some cases where this is true, one of the most celebrated being the 9g-9 Theorem.
The goal of this project is to determine length spectrum rigidity for a class of spaces modelled on real projective space. These spaces are constructed using the theory of linear representations of coxeter groups, which will allow us to use tools from linear algebra such as eigenvalues and characteristic polynomials to attempt to establish a result. We will use software, such as Mathematica, to do computations and create visualizations.
Faculty Mentor: Dr. Harrison Bray
Graduate Mentors: Anunoy Chakraborty & Gabe Lumpkin
Pre-requistes: Math 203 and Math 300
Continuing Project Accepting New Members
This project aims to study the dynamics of tumor growth using models based on partial differential equations for each constituent variable and interactions. The emphasis is on developing and analyzing models that describe the evolution of cellular populations (normal tissue, tumor tissue, and extracelluar tissue) for growth, diffusion, and eventually possible treatment inputs. Analytical tools from differential equations, dynamical systems, numerical methods, and numerical continuation methods will be used to investigate steady states, stability, and bifurcations of the solution space. The main goal is understanding of how qualitative behavior of solutions depends on parameters and to develop computational and analytical methods to continue exploring more complicated models.
Faculty Mentors: Dr. Scott Cochran & Dr. Daniel Anderson
Pre-requisites: Math 203, 213, 214. Experience in python and/ or other programming languages.
To start, we will demonstrate how mortality dynamics can be visualized using the “Lexis diagram”. This will be done for a list of chosen countries of interest. We will also apply actuarial mathematics, in particular life contingencies, to this.
For example, a requirement for a stable population is that l* x+1 = (px ) ( l*x) for all ages x. Here, px comes from an actuarial table and l*x is a real-world population count. This will not hold in reality of course, but it is a valuable reference point.
Further goals that we will have:
- Build the bridge from Lexis diagrams to population pyramids
- Extend to more detailed analyses; such as (1) age-specific contributions to higher life expectancy; (2) rates of mortality improvement for specific causes of death,
- Search for and analyze any trends that we find in working with the diagrams and pyramids
- Develop a measure of the degree of population stability inherent in a population pyramid. To my knowledge, no such measure yet exists.
Faculty Mentor: Dr. Douglas Eckley
Pre-requisites: Math 352
In this project, we develop rigorous techniques and apply them to understand various models of computation. We will investigate real and theoretical devices, as well as heuristic algorithms, for understanding models such as reversible computing, quantum computing, cellular automata, probabilistic, and potentially analog computation.
Faculty Mentor: Dr. Michael Jarret
Graduate Mentor: Anthony Pizzimenti
Pre–requisites: Linear algebra, some combinatorics and/or graph theory would be nice
Continuing Project Accepting New Members
Probabilistic, time-dependent models can characterize the variability inherent in antibody response post infection or vaccination for diseases such as the flu or COVID-19. Current models have been fit to data from different demographic groups to identify differences in antibody response, but have yet to be analyzed rigorously for structural identifiability (i.e., parameters can be uniquely identified from “perfect” data with no noise).
The goal of this project is to perform a parameter identifiability study on models for time-dependent antibody response. Resulting updated models will then be fit to experimental data from a long COVID-19 cohort. Time permitting, students will use bootstrapping or other data-driven approaches to identify minimal sample sizes for demographic sub-model trustworthiness.
Faculty Mentor: Dr. Rayanne Luke
Pre-requisites: basic programming experience; some knowledge of probability preferred
If you pick a number between 0 and 1 at random and look at its first 1,000,000,000 base-10 digits, approximately 100,000,000 of them will be 6s. If you instead look at its continued fraction expansion, approximately 29,747,343 of the digits will be 6s. The underlying digit frequency was figured out by Gauss and can be calculated from the probability distribution 1/(ln 2) 1/(1+x). We will adjust the continued fraction algorithm, run extensive computer computations to approximate the Taylor series for the new probability distribution, and then search for a simple description like the one Gauss found – or prove that they don’t exist by tracking how precise our estimates are and therefore ruling out specific types of expressions.
Faculty Mentor: Dr. Anton Lukyanenko
Graduate Mentor: Nicole Savir
Current Project, Not Accepting New Students
Continued fractions are a way to represent numbers as $a_0+\cfrac{\pm 1}{a_1+\cfrac{\pm 1}{a_2+\dots}}$, where the $a_i$ are positive integers. They appear in various areas of mathematics, from number theory to dynamical systems to geometry. We can use different rules or functions to “generate” different types of continued fraction. This project will explore geometric descriptions of the “nearest integer” continued fractions, which rounds to the nearest whole number at every step. For example, $\dfrac{5}{3} = 2 \text{ – } \dfrac{1}{3}$ and $\dfrac{8}{5} = 1 + \dfrac{3}{5} = 1 + \dfrac{1}{2 \text{ – } \dfrac{1}{3}}$.
The continued fraction expansions provide a nice description of paths on the geometric surfaces. The geometric properties of these pictures help us to describe patterns in the continued fraction expansions, and the continued fraction expansions provide a compact description of the geometry. This project will likely explore modular surfaces, but there are other directions based on student interest.
Faculty Mentor: Dr. Claire Merriman
Graduate Mentor: Tim Banks
Current Project, Not Accepting New Students
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 & Morgan Shuman
Continuing Project Accepting New Members
The Bergman projection is a very important operator on spaces of analytic functions in complex analysis. It takes an arbitrary square-integrable function (which can be quite rough) and projects it onto a very nice subspace which consists of analytic (complex differentiable) functions. This project will involve 3 main components, perhaps spread over multiple semesters:
Faculty Mentor: Dr. Nathan Wagner
Pre-requisites: Math 315 and Math 411