Fall 2022

To learn more, click on the project below for a full description. Project difficulty is out of three options: challenging, moderately difficult, very difficult. Completion of Math 213 is required for all projects, even if not specifically listed.

In choosing a project, students should do some preliminary research (e.g. read up on relevant topics on Wikipedia) to gain a basic understanding of the project description and whether they are interested in spending a semester learning about it in-depth. The application should demonstrate this understanding and include an explanation of why your choice of project is a good fit for your background and interests.

Most of our positions have now been filled. However, we are still looking to fill a few roles.

Faculty Mentor: Professor Daniel Anderson and Professor Evelyn Sander

Description: Computing quantities like the center of mass, the center of buoyancy, and another quantity called the metacenter.  Archimedes’ Principle plays a role here.  This project will involve predicting stability and floating orientations of shapes with simple and complex geometries using numerical and analytical approaches  Some experimentation will likely be involved as well.

Background/Prerequisite: Math 213. Some numerical experience (or willingness to learn) will be useful.

Difficulty level: Challenging

Faculty Mentor: Professor Ben Schweinhart and Professor Erika Roldan

Description: This project will study a class of three-dimensional shapes called random polyforms in computational experiments. A polyform is a collection of cubes that is strongly connected in the sense that every pair of cubes is connected by a path of cubes sharing square two-dimensional faces. Random polyforms are important examples of random polymers in statistical physics.

The students will learn how to use an important sampling technique called the Metropolis-Hastings Algorithm (a special case of Markov Chain Monte Carlo) to generate random polyforms, and they will perform data analysis to study the resulting structures. This will involve methods from topological and geometric data analysis (TGDA). Random polyforms have intricate geometry, and properties of interest will include the size of the perimeter, the radius, and the number of holes (i.e. the ranks of the homology groups).

The project builds on work of David Aristoff (Colorado State) and Erika Roldán (TU Munich/EPFL) to study random two-dimensional polyforms (called polyominoes), and will be conducted in collaboration with Dr. Roldán.

Background/Prerequisite: Programming experience, or a strong motivation to learn programming.

Difficulty level: Moderately Difficult

Faculty Mentor: Professor Rebecca Goldin

Description: We will be looking at a ring map associated to the inclusion of the Peterson variety into the flag manifold, mainly the induced restriction in the S1-equivariant cohomology ring. We will study the restriction of geometrically represented classes called Schubert classes to the Peterson variety.

Background/Prerequisite: Completion of Math 321, Abstract Algebra. Combinatorics.

Difficulty level: Challenging

Faculty Mentor: Professor Rebecca Rebhuhn-glanz and Professor Hugh Geller

Description: Given a stimuli space, one can look at all possible neuron firings to create binary strings known as neural codes. We study neural codes using an algebraic object called a neural ideal. In this project, we will construct free resolutions of neural ideals, homological tools that may give us information about the original neural code.

Background/Prerequisite: Math 300 required, Math 321 and programming experience recommended

Difficulty Level: Moderately Difficult

Faculty Mentor: Professor Matthew Holzer

Description: Geometric desingularization is a method to analyze non hyperbolic fixed points. It involves “blowing up” the non hyperbolic point to a higher dimensional manifold where hyperbolicity is regained. This project will apply these methods to understand the dynamics of singularly perturbed differential equations in some applied problems.

Background/Prerequisite: Math 214

Difficulty Level: Moderately Difficult

Faculty Mentor: Professor Sean Lawton

Description: We will be exploring the idea of a “moduli space”; that is, a “space of spaces”. We will begin with toy examples; literally explored with toys. Then students will creatively make their own example moduli spaces with computer experimentation/visualization, 3D printing, or physical manipulatives. We will then start to learn about some of the mathematics behind the simplest of the examples and build from that point of common understanding. Once the team constructs enough examples, questions about the examples will naturally arise. The team will gravitate towards one or a few of said questions and then we will start to try to answer those questions or gather compelling evidence to justify a conjecture.

Background/Prerequisite: Math 300 with a B or better

Difficulty Level: Challenging

Faculty Mentor: Professor Emmanuel Fleurantin and Professor Chris Jones

Description: Tipping from one apparently stable state of a system to another can be activated by a too rapid rate of change in some underlying parameter. The presence of noise can also force tipping to occur and, moreover, these two effects can work together. The rate change will come from either the background warming of the climate or some sudden shift in conditions triggered by climate change. Tipping between two saddle equilibria is associated with a heteroclinic connection between these saddle points. Particular challenges emerge in the complex systems that arise in climate where the basin boundary (tipping threshold) becomes intricate, such as the stable manifold of a periodic orbit. In this project, we will seek to capture the connecting orbit by deliberately computing invariant sets. This will involve rigorous computations of invariant sets and their visualization. There is the possibility of enrolling in an Udemy course to get more familiar with specific software for computational purposes. We will develop the project in models that arise in areas such as the carbon cycle, permafrost melting, peat fires and hurricanes.

Background/Prerequisite: Math 203

Difficulty Level: Moderately Difficult

Faculty Mentor: TBD

Description: MEGL offers outreach activities to the community every semester. To fulfill our mission of outreach, we need your help. Participants will deliver, refine and help develop mathematical outreach activities which inspire a passion for mathematics. Check out our website for examples of MEGL outreach activities.

Note: This is a paid outreach opportunity, paying approximately $13/hour.

Background/Prerequisites: A passion for mathematics

Difficulty level: Challenging