Spring 2022

Covid plan The lab as a whole will operate in a hybrid capacity this spring: access to the physical space and resources will be available, but we will also make remote participation possible in all full-lab activities (orientation, mid-semester meeting, final day activities, and seminar). Individual projects will run based on the mentor’s preferences, as listed below.

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 Math213 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.

Faculty Mentor: Professor Daniel Anderson

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

Participation: In-person/online hybrid, as needed.

Faculty Mentor: Professor Ben Schweinhart

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

Participation: In-person/online hybrid, as needed.

Faculty Mentor: Professor Brent Gorbutt

Description: We’ll be trying to understand the relationships between binomial and multinomial coefficients in equations such as Vandermonde’s identity, the Pfaff-Saalschutz equation, and a recent identity found by Goldin and Gorbutt. The goal is to find a set of parameters that describes a “family” of such formulas. The primary tool is the proof of the formula found by Goldin-Gorbutt that uses a method called “bike lock moves” on sequences which are counted by the binomial and multinomial coefficients in the previously mentioned formulas.

I would expect at least be able to find additional identities. The (maybe tenuous) link to geometry is that the formula found by Goldin and Gorbutt was used to find a formula for structure constants in the equivariant cohomology of Peterson varieties.

Background/Prerequisites: Math 125, really just some familiarity with binomial coefficients. Some programming experience could be helpful.  

Difficulty level: Challenging 

Participation: In-person/online hybrid, as needed.

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

Participation: In-person/online hybrid, as needed.

Faculty Mentor: Professor Valeriu Soltan

Description: A well-known result of convex and discrete geometry says that a convex polygon P in the plane allows a tiling of the plane by translates of P is and only if P is either a parallelogram or a centrally symmetric hexagon. The project goal is to solve the following new problem: Describe all convex polygons P which allow a locally finite tiling of the plane by homothetic copies of P, that is by polygons of the form x + t P, where x is a point and t = t(x) is a nonzero scalar.

Background/Prerequisite: Taste of geometry

Difficulty level: Challenging

Participation: In-person/online hybrid, as needed.

Faculty Mentor: Professor Gabriela Bulancea

Description: The Dirichlet problem asks for a harmonic function on a domain D with prescribed values on the boundary of D. The Dirichlet problem plays an important role in mathematics, physics, and engineering. In this project, we will study the solutions to the Dirichlet problem on the unit disk and on other domains in the plane in the case when the given boundary function is polynomial or rational. We will use properties of harmonic and complex analytic functions to find the solutions in these particular cases.

Background/Prerequisites: Math 213, Math 203 (Math 322 preferred), basic knowledge of complex numbers

Difficulty level: Challenging

Participation: In-person/online hybrid, as needed.

Faculty Mentor: Professor Matt Holzer

Description: We will study the evolution of epidemics on networks. Our goal is to understand how the topology of the network influences the speed at which epidemics spread through the network. We will use the theory of graphons — nonlocal operators that approximate graph Laplacians — to hopefully analyze these problems in a rigorous fashion and to help make predictions for the dynamics.

Background/Prerequisite: Math 203 and 214.

Difficulty level: Moderately Difficult

Participation: In-person.

Faculty Mentor: Professor Harrison Bray

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

Participation: In-person/online hybrid, as needed.