During Fall 2021 MEGL ran a 6 project program along with aiding 3 honor thesis projects with 33 participants (faculty, graduate, and undergraduate students). The six research/visualization groups are titled:
- The stability of icebergs and other floating objects
- Random 3D Polyforms
- Combinatorics of Cohomology Rings of Peterson Varieties
- Peterson Schubert Calculus
- Harmonic Functions and the Dirichlet Problem with Polynomial or Rational Data on Certain Domains in the Plane
- Networks, Graphons and Epidemics
The three honor thesis projects are titled:
- Speed and stability of traveling waves in population growth models
- Persistent homology
- Vertex algebras
In addition to these projects, MEGL ran a public engagement group which we refer to below as Outreach.
The research/visualization groups engaged in experimental explorations involving faculty, graduate students, and undergraduates. Teams met weekly to conduct experiments generating data, to make conjectures from data, and to work on theory resulting from conjectures. The outreach group involved faculty, graduate students, and undergraduates to develop and implement activities for elementary and high school students that were presented at local schools and public libraries.
We concluded with an end of term symposium and a poster session, sharing progress and results of our research. To learn more, see below:
Faculty Mentor: Professor Daniel Anderson
Graduate Research Assistant: Will Howard
Undergraduate Research Assistants:
- Brandon Barreto
- Joshua Calvano
- Lujain Nsair
We explore the stability of floating objects through mathematical modeling and experimentation. Our models are based on standard ideas of center of gravity, center of buoyancy, and Archimedes’ Principle. We investigate a variety of floating shapes with two-dimensional cross sections and identify analytically and/or computationally a potential energy landscape that helps identify stable and unstable floating orientations. We compare our analyses and computations to experiments on floating objects designed and created through 3D printing. In addition to our results, we provide code for testing the floating configurations for new shapes, as well as giving details of the methods for 3D printing the objects. The paper includes conjectures and open problems for further study.
Faculty Mentor: Faculty Mentor: Professor Ben Schweinhart and Professor Erika Roldan
Graduate Research Assistant: Aleyah Dawkins
Undergraduate Research Assistants:
- Dhruv Gramopadhye
- James Serrano
- Khoi Tran
Polyforms are a class of shapes constructed by joining together identical polytopes connected by at least one of their faces. Specifically, This project studies random polyforms composed of cubes strongly connected by their two dimensional square faces. We develop and utilize a series of algorithms for generation, shuffling, enumeration, and cataloging both two-dimensional polyominoes and three-dimensional polyforms. Our work concurs with previous research done in the two dimensional case, finding shuffle time to be between quadratic and cubic time. We conjecture significant speedups in the shuffle algorithm by optimizing connectedness checks, which currently operate in linear time.
Faculty Mentor: Faculty Mentor: Professor Brent Gorbutt
Graduate Research Assistant: Martha Hartt
Undergraduate Research Assistants:
- Aidan Donahue
- Ryan Simmons
- Ziqi Zhan
Our project looks for the relationships between binomial and multinomial coefficients in equations such as the combinatorial formula found by Goldin and Gorbutt that generalizes Vandermonde’s Identity. We attempt to expand on Goldin/Gorbutt’s work by finding more identities for binomial and multinomial coefficients in equations. To do this, we use a method called bike lock moves on sequences which are counted by the binomial and multinomial coefficients.
Faculty Mentor: Faculty Mentor: Professor Rebecca Goldin
Graduate Research Assistant: Quincy Frias
Undergraduate Research Assistants:
- Swan Klein
- Connor Mooney
Our goal was to verify a conjecture about the decomposition of the restriction of Schubert classes associated with transpositions to the Peterson variety into a linear combination of Peterson classes. Using a corollary to Billey’s formula, we reduced the conjecture to a more concise combinatorial question about counting reduced words for transpositions embedded into long words. We uncovered an elegant visual framework for understanding these combinatorial questions and proved our conjecture in a specific subcase. Future work will involve proving the remaining cases of the conjecture and extending our combinatorial strategy to as many types of Schubert classes as possible.
Faculty Mentor: Professor Gabriela Bulancea
Graduate Research Assistant: Abigail Friedman
Undergraduate Research Assistants:
- George Andrews
- Justin Cox
- Ryan Nguyen
The Dirichlet problem is concerned with finding harmonic functions on certain domains that satisfy given boundary data. It has applications in the physics of heat flow, electrostatics, and other fields. Our project focuses on domains Ω ⊂ R2 with polynomial or rational data. Our work consists of an attempt to extend the linear algebraic approach with Fischer’s lemma to special cases of rational boundary data and writing the Poisson integral as a finite sum at the center of the unit disk with polynomial data.
Faculty Mentor: Professor Matt Holzer
Graduate Research Assistant: John Kent
Undergraduate Research Assistants:
- Marcelo Montan ̃ez-Collado
- Raina Joy Saha
- Caleb Schear
The spread of infection can be modeled by a graph where each vertex is an SIR model and each edge represents transportation between population centers as governed by a global diffusion parameter. More transportation between vertices intuitively would lead to local infected populations growing faster. We study the relationship between the diffusion parameter and how long it takes for infection in a given vertex to reach a critical threshold, identify special cases of interest, and derive some conditions for observing unintuitive behavior.
Faculty Mentor: Professor Harry Bray
Graduate Assistant:
- Aleyah Dawkins
- Martha Hartt
Undergraduate Assistants:
- Aidan Donahue
- Lujain Nsair
- Joanna Ro
This semester, we reached 1,247 students. We were able to run 39 activities, reaching students of all ages from 10 different organizations. We focused on the activities “You Can Count on Monsters”, “Really Big Numbers”, and our newest activity “Irrational Thinking”. Check out our website for more information on MEGL outreach activities.
