To learn more, click on the project below for a full description.

**Faculty Mentor:** Professor Sean Lawton

**Description: **In fall 2022 we studied families of dynamical systems over finite fields. In particular we looked at an action of the classical groups (with an emphasis on the general linear group) on multivariate polynomials. We focused on describing the orbits in the linear and quadratic cases, as well as finding forms for fixed points, describing maximal orbits, and outlining asymptotic behaviors. In the spring we will continue this work by finishing the quadratic case and examining higher degrees, describing fixed points in full, sharpening bounds on when maximal orbits occur, and fully describing asymptotic behavior. In addition to these extensions of the current work, we aim to look at the “local dynamics” of our system. If time permits we may examine the action of other classical groups in more depth, examine other fields, and look at the behavior in projective space.

This is a continuing project.**Graduate Research Assistant: **Michael Merkle**Research Interns: **

- Holly Miller
- Violet Nguyen

**Faculty Mentor**: Harry Bray

**Description**: We will explore the three constant curvature geometries of surfaces and their tilings via crochet. This project will occur in collaboration with a team in the Geometry Lab at UVA. No experience with crochet or any kind of knitting is necessary. Materials will be provided to students.

Prerequisites: At least one of math 125 or math 114, OR an artistic/creative background and enthusiasm to learn and explore mathematics and geometry. A willingness to learn and do a lot of crochet is a necessary prerequisite.

**Graduate Research Assistant: **Madeline Horton **Research Interns:**

- Nhuphuong Au
- Merold Saffa
- Aidan Self
- Sydney Thu

**Faculty Mentor: **Douglas Eckley

**Description: **This project will study the Polar Method for simulating draws from the standard normal distribution. There are various applications of interest including wine production.

The goal of the project is to bring together several concepts from statistics and analytic geometry in a way that is conducive to visualization. The project can be done entirely either in spreadsheet or in R. Students will be expected to, for example, do research online on the polar method for creating random standard normal variables, apply the polar method in spreadsheet or R, and test the simulated values for normality using D’Agostino’s K-squared test (see https://en.wikipedia.org/wiki/D%27Agostino%27s_K-squared_test for example).

The prerequisites are: 1) some exposure to expectation/variance; 2) some exposure to continuous probability distributions; and 3) some familiarity with spreadsheets or the R statistical package. Familiarity with both spreadsheets and R is helpful but not necessary for students to apply – students need a willingness to learn and work with these technologies. **Research Interns: **

- Aleksei Miles
- Jessica Nguyen
- Salina Tecle
- Ethan Walter

**Outreach Director: **Professor Rosemberg Toala

**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.

**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.

This is a continuing project.

**Graduate Research Assistant: **Shrunal Pothagoni **Research Interns: **

- Dhruv Gramopadhye
- James Serrano
- Khoi Tran

**Faculty Mentor: **Yiannis Loizides

**Description: **We’ll start by getting a feel for foliations (including singular ones) on surfaces. We will draw lots of examples and (perhaps) build 3-d models. We’ll experiment with some mathematical tools for building interesting foliations. Groupoids are objects that generalize groups and that, like groups, can be used to describe symmetries. There is a groupoid one can associate to a foliation on a surface, which encodes some of its properties and partially desingularizes it. We will work out what these groupoids are in some of the examples, formulate and attempt to prove conjectures about them.

Recommended prerequisites: at least one of Math 321 (abstract algebra) or Math 431 (topology). **Graduate Research Assistant: **Quincy Frias**Research Interns: **

- Deven Linthicum
- Gabe Lumpkin

**Faculty Mentors**: Evelyn Sander and Dan Anderson

**Description**: We will be performing modeling, analysis, and experiments to understand the effects of surface tension on 3D printed floating objects. **Graduate Research Assistant: **Patrick Bishop**Research Interns:**

- Aiden Dunlop
- Phap “James” Nguyen
- Max Werkheiser