# Project: Finding Topological Chaos

**Graduate supervisor:** Edgar Bering

**Project type:** Semester undergraduate team research

**Description:** Braided twisting of three mixing arms (as in the “taffy pulling” mechanism depicted schematically above) can be used to efficiently mix viscous fluids. This is an example of a two-dimensional mixing process that is *topologically chaotic*, in the sense that it creates exponentially increasing amounts of winding of strands within the medium. This project will focus on creating a computer program to identify whether or not this specific type of topological chaos (known as *pseudo-anosov dynamics*) is present in any given two-dimensional process involving twisting along curves in a surface.

**Prerequisites:** Multivariable calculus and linear algebra. Python programming experience is helpful but not required.

# Project: 3D Printing Surfaces

**Faculty supervisor:** Daniel Groves

**Graduate mentor:** Christopher Perez

**Project type:** Semester undergraduate team research

**Description:** This project is to explore surfaces in three-dimensional space by 3D-printing models of them. The goal is to start by choosing appropriate equations and end with physical models which would be appropriate for use in multivariable calculus classes. This will involve learning about function graphing software, 3D-modeling software and 3D-printing. Time and student interest permitting, we can investigate such things as curvature, ruled surfaces and other things which touch upon ideas from topology, geometry and algebraic geometry.

**Prerequisites:** Multivariable calculus would be ideal, though it would be possible to succeed in this project whilst enrolled in Math 210.

# Project: Deforming Algebraic Geometric Configurations with phcpy in Sage

**Faculty supervisor:** Jan Verschelde

**Graduate mentors:** Nathan Bliss and Jeff Sommars

**Project type:** Semester undergraduate team research

**Description:** The free and open source computer algebra system Sage has Python as its scripting language. The Python package phcpy solves polynomial systems via the polynomial homotopy continuation methods in PHCpack. One of the optional packages in Sage is phc.py, which offers an interface to the executable phc of PHCpack. The goal of the project is to replace phc.py with phcpy. Use cases, such as the circle problem of Appolonius, will guide the development of an object oriented interface to the functions in the phcpy package.

**Prerequisites:** Good knowledge of Python (e.g. MCS 275) and familiarity with computer algebra (e.g. MCS 320).

# Project: Hyperbolic Racquetball

**Faculty supervisor:** David Dumas

**Project type:** Semester undergraduate team research

**Description:** We will build a virtual reality game to demonstrate features of hyperbolic geometry (to students, researchers, or anyone else) using the Oculus Rift VR headset and touch controllers. This game will give the user a “first person experience” of hyperbolic geometry in which they play racquetball with infinitely many copies of themselves, all arranged according to a polyhedral tiling of 3-dimensional hyperbolic space.

**Prerequisites:** Applicants should have some experience with 3D graphics programming.