Project: Topology of algebraic varieties
The goal of this project is to use persistent homology to study the topology of random complex algebraic varieties, geometric figures described as the solution sets of system of polynomial equations. Several students will work together to create a suite of software packages to numerically find solutions of polynomial equations, compute the persistent features of the resulting cloud of points, and visualize the results, using a virtual reality platform such as Oculus Rift.
Project: Immersive visualization of data sets in the 3-sphere
In this project, a team of students will develop real-time 3D graphics programs for visualization of point-cloud data sets in the 3-dimensional sphere, targeting both virtual reality platforms (such as Google Cardboard and Oculus Rift) and desktop computers. The primary focus will be on datasets related to the geometry and combinatorics of curves on compact hyperbolic surfaces. Students will have the chance to learn about hyperbolic geometry and hyperbolic surfaces—the theoretical background underlying the data sets in this visualization project—in parallel with their software development efforts.
For this project, we are seeking applicants with some computer programming experience. Experience with 3D graphics is a plus. No background in hyperbolic geometry is expected or required.