I am happy to have small teams working on proects together, reading and talking about papers, trying different approaches, ect.

In terms of project choices, you should work on something related to your research or planned research area. I can help you focus in on some good papers to read and maybe some questions to think about or experiment with. Please talk to me.

If nothing immediately springs to mind, here are some comparatively well-thought-out projects I think might lead to papers. These are areas my students, my colleagues and I are currently working in.

• ChengCheng Hu: There has been interest lately in computer graphics in producing high-quality tetrahedral meshes by placing the vertices nicely and then taking the Delaunay triangulation. One method centers on a particular mesh-smoothing technique. While it works well in practice, there are no theoretical results connecting the method to any "nice" properties of the triangulation. In 2D, we could instead explictly place vertices so as to maximize the minimum angle, which should make lovely triangle meshes. We wrote a paper many years ago describing how to do this using LP-type algorithms.
• Fatemeh Abbasinejad, Shubho Sengupta: There is a construction, due to Erickson, of a set of points on a smooth surface in R³ with a Delaunay triangulation of size O(n^^3/2). Make a conjecture about how this example extends to higher dimensions, and prove it, or test it experimentally.
• Paul Mach, Shengyin Gu: We will study an interesting structure called the mixed complex, used for modeling molecules. Point location in Delaunay triangulations is well studied, in Voronoi diagrams less so. What can we prove about point location in the mixed complex? What would be the best approach in practice? One good approach might be jump and walk.
• Ranjan Pal, Wei Yu: Nearest-neighbors in high-dimensional space. One now-classic algorithm is Locality-sensitive hashing. From this page, I would recommend the book chapter and the GIM'99 paper.
• Anna Tikhonova, Michael Ogawa: Large graph layout. Some papers that look substantial and interesting:
  A multi-dimensional approach to force-directed layouts of large graphs
  FADE: Graph Drawing, Clustering, and Visual Abstraction
  A fast multi-scale method for drawing large graphs
• Michael Byrd: Kinetic algorithms. Reading?
• David Haws: Minimum-weight triangulation approximation.
• Dan Alcantara: Path planning in animation?
• Kenny Huang:??
• Eddie Kim:??

• Find examples of four-dimensional polytopes, or three-dimensional spatial decompositions, with many faces and edges but few vertices and cells. See Jeff Erickson's page on intricate polytopes. This is a hard problem, so a reasonable project would be to study and write a survey paper on the results know so far.