Geometric Calculus of Variations PFUG

Rice University is currently receiving funding from the National Science Foundation (NSF) through a grant for Vertical Integration of Research and Education in the Mathematical Sciences, or VIGRE for short. The idea behind VIGRE is to bring together mathematics and potential mathematicians from all stages of their educations and careers to study and to do research on mathematics as a group. Our philosophy is that everyone, from the youngest undergraduates to the oldest and wisest full professors, can contribute to the mathematics community and benefit from each other's contributions.

At Rice, we have used this funding in part to organize research groups of postdoctoral fellows, faculty members, undergraduate students, and graduate students, called PFUGs (and pronounced "fugue," like the musical composition) to work on specific open problems in pure mathematics, applied mathematics, and statistics.

The Geometric Calculus of Variations group is one of the PFUGs working on pure mathematics problems. We study problems of minimization in geometry: we look for interesting quantitative measures, called functionals, of the properties of some class of geometric objects, and then search for the objects which minimize those functionals.

The Geometric Calculus of Variations PFUG has been in continuous operation since January 2005, and typically has ten to thirteen participants from all four strata of mathematicians and potential mathematicians. This PFUG places a great deal of emphasis on its undergraduate participants, traditionally giving them the role of spearheading our research projects.

Meeting Time: TTh 9:40 AM -- 10:40 AM
Location: Herman Brown 423

PFUG Leader: Dr. Daniel R. Cole (postdoctoral fellow)
Email: dcole at rice dot edu
Office: Herman Brown 454
Office Hours: T 3:30 PM -- 4:30 PM, WTh 2:30 PM -- 3:30 PM, or by appointment

Faculty Members: Professor Robert Hardt, Professor Michael Wolf
Graduate Student Members (Fall 2006): Barbara Chervenka Paier, Casey Douglas, Ryan Dunning
Undergraduate Members (Fall 2006): Max Glick, Carl Hammarsten, Matthew Patterson, James Winkler

Current Research Project: Edge Length Minimizing Polyhedra
For every polyhedron in three-dimensional space, we can measure the volume of that polyhedron and the total length of its edges. For example, a cube with unit length edges has volume 1 and total edge length 12. A natural question to ask is the following: if a polyhedron has volume 1, how small can its total edge length be, and does there exist a polyhedron which achieves this minimum (and if so, what does it look like)? This deceptively simple question is called Melzak's Problem, and it is the current focus of our PFUG. We are currently developing techniques to study how edge length and volume change as we vary the shape of a polyhedron, modelling our ideas off of the theory of moduli spaces. Our hope is that these techniques will allow to prove some general theorems about what a minimizer of total edge length must look like if it exists.