Rice University is currently receiving funding from the National Science Foundation (NSF) through a grant for

At Rice, we have used this funding in part to organize research groups of

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

Website: http://math.rice.edu/~dcole/VIGRE

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:

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.