I will adapt a recent paper with C.B. Muratov (arXiv:1403.7808) using the help of Maple.  The paper concerns a nonlocal isoperimetric poblem: Find the 2-D region R of fixed area m which minimizes the energy E(R) := P(R) + \int_R \int_R |x-y|^-alpha dx dy, where P is the perimeter and alpha>0 is fixed.  In the paper we find quantitative regions in the parameter space (m and alpha) for which minimizers fail to exist, are convex, and are balls if they exist.  My first goal will be to translate the existing calculations of the paper to Maple code.  If time permits, I will try to approximate shapes of minimizers by searching over regions with polygonal boundaries. 

Note:  I may also explore Hofstadter's conjectures on the side.