Links

• Fractals. Any old fractal. Right now, we can't really say anything about differential equations on arbitrary fractals.
• Iterated Function Systems. These are the fractals obtained from iterated contractive mappings. In general, we only understand these through computational approximations by open sets, for which differential equations can be computed as in the non-fractal case. For some very interesting results on the Sierpinski Carpet and the Octagasket, see Stacey's webpage.
• Post-Critically Finite Iterated Function Systems. This essentially means that the ifs is finitely ramified, although the actual definition is a little more involved. There are many abstract analytic results for these fractals. Work by Sabot and Kigami in particular was consulted for my project. There are very few special cases, however, where much more is known than these general results..
• Maximally Connected Post-Criticallly Finite Iterated Function Systems. This was my project. If anyone other than Luke Rogers, Bob Strichartz, and myself is studying these fractals, I would like to know about it.
• The Sierpinski 3-Gasket...
  • ... with an Energy measure. Alex and Naotaka studied this extensively from a computational perspective and have some excellent conjectures about the spectrum of the Kusuoka Laplacian. Their website is not up yet, so far as I know.
  • ... with the standard graph energy with conductances and varied contractive mappings. Anna and Baris did extensive work on the spectrum of the Laplacian for various twisted gaskets. Their work can be found here.
    • ... with constant conductances. A great deal is understood about the Laplacian and a select few other differential equation concepts on these gaskets. Jessie worked on defining and computing a gradient for eigenfunctions of the Laplacian. Zuhair and Edward investigated Schroedinger equations with various potentials, and in particular did a promising study on what exactly x^2 is as a function on SG3. I haven't seen websites for any of these yet.

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