Matt Begue, Dan Kelleher, Gabe Khan, and Diwakar Raisingh
The Strichartz hexacarpet is a fractal which can be obtained by repeated barycentric subdivisions of a triangle. This group examined properties of graph approximations and looked at analogous fractals obtained by subdivisions of higher dimensional triangles — n-simplexes. They studied properties of random walks on these fractals, obtaining heat kernel estimates and resistance factors. See also Random Walks on Barycentric Subdivisions and the Strichartz Hexacarpet Experiment. Math. Volume 21, Issue 4 (2012), 402-417.
Random Walks on Barycentric Subdivisions and the Strichartz Hexacarpet
Even in low dimensional examples, basic object in differential geometry, such has tangent spaces and bundles, can only be perceived in at least 4-dimensions. This makes intuitive understanding of these objects difficult. This group created some tools for understanding these objects a little better.
In these “tangent space visualisers,” we explore the concept of tangent vectors and tangent spaces. Below we have two examples, the figure-8 space (left) and the “theta” space (right). Click on the images to download the applets. The arrows indicate the tangent vectors which span the tangent space at any given point, click on any point in the space to move the arrow.
Christopher Kauffman, Robert Kesler, Amanda Parshall, Evelyn Stamey, and Benjamin Steinhurst
This group first reviewed the spectrum of the Laplacian operator on a general Laakso space before considering modified Hamiltonians for the infinite square well, parabola, and Coulomb potentials. Additionally, they computed the spectrum for the Laplacian and its multiplicities when certain regions of a Laakso space are compressed or stretched and calculated the Casimir force experienced by two uncharged conducting plates by imposing physically relevant boundary conditions and then analytically regularizing the resulting zeta function. Lastly, they derived a general formula for the spectral zeta function and its derivative for Laakso spaces with strict self-similar structure before listing explicit spectral values for some special cases.
See also B. Steinhurst, Dirichlet Forms on Laakso and Barlow-Evans Fractals of Arbitrary Dimension arXiv:0811.1378
AIP | Journal of Mathematical Physics
YMC 2012 at Ohio State University – Won third prize for presentation.
Presented at the 2012 Mt. Holyoke REU Mini-Symposium