Contributors

Tyler Reese, Dylan Yott, Antoni Brzoska, and Daniel Kelleher

Overview

The brain is one of the most studied and highly complex systems in the biological world. This group analyzed the brain of the nematode Caenorhabditis elegans. They used eigenvalues and eigenvectors of the Laplacian matrix — a matrix representation — of the neural network of the nematode brain with an eye on indicators of self-similarity.

The following is a list of the programs this group used in their research. Most of them were written in Matlab.

• Energy.m
• Energy2.m
• Harmonic.m
• RandomGraph.m
• RandomPick.m
• Resistance.m
• Trees.m
• TRewire.m
• Varz.m

Publications

arXiv.org
PLoS ONE

Contributors

Matt Begue, Dan Kelleher, Gabe Khan, and Diwakar Raisingh

Overview

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.

Presentation

Random Walks on Barycentric Subdivisions and the Strichartz Hexacarpet

Contributors

Christopher Kauffman, Robert Kesler, Amanda Parshall, Evelyn Stamey, and Benjamin Steinhurst

Overview

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

Publications

arXiv.org
AIP | Journal of Mathematical Physics