Generalized and Higher Dimensional Apollonian Packings

In this thesis, we show that circle, sphere, and higher dimensional sphere packings may be realized as subsets of the boundary of hyperbolic space, subject to certain symmetry conditions based on a discrete group of motions of the hyperbolic space. This leads to developing and applying counting methods which admit rigorous upper and lower bounds on the Hausdorff (or Besikovitch) dimension of the residual set of several generalized Apollonian circle packings. We find that this dimension (which also coincides with the critical exponent of a zeta-type function) of each packing is strictly greater than that of the Apollonian packing, supporting the unsolved conjecture that, among the many possible disk tilings of the plane, the Apollonian packing has the smallest possible residual set dimension. The obtained rigorous bounds are also consistent with the heuristic estimates calculated herein

A Note on Fourier Eigenfunctions in Four Dimensions

In this note, we exhibit a weakly holomorphic modular form for use in constructing a Fourier eigenfunction in four dimensions. Such auxiliary functions may be of use to the D4 checkerboard lattice and the four dimensional sphere packing problem.Comment: 5 pages, comments welcom

Charter Schools as Nation Builders: Democracy Prep and Civic Education

This policy brief is the first in a series of in-depth case studies exploring how top-performing charter schools have incorporated civic learning in their school curriculum and school culture. For more information about AEI's Program on American Citizenship