Abstract:

It has long been known that geometric symmetry yields extreme spectral properties of the underlying Laplacian; projective planes and symmetric designs are immediate examples. We list other infinite families with such properties. The focus becomes the least nonzero eigenvalue of the Laplacian which measures connectivity of the configuration. Primarily the isoperimetric inequalities of Cheeger point us in the direction of how such structures could be constructed. A main focus are pseudo-random sequences and a study of the Goethals-Seidel appoach to constructing Hadamard designs through such sequences. In this respect we highlight partial results and mounting conjectural support. Related applications to spectral clustering in machine learning, optimal statistical design, and construction of nonlinear codes will be described.

427 Thackeray Hall