Spectral statistics of permutation matrices.

We compute the mean two-point spectral form factor and the spectral number variance for permutation matrices of large order. The two-point correlation function is expressed in terms of generalized divisor functions, which are frequently discussed in number theory. Using classical results from number theory and casting them in a convenient form, we derive expressions which include the leading and next to leading terms in the asymptotic expansion, thus providing a new point of view on the subject, and improving some known results.

Topological resonances in scattering on networks (graphs).

We report on a hitherto unnoticed type of resonances occurring in scattering from networks (quantum graphs) which are due to the complex connectivity of the graph-its topology. We consider generic open graphs and show that any cycle leads to narrow resonances which do not fit in any of the prominent paradigms for narrow resonances (classical barriers, localization due to disorder, chaotic scattering). We call these resonances "topological" to emphasize their origin in the nontrivial connectivity.

Studying post depositional damage on Acheulian bifaces using 3-D scanning.

In this study, we explore post-depositional damage observed on Acheulian bifacial tools by comparing two assemblages: a collection of archaeological handaxes which shows pronounced damage marks associated with high energy water accumulation system, and an experimental assemblage that was rolled and battered in a controlled simulation experiment. Scanning the two assemblages with a precise 3-D optical scanner and subjecting the measured surfaces to the same mathematical analysis enabled the development of quantitative measures assessing and comparing the degree of damage observed on archaeological and experimental tools. The method presented here enables the definition of morphological patterns typically resulting from battering and different from intentional controlled knapping.

Can one count the shape of a drum?

Sequences of nodal counts store information on the geometry (metric) of the domain where the wave equation is considered. To demonstrate this statement, we consider the eigenfunctions of the Laplace-Beltrami operator on surfaces of revolution. Arranging the wave functions by increasing values of the eigenvalues, and counting the number of their nodal domains, we obtain the nodal sequence whose properties we study.