Thursday, October 5, 2023
10/05/2023 - 4:00pm
Our intuition regarding waves suggests that the k-th eigenvector of a graph Laplacian L (or any Discrete Schrodinger operator) should exhibit greater fluctuations as k increases. In this context, the "nodal count" is the number of edges on which the eigenvector changes sign. The works of Fiedler (1975) and Berkolaiko (2007) show that the nodal count is bounded between k-1 and k-1+b, where b is the first Betti number of the graph. We establish that these bounds hold for signed graphs as well when considering sign changes accordingly. The “nodal surplus”, the deviation from k-1, is expected to concentrate around b/2. Numerical observations indicate that the distribution of the nodal surplus, across all eigenvectors and different signings of the graph, resembles a Gaussian distribution centered at b/2, regardless of the graph's characteristics. We prove that it is precisely binomial with mean b/2 in the case of operators on complete graphs with sufficiently high potential.
This outcome, among others, stems from a noteworthy relationship. The magnetic perturbations of L are achieved by multiplying the off-diagonal entries of L by phases (in a Hermitian manner), modulo gauge invariance. The eigenvalues of L extend to piece-wise analytic functions of the phases. At non-degenerate critical points, the Morse index is equal to the associated nodal surplus. If time permits, I will draw the line connecting this work to spectral gaps of periodic operators.
10/05/2023 - 4:00pm
In the theory of discrete subgroups of Lie groups, given a length function on the Lie group G, one popular object of study is the asymptotic when R goes to infinity of the number of loxodromic elements with length less R in a discrete subgroup H, and the distribution of the fixed points of these loxodromic elements.
When G is the group of (real) projective transformations and H acts properly discontinuously and cocompactly on a strictly convex domain O of the projective space, Yves Benoist noticed that one can use the Hilbert metric of O, and the associated Hilbert geodesic flow, to estimate the above counting function, for a special length function called the Hilbert length.
An important ingredient in Benoist’s result is the uniform hyperbolicity of the Hilbert geodesic flow, which does not hold when O is not strictly convex.
In this talk we will develop and use instead the theory of Patterson–Sullivan measures for Hilbert geometries O/H that satisfy a mild rank-one assumption, and are not necessarily strictly convex nor compact.
This will yield counting results for H in the cases where the induced Bowen–Margulis measure is finite. (This is joint work with Feng Zhu).