Spectral Gap for Weil–Petersson Random Surfaces with Cusps

Spectral Asymptotics of Laplace Operators on Surfaces with Cusps

K3 Surfaces with Nine Cusps

By a K3-surface with nine cusps I mean a surface with nine isolated double points A 2 , but otherwise smooth, such that its minimal desingularisation is a K3-surface. It is shown, that such a surface admits a cyclic triple cover branched precisely over the cusps. This parallels the theorem of Nikulin, that a K3-surface with 16 nodes is a Kummer quotient of a complex torus.

On Enriques Surfaces with Four Cusps

We study Enriques surfaces with four disjoint A2-configurations. In particular, we construct open Enriques surfaces with fundamental groups (Z/3Z) × Z/2Z and Z/6Z, completing the picture of the A2-case from [10]. We also construct an explicit Gorenstein Q-homology projective plane of singularity type A3 + 3A2, supporting an open case from [7].

Cusps of Picard modular surfaces

We determine the number of cusps of minimal Picard modular surfaces. The proof also counts cusps of other Picard modular surfaces of arithmetic interest. Consequently, for each N > 0 there are finitely many commensurability classes of nonuniform arithmetic lattices in SU(2, 1) that contain an N -cusped surface. We also discuss a higher-rank analogue.

The Spectral Gap of Random Graphs with Given Expected Degrees

We investigate the Laplacian eigenvalues of a random graphG(n,d) with a given expected degree distribution d. The main result is that w.h.p. G(n,d) has a large subgraph core(G(n,d)) such that the spectral gap of the normalized Laplacian of core(G(n,d)) is ≥ 1− c0d̄ min with high probability; here c0 > 0 is a constant, and d̄min signifies the minimum expected degree. The result in particular appli...