Dehn fillings for relatively hyperbolic groups generalize the topological Dehn surgery on a noncompact hyperbolic $3$-manifold such as hyperbolic knot complements. We prove a rigidity result saying that if two nonelementary relatively hyperbolic groups without certain splittings have sufficiently many isomorphic Dehn fillings, then these groups are in fact isomorphic. Our main application is a solution to the isomorphism problem in the class of nonelementary relatively hyperbolic groups with residually finite parabolic groups and with no suitable splittings.

In this article, we investigate the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according to the circular beta ensemble (C$\beta$E). More precisely, assuming that $X_n$ is this characteristic polynomial and $\mathbb{U}$ is the unit circle, we prove that

$${sup\hspace{0.17em}}_{z\in \mathbb{U}}\Re log{X}_n\left(z\right)=\sqrt{\frac{2}{\beta }}(logn-\frac{3}{4}loglogn+\mathcal{O}(1\left)\right),$$ as well as an analogous statement for the imaginary part. The notation $\mathcal{O}\left(1\right)$ means that the corresponding family of random variables, indexed by $n$, is tight. This answers a conjecture of Fyodorov, Hiary, and Keating, originally formulated for the $\beta =2$ case, which corresponds to the circular unitary ensemble (CUE) field.

We describe the closure of the strata of Abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne–Mumford moduli space of stable curves with marked points. We provide an explicit characterization of pointed stable differentials in the boundary of the closure, both a complex analytic proof and a flat geometric proof for smoothing the boundary differentials, and numerous examples. The main new ingredient in our description is a global residue condition arising from a full order on the dual graph of a stable curve.