We study the asymptotic behavior of the discrete analogue of the holomorphic map $z^a$. The analysis is based on the use of the Riemann–Hilbert approach. Specifically, using the Deift–Zhou nonlinear steepest descent method we prove the asymptotic formulas which were conjectured in 2000.

The dimension datum of a closed subgroup of a compact Lie group is the sequence of invariant dimensions of irreducible representations by restriction. In this article we classify connected closed subgroups of a given compact Lie group with equal dimension data or linearly dependent dimension data. We also study the equalities and linear relations among dimension data of closed subgroups of a unitary group which act irreducibly on the natural representation.

We generalize Bogomolov’s inequality for Higgs sheaves and the Bogomolov– Miyaoka–Yau inequality in positive characteristic to the logarithmic case. We also generalize Shepherd-Barron’s results on Bogomolov’s inequality on surfaces of special type from rank $2$ to the higher-rank case. We use these results to show some examples of smooth nonconnected curves on smooth rational surfaces that cannot be lifted modulo $p^2$. These examples contradict some claims by Xie.

We completely characterize the simultaneous membership in the Schatten ideals $S_p$, $0<p<\infty$ of the Hankel operators $H_f$ and $H_{\overline{f}}$ on the Bergman space, in terms of the behavior of a local mean oscillation function, proving a conjecture of Kehe Zhu from 1991.

We prove an optimal exponential lower bound on the width of the resonance associated to the first eigenvalue of the cavity for a general Helmholtz resonator with straight neck, in any dimension.