Higgs bundles, pseudo-hyperbolic geometry and maximal representations

These notes are an extended version of a talk given by the author in the seminar "Theorie Spectrale et Geometrie" at the Institut Fourier in No- vember 2016. We present here some aspects of a work in collaboration with B. Collier and N. Tholozan (arXiv:1702.08799). We describe how Higgs bundle theory and pseudo-hyperbolic geometry interfere in the study of maximal representations into Hermitian Lie groups of rank 2

The n-th prime asymptotically

A new derivation of the classic asymptotic expansion of the n-th prime is presented. A fast algorithm for the computation of its terms is also given, which will be an improvement of that by Salvy (1994). Realistic bounds for the error with \li^{-1}(n), after having retained the first m terms, for $1\le m\le 11$, are given. Finally, assuming the Riemann Hypothesis, we give estimations of the best possible $r_3$ such that, for $n\ge r_3$, we have $p_n> s_3(n)$ where $s_3(n)$ is the sum of the first four terms of the asymptotic expansion

Maximal surfaces in the pseudo-hyperbolic space

The pseudo-hyperbolic space $\mathbb{H}^{p,q}$ is a pseudo-Riemannian analogue of the classical hyperbolic space. In this talk, I will explain how equivariant maximal surfaces in $\mathbb{H}^{2,n}$ are related to harmonic maps into some non-compact symmetric space (so in particular to Higgs bundles). This link provides a powerful tool to study maximal representations in rank 2 Hermitian Lie groups. This is a joint work with Brian Collier and Nicolas Tholozan.Non UBCUnreviewedAuthor affiliation: USC DornsifeResearche