Johannes Linn: Bounds for Kloosterman Sums for $\mathrm{GL}_n$

Classical Kloosterman sums defined by $S(m,n;c):=\sum_{x\in (\mathbb{Z}/c\mathbb{Z})^*}e\Big(\frac{mx+n\overline{x}}{c}\Big)$ for $m,n\in\mathbb{Z}$ and $c\in\mathbb{Z}^+$ have become ubiquitous in Number Theory appearing for example in Fourier coefficients of classical Poincaré series and therefore in the geometric side of relative trace formulae of Petersson-Kuznetsov type. Working with relative trace formulae over $\mathrm{GL}_n$ requires understanding of more general Kloosterman sums. In this talk, I will present a method to parametrize and bound the generalized Kloosterman sums for $\mathrm{GL}_n$ obtaining a power saving compared to the trivial bound.