My research revolves around basic inequalities in harmonic analysis, in particular inequalities which either possess a large amount of symmetries or have some semblance of such symmetries. Singular integrals and many maximal operators relate to translation and dilation symmetries. Many objects appearing in my work have in addition modulation symmetries, which necessitates to study them with a tool called time frequency analysis. Early examples of this theory are Carleson's theorem on almost everywhere convergence of Fourier series and Lp bounds on the bilinear Hilbert transform. More recently, time frequency analysis was recognized as closely connected with an Lp theory of outer measures.