Mappings with integrally controlled $p$-moduli

We consider classes of mappings (with controlled moduli) whose $p$-module of the families of curves/surfaces

is restricted by integrals containing measurable functions and arbitrary admissible metrics. In the talk we

discuss various properties of mappings with controlled moduli including their differential features (Lusin's

$N-$ and  $N^{-1}$-conditions, Jacobian bounds, estimates for distortion dilatations, H\"older/logarithmically

H\"older continuity) and the topological structure (openness, discreteness, invertibility, finiteness of the

multiplicity function). This allows us to investigate the interconnection between mappings of bounded and finite

distortion defined analytically and mapping with controlled moduli having no analytic assumptions.