In this thesis, we study the reduced coaction in noncommutative geometry and explore its applications in associator theory and the theory of multiple zeta values. The main results are as follows. We generalize Drinfeld's Knizhinik-Zamolodchikov (KZ) associator theory to paths that start and end at tangential base points that may have a finite number of transversal self-intersections. We derive explicit closed formulas for the reduced coaction maps of open path regularized holonomies of the KZ equation. And explicit formula for the Poisson bracket of matrix entries of regularized holonomies of a pair of loops starting and ending at the same tangential base point. We present a formula that relates the Turaev coaction and the Goncharov–Brown coaction. Motivated by this relation, we introduce the reduced coaction equation. The skew-symmetric solutions to this equation form a Lie algebra under Ihara bracket.