UCLA

This thesis presents effective quantitative bounds for l^2 decoupling for the parabola. We first make effective the argument of Bourgain and Demeter for the case of the parabola. This allows us to improve upon the bound of O_{\epsilon}(\delta^{-\epsilon}) on the decoupling constant. Next, we give a new proof of l^2 decoupling for the parabola inspired from efficient congruencing. We also mention how efficient congruencing relates to decoupling for the cubic moment curve. This chapter contains the first known translation of an efficient congruencing argument into decoupling language. Finally, we discuss equivalences and monotonicity of various parabola decoupling constants and a small ball l^2 decoupling problem.