Title: Dyadic Harmonic Analysis and Weighted Inequalities

Abstract: In this talk we will present some basic ideas in harmonic analysis via simpler dyadic models. We will show how they can be used to describe important continuous objects such as the Hilbert transform or more generally singular integral operators. We will introduce the dyadic setting, Haar functions, and basic dyadic operators. We will illustrate the power of this approach  in  the study of the boundedness of these operators on Lebesgue spaces and on weighted Lebesgue spaces, both one weight and two-weight problems. There has been a lot of activity in this area recently, including the resolution of the A_2 conjecture by Hytonen in 2012 and the resolution of the two-weight problem for the Hilbert transform by Lacey, Sawyer, Shen and Uriarte-Tuero in 2014. We will touch on those as well as on the corresponding story for the dyadic square function, and the dyadic paraproduct.